Central European Journal of Mathematics

CEJM 2(3) 2004 339–361 Discrete limit theorems for general Dirichlet series. III A. Laurinˇcikasa∗ , R. Macaitien˙e† De...

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CEJM 2(3) 2004 339–361

Discrete limit theorems for general Dirichlet series. III A. Laurinˇcikasa∗ , R. Macaitien˙e† Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

Received 24 February 2004; accepted 23 April 2004 Abstract: Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given. c Central European Science Journals. All rights reserved. Keywords: Dirichlet series, probability measure, random element, weak convergence MSC (2000): 11M41, 30B50, 60B10 a

1

Partially supported by Lithuanian Foundation of Studies and Science

Introduction

Let s = σ + it be a complex variable, and let R and C denote the set of real and complex numbers, respectively. The series ∞ am e−λm s , (1) m=1

where am ∈ C and λm ∈ R, 0 < λ1 < λ2 < . . . , limm→∞ λm = +∞, is called a general Dirichlet series with coefficients am and exponents λm . If λm = logm, then we have the ordinary Dirichlet series ∞ am . ms m=1 ∗ †

E-mail: [email protected] E-mail: [email protected]

340

A. Laurinˇcikas, R. Macaitien˙e / Central European Journal of Mathematics 2(3) 2004 339–361

Suppose that (1) absolutely converges for σ > σa and has a sum f (s). Then the function f (s) is regular in the half-plane σ > σa . The value-distribution of the function f (s) is complicated, therefore probabilistic methods were used to begin attacking the problem. It is convenient to state probabilistic results for functions given by Dirichlet series in the form of limit theorems in the sense of the weak convergence of probability measures. This idea belongs to H. Bohr who jointly B. Jessen [2], [3] obtained limit theorems for the Riemann zeta-function. Later many mathematicians (A. Wintner, V. Borchsenius, A. Selberg, A. Ghosh, P.D.T.A. Elliott, E. Stankus, D. Joyner, D. Hejhal, E. M. Nikishin, B. Bagchi, ˇ zeviˇcien˙e, R. Kaˇcinskait˙e, K. Matsumoto, R. Garunkˇstis, W. Schwarz, J. Steuding, R. Sleˇ J. Ignataviˇci¯ ut˙e, I. Belov, the authors and others) extended and generalized Bohr–Jessen’s results. In general, the probabilistic value-distribution of the ordinary Dirichlet series has been widly studied, while results of such a kind for general Dirichlet series are not numerous. Denote by meas{A} the Lebesgue measure of the set A ⊂ R, and let νTt (...) = 1 meas{t ∈ [0; T ] : ....}, where in place of dots, a condition satisfied by t is written. In T [14] the distribution function νTt (F (t) < x), where F (t) = f (σ+it) or F (t) = f (σ+it), was considered in connection with the Besicovitch classes. Let G be a region on the complex plane C. Denote by H(G) the space of analytic functions on G equipped with the topology of uniform convergence on compacta, and let B(S) stand for the class of Borel sets of the space S. The paper [6] is devoted to weak convergence of the probability measure νTτ f (s + iτ ) ∈ A , A ∈ B(H(Da )), where Da = {s ∈ C : σ > σa }. It was proved that the latter measure converges weakly to some probability measure on H(Da ), B(H(Da )) as T → ∞, while in the case of a linear independence of the system {λm } over the field of rational numbers the explicit form of the limit measure was given. The further investigations in this field are related to limit theorems in the space of meromorphic functions. Denote by M (G) the space of meromorphic functions on G equipped with the topology of uniform convergence on compacta. Suppose that the function f (s) is meromorphically continuable to the region σ > σ1 with some σ1 < σa , all poles in this region being included in a compact set. Moreover, we assume that, for σ > σ1 , the estimates f (σ + it) = O(|t|α ),

|t| ≥ t0 , α > 0,

and t0 is a fixed positive number, and T 1 f (σ + it)2 dt = O(T ), 2T −T

T → ∞,

(2)

(3)

are satisfied. Let D = {s ∈ C : σ > σ1 }. Then in [7] it was proved that the probability measure νTτ f (s + iτ ) ∈ A , A ∈ B(M (D)),


341

weakly converges to some probability measure P on M (D), B(M (D)) , however, the explicit form of this measure was not indicated. The first attempt to find the limit measure P in the theorem of [7] was made in [9]. Assuming additionally that λm ≥ c(log m)δ , (4) where c and δ are some positive constants, and that the set {log 2} ∪

∞

{λm }

(5)

m=1

is linearly independent over the field of rational numbers; in [9] the explicit form the limit measure P was obtained. Finally, in [4] the number log 2 from (5) was removed, so the explicit form of the measure P is known if the system of exponents {λm } is linearly independent over the field of rational numbers. In [8] the weak convergence for σ > σ1 of the probability measure νTt f (σ + it) ∈ A , A ∈ B(C), was considered as T → ∞. All mentioned above limit theorems for the function f (s) have a continuous character: in these theorems the studied measures are defined by translations f (σ + it) or f (s + iτ ), where t and τ vary continuously in the interval [0; T ]. An another discrete statement of the problem is also possible. In this case t or τ takes values in some arithmetical progression. In [10] we began the investigation of the discrete value-distribution of (1) by probabilistic methods, and we proved for it limit theorems in the sense of the weak convergence of probability measures on the complex plane. Discrete limit theorems for f (s) in the space of analytic functions were obtained in [12]. Let 1 µN (...) = #{0 ≤ m ≤ N : ...}, N +1 where in place of dots a condition satisfied by m is written. Let γ denote the unit circle on C, i.e. γ = {s ∈ C : |s| = 1}, and let Ω=

∞

γm ,

m=1

where γm = γ for all m ∈ N. With the product topology and pointwise multiplication, the infinite-dimensional torus Ω is a compact topological Abelian group (proof can be found in [5]). Therefore, there exists the probability Haar measure mH on (Ω, B(Ω)), and this gives a probability space (Ω, B(Ω), mH ). Let ω(m) stand for the projection of ω ∈ Ω to the coordinate space γm . Suppose, as above, that the function f (s) is meromorphically continuable to the halfplane σ > σ1 , σ1 < σa , and that all poles in this region belong to a compact set. Moreover, we require that, for σ > σ1 , the estimates (2) and (3) should be is satisfied.

342


Now, for σ > σ1 , on the probability space (Ω, B(Ω), mH ) we define a complex-valued random element f (σ, ω) by f (σ, ω) =

∞

am ω(m)e−λm σ ,

m=1

and denote by Qf the distribution of the random element f (σ, ω). In this paper we suppose that h > 0 is fixed and such that exp{ 2π } is a rational number. Then in [10] the h following theorem was proved. Theorem 1.1. Suppose that the function f (s) satisfies conditions (2) and (3), {λm } is a sequence of algebraic numbers linearly independent over the field of rational numbers, and satisfies condition (4). Then the probability measure QN (A) = µN (f (s + imh) ∈ A),

A ∈ B(C),

converges weakly to Qf as N → ∞. Now define an H(Da )-valued random element f (s, ω) on (Ω, B(Ω), mH ) by f (s, ω) =

∞

am ω(m)e−λm s ,

s ∈ Da ,

m=1

f its distribution. Then in [12] the following theorem was obtained. and denote by Q Theorem 1.2. Suppose that {λm } is a sequence of algebraic numbers linearly independent over the field of rational numbers. Then the probability measure N (A) = µN (f (s + imh) ∈ A), Q

A ∈ B(H(Da )),

f as N → ∞. converges weakly to Q The aim of this paper is to obtain a discrete limit theorem for the function f (s) in the space of meromorphic functions. Let C∞ = C ∪ {∞} be the Riemann sphere, and let d(s1 , s2 ) be a metric given by the formulae d(s1 , s2 ) =

2|s1 − s2 | , 1 + |s1 |2 1 + |s2 |2

d(s, ∞) =

2 1 + |s|2

,

d(∞, ∞) = 0,

where s, s1 , s2 ∈ C. This metric is compatible with the topology of C∞ . Denote by M (G) the space of meromorphic functions g : G → (C∞ , d) equipped with the topology of uniform convergence on compacta. In this topology, a sequence {gn : gn ∈ M (G)} converges to the function g ∈ M (G) if d(gn (s), g(s)) → 0,

n → ∞,


343

uniformly on compact subsets of G. Consider the weak convergence of the probability measure PN (A) = µN f (s + imh) ∈ A , A ∈ B(M (D)). On the probability space (Ω, B(Ω), mH ) define an H(D)-valued random element f (s, ω) by the formula ∞ f (s, ω) = am ω(m)e−λm s , s ∈ D. (6) m=0

Denote by Pf the distribution of the random element f (s, ω). Then we have the following statement. Theorem 1.3. Suppose that {λm } is a system of algebraic numbers linearly independent over the field of rational numbers and satisfies condition (4), and that the function f (s) satisfies conditions (2) and (3). Then the probability measure PN converges weakly to Pf as N → ∞.

2

Limit theorems for Dirichlet polynomials

Since all poles of the function f (s) in the region D are included in compact set, the number of these poles is finite. Denote them by s1 , . . . , sr , and let f1 (s) =

r

1 − eλ1 (sj −s) .

j=1

Then, clearly, f1 is a Dirichlet polynomial, and f1 (sj ) = 0 for j = 1, . . . , r. Moreover, let f2 (s) = f1 (s)f (s). Then we have, that the function f2 (s) is regular on D. Denote by |A| the number of elements of the set A. Then, for σ > σa , we have f2 (s) =

r

1−e

j=1

= =

λ1 (sj −s)

∞

∞

am e−λm s

m=1

am e−λ1 sj (−1)|A| e−(λm +|A|λ1 )s

A⊆{1,...,r} m=1 r ∞

amj e−(λm +jλ1 )s ,

j=0 m=1

with some coefficients amj satisfying amj = O |am | for m ∈ N and j = 0, 1, . . . , r. Here the first sum runs over all the subsets A of {1, . . . , r}. Obviously, the definition of f2 (s) and conditions (2) and (3) imply, for σ > σ1 , the estimates |t| ≥ t0 , α > 0, (7) f2 (s) = O |t|α ,

344


and

T

−T

f2 (σ + it)2 dt = O(T ),

T → ∞.

(8)

Let S1 and S2 be two metric spaces. For us the following statement will be useful. Lemma 2.1. Let P, Pn be probability measures on (S1 , B(S1 )), and h : S1 → S2 be a continuous function. Then the weak convergence of Pn to P implies the weak convergence of Pn h−1 to P h−1 on (S2 , B(S2 )) as n → ∞. The lemma is a particular case of Theorem 5.1 from [1]. We begin with a limit theorem for the Dirichlet polynomial pn (s) =

r n

amj e−(λm +jλ1 )s .

j=0 m=1

Lemma 2.2. Suppose that {λm } is a sequence of algebraic numbers linearly independent over the field of rational numbers. Then there exists a probability measure Ppn on (H(D), B(H(D))) such that the probability measure PN,pn (A) = µN pn (s + imh) ∈ A),

A ∈ B(H(D)),

converges weakly to Ppn as N → ∞. Proof. Let Ωn =

n

γm ,

m=1

where γm = γ for all m = 1, . . . , n. Define the function u : Ωn → H(D) by the formula u(x1 , . . . , xn ) =

r n

−1 amj e−(λm +jλ1 )s x−j 1 xm ,

(x1 , . . . , xn ) ∈ Ωn .

j=0 m=1

Then we have that u is a continuous function, and pn (s + imh) = u(eiλ1 mh , . . . , eiλn mh ). Consider the probability measure PN on (Ωn , B(Ωn )) defined by PN (A) = µN (eiλ1 mh , . . . , eiλn mh ) ∈ A . The Fourier transform gN (k1 , . . . , kn ), k1 , . . . , kn ∈ Z, of the measure PN is gN (k1 , . . . , kn ) =

Ω

1 xik 1

n . . . xik n dPN

n

N 1 imh l=1 kl λl = e . N + 1 m=0

(9)


345

Since {λm } is a system of algebraic numbers linearly independent over the field of rational numbers and h has the above properties, we find [11] that   if (k1 , . . . , kn ) = (0, . . . , 0),  1 gN (k1 , . . . , kn ) =

1    N +1

1−exp{i(N +1)h 1−exp{ih

n

l=1

n

l=1

kl λl }

kl λl }

if (k1 , . . . , kn ) = (0, . . . , 0).

Consequently, lim gN (k1 , . . . , kn ) =

N →∞

1 if (k1 , . . . , kn ) = (0, . . . , 0), 0 if (k1 , . . . , kn ) = (0, . . . , 0).

By Theorem 1.3.19 of [5] this implies that the measure PN converges weakly to the Haar measure mnH on (Ωn , B(Ωn )) as N → ∞. Hence, the continuity of the function u, Eq. 9, and Lemma 2.1 yield that the probability measure PN,pn converges weakly to Ppn = mnH u−1 as N → ∞. Now let g(m), m ∈ N, be complex numbers such that |g(m)| = 1 for all m ∈ N. Define pn (s, g) =

r n

amj g(m)e−(λm +jλ1 )s .

j=0 m=1

Lemma 2.3. The probability measure PÑ,pn (A) = µT pn (s + imh, g) ∈ A ,

A ∈ B(H(D)),

also converges weakly to the measure mnH u−1 as N → ∞. Proof. Let θm = arg g(m), m = 0, 1, ..., n, and let the function u1 : Ωn → Ωn be given by the formula u1 (x1 , . . . , xn ) = (x1 e−iθ1 , . . . , xn e−iθn ), Then we have that

(x1 , . . . , xn ) ∈ Ωn .

pn (s + imh, g) = u u1 (eiλ1 mh , ..., eiλn mh ) .

Therefore, by the proof of Lemma 2.2, the measure PT,pn converges weakly to the measure mnH (uu1 )−1 as N → ∞. Since the Haar measure is invariant with respect to translations −1 by points from Ωn , PN,pn converges weakly to (mnH u−1 = mnH u−1 as N → ∞. 1 )u

3

Approximation in the mean

In this section we approximate the function f2 (s), defined at the beginning of Section 2, by an absolutely convergent Dirichlet series in the mean. Let σ2 = σa − σ1 . For σ ∈ [−σ2 , σ2 ] define s s (λn +jλ1 )s e , ln (s) = Γ σ2 σ2

346


where, as usual, Γ(s) is the gamma-function. Obviously, σ2 > 0. For σ > σ1 consider the function σ2 +i∞ 1 dz f2 (s + z)ln (z) . gn (s) = 2πi σ2 −i∞ z Lemma 3.1. The function gn (s) has the expansion gn (s) =

r ∞

amj exp − e−(λm −λn )σ2 e−(λm +jλ1 )s ,

(10)

j=0 m=1

the series being absolutely convergent for σ > σ1 . Proof. Since σ2 = σa − σ1 , it follows that σ + σ2 > σa for σ > σ1 . Therefore, for z = σ2 , r ∞

f2 (s + z) =

amj e−(λm +jλ1 )(s+z) .

j=0 m=1

We put

1 kn (m) = 2πi and consider the series

r ∞

σ2 +i∞

σ2 −i∞

ln (s)e−(λm +jλ1 )s

ds , s

amj kn (m)e−(λm +jλ1 )(s+z) .

(11)

j=0 m=1

Since

kn (m) = O e

−(λm +jλ1 )σ2

∞

−∞

−(λ +jλ )σ m 1 2 ln (σ2 + it) dt = O e ,

the series (11) converges absolutely for σ > σa − σ2 , i. e., for σ > σ1 . Hence we may interchange summation and integration in the definition of gn (s). This gives

gn (s) = =

r ∞ j=0 m=1 r ∞

amj e

−(λm +jλ1 )s

1 2πi

σ2 +i∞

σ2 −i∞

ln (z)e−(λm +jλ1 )z

dz z

amj kn (m)e−(λm +jλ1 )s .

(12)

j=0 m=1

Using the equality 1 2πi we find that

c+i∞

Γ(s)b−s ds = e−b ,

c > 0, b > 0,

c−i∞

σ2 +i∞ 1 s s −(λm −λn )s ds e Γ 2πi σ2 −i∞ σ2 σ2 s σ2 +i∞ 1 s −(λm −λn ) − σs σ2 s 2 = e Γ d = exp − e(λm −λn )σ2 . 2πi σ2 −i∞ σ2 σ2


347

Hence and from Eq. 12 the lemma follows. Lemma 3.2. Let T0 and T ≥ δ > 0 be real numbers, T be a finite set in the interval [T0 + 2δ , T0 + T − 2δ ]. Moreover, let

Nδ (x) =

1,

t∈T |t−x| σ1 , T

|f2 (σ + it)|2 = O(T ).

0

Proof. By the Cauchy formula f2 (s)

1 = 2πi

|z−s|=δ

f2 (z) dz, (z − s)2

where the circle |z − s| = δ lies in the half-plane σ > σ1 . Then for some σ > σ1 and bounded τ by Eq. 8 T 0

T 1 |f2 (σ + it)|2 dt = 2πi 0

|z−s|=δ

2 T 2 f2 (z) dz dt = O f2 (σ + it + iτ ) dt = O(T ). (z − s) 0

Lemma 3.4. Let K be a compact subset of D. Then N 1 lim lim sup sup f2 (s + imh) − gn (s + imh) = 0. n→∞ N →∞ N + 1 m=0 s∈K

Proof. We change the contour in the integral for gn (s). The integrand has a simple pole at the point z = 0. Suppose σ ∈ [σ1 + η, σ4 ], η > 0, when s ∈ K, and put σ3 = σ1 + η2 .

348


Then, by the residue theorem, we find σ3 −σ+i∞

1 gn (s) = 2πi

f2 (s + z)ln (z) σ3 −σ−i∞

dz + f2 (s). z

(13)

Now let L be a simple closed contour lying in D and enclosing the set K, and let δ stand for the distance of L from the set K. Then by the Cauchy formula, for s ∈ K, we have f2 (z + imh) − gn (z + imh) 1 f2 (s + imh) − gn (s + imh) = dz. 2πi z−s L

Therefore, 1 sup |f2 (s + imh) − gn (s + imh)| ≤ 2πδ s∈K

|f2 (z + imh) − gn (z + imh)||dz|. L

Hence, for sufficiently large N , we obtain N 1 sup |f2 (s + imh) − gn (s + imh)| = N + 1 m=0 s∈K

N 1 1 =O |f2 (z + imh) − gn (z + imh)||dz| N m=0 2πδ

1 =O N

L

1 =O Nδ

L 2N

1 |dz| |f2 ( z + imh) − gn ( z + imh)| + O Nδ m=0 2N 1 |f2 (σ + imh) − gn (σ + imh)| . +O sup σ N s∈L m=0

(14)

Remembering Eq. 13, we have ∞ f2 (s + imh) − gn (s + imh) = O

|f2 (σ3 + imh + iτ )||ln (σ3 − σ + iτ )|dτ .

−∞

Hence, taking u = [ |τh| ] + 1, where [x] denotes the integer part of x, we obtain that 2N 2N +u 1 1 |f2 (σ + imh) − gn (σ + imh)| = O |ln (σ3 − σ + iτ )| |f2 (σ3 + imh)|dτ . N m=0 N m=−u ∞

−∞

By Lemmas 3.2 and 3.3, and estimate (8) 2N +u

1 |f2 (σ + imh)| ≤ h m=−u 2

(2N +u)h

|f2 (σ + it)|2 dt

−uh (2N +u)h

2

(2N +u)h

|f2 (σ

|f2 (σ + it)| dt

+ −uh

−uh

12 + it)| dt = O(2N + 2u). 2


349

Therefore the Cauchy - Schwarz inequality yields 2N 1 |f2 (σ + imh) − gn (σ + imh)| = sup σ N + 1 s∈L m=0 12 ∞ 2N +u 1 2 = |ln (σ3 − σ + iτ )|dτ |f2 (σ3 + imh)| = O sup σ N + 1 m=−u s∈L −∞ ∞

= O sup σ s∈L

−∞ ∞

=

|ln (σ3 − σ + iτ )|(1 + |τ |)dτ .

= O sup σ s∈L

2N + 2u |ln (σ3 − σ + iτ )| dτ N +1

(15)

−∞

We can choose the number δ so that the inequality σ3 − σ ≤ − η4 , s ∈ L, should be satisfied. In this case by definition of ln (s), we have that ∞ |ln (σ + it)|(1 + |t|)dt = 0. lim sup n→∞ σ≤− η

4

−∞

The above along with estimates (14) and (15) prove the lemma. Let ah = {e−iλm h : m ∈ N}, h > 0. Then ah is a one-parameter group. We define the one-parameter family ϕh of transformations on Ω by ϕh (ω) = ah ω for ω ∈ Ω. Then ϕh is a measurable measure preserving transformation on the probability space (Ω, B(Ω), mH ). Now we recall some basic facts of ergodic theory, see for example [15]. Let G be a compact topological Abelian group with Haar measure mG . Let {Gh } be an one-parameter group of measurable transformations on G. A set A ∈ B(G) is called an invariant set with respect to the group {Gh } if, for each h, the sets A and Ah = Gh (A) differ from one another by a set of zero mG -measure, i.e. mG (A∆Ah ) = 0, where A∆Ah denotes the symmetric difference of sets A and Ah . A one-parameter group {Gh } is called ergodic if its σ-field of invariant sets consists only of sets having mG -measure equal to 0 or 1. For these and other facts of ergodic theory, see for example, [15]. Lemma 3.5. The one-parameter group ϕh is ergodic. Proof. This is Lemma 7 from [10]. Let, for s ∈ D, f2 (s, ω) =

r ∞

am ω j (1)ω(m)e−(λm +jλ1 )s .

j=0 m=1

Then we have that f2 (s, ω) is a product of two H(D)-valued random elements, r 1 − ω(1)eλ1 (sj −s) f1 (s, ω) = j=1

350


and f (s, ω), and therefore it is an H(D)-valued random element. Denote by Eξ the mean of the random element ξ. Lemma 3.6. Let T be a measurable measure preserving ergodic transformation on prob F, m). Then for every g ∈ L1 (Ω, F, m) ability space (Ω, 1 g(T k ω) = Eg lim n→∞ n k=0 n−1

for almost all ω ∈ Ω. The lemma is the well-known Birkhoff-Khinchine theorem, see for example [16]. Lemma 3.7. Let σ > σ1 and N → ∞. Then N f2 (σ + imh, ω)2 = BN m=0

for almost all ω ∈ Ω. Proof. Let, for m ∈ N, fmj (σ, ω) = amj ω j (1)ω(m)e−(λm +jλ1 )σ , and let, for a fixed j, fj (σ, ω) =

∞

fmj (σ, ω).

m=1

Then we have E(fmj , fkj ) = amj akj e

=

−(λm +jλ1 )σ −(λk +jλ1 )σ

e

ω j (1)ω −j (1)ω(m)ω(k)mH (dω)

Ω 2 −2(λm +jλ1 )σ , m = k, |amj | e

since

m = k,

0,

ω(m)ω(k)mH (dω) = Ω

1, m = k, 0, m = k.

Thus we have that the random variables fmj (σ, ω) are orthogonal. In [9] it was proved that, for σ > σ1 , ∞ |am |2 e−2λm σ < ∞. m=1

Consequently,

2

E|fj (σ, ω)| =

∞ m=1

2

E|fm,j (σ, ω)| = O

∞ m=1

2 −2(λm +jλ1 )σ

|am | e

< ∞,

j = 0, .., r.


Hence,

r 2 E f2 (σ, ω) = E|fj (σ, ω)|2 < ∞.

351

(16)

j=0

Clearly,

2 2 = f2 (σ + imh, ω)2 . |f2 (σ, ϕm h (ω))| = f2 (σ, amh ω)

(17)

Therefore, in view of (16) and Lemmas 3.5 and 3.6, we find N N 2 2 1 1 f2 (σ + imh, ω) = lim f2 (σ, ϕm lim h (ω)) N →∞ N + 1 N →∞ N + 1 m=0 m=0 2 = E f2 (σ, ω) < ∞

for almost all ω ∈ Ω. Hence the lemma follows. Let, for ω ∈ Ω, gn (s, ω) =

r ∞

amj ω j (1)ω(m) exp{−e−(λm −λn )σ2 }e−(λm +jλ1 )s .

j=1 m=1

Clearly, the last series converges absolutely for σ > σ1 . Lemma 3.8. Let K be a compact subset of D. Then N 1 sup |f2 (s + imh, ω) − gn (s + imh, ω)| = 0 lim lim sup n→∞ N →∞ N + 1 s∈K m=0

for almost all ω ∈ Ω. Proof. In virtue of Lemma 3.7, the proof is similar to that of Lemma 3.4.

4

Limit theorems for gn (s)

In this section we consider the weak convergence of two probability measures on (H(D), B(H(D))), namely, PN,n (A) = µN gn (s + imh) ∈ A , and

PN,n (A) = µN gn (s + imh, ω) ∈ A

as N → ∞. To investigate the weak convergence of these measures we need a metric on H(D) which induces its topology. It is known, (see, for example, Lemma 1.7.1 of [5]), that there exists a sequence {Kn } of compact subsets of D such that D = ∪∞ n=1 Kn , Kn ⊂ Kn+1 , and if K is a compact subset of D, then K ⊆ Kn for some n. Then 8(f, g) =

∞ n=1

8n (f, g) , 1 + 8n (f, g)

f, g ∈ H(D),

352


where 8n (f, g) = sup |f (s) − g(s)| s∈Kn

is a metric on H(D) which induces its topology. Lemma 4.1. There exists a probability measure Pn on (H(D), B(H(D))) such that both the measures PN,n and PN,n converge weakly to Pn as N → ∞. Proof. Define, for a positive integer M , −(λ +jλ )s

(λm −λn )σ2 e m 1 , gn,M (s) = rj=0 M m=1 amj exp − e −(λ +jλ )s

j (λm −λn )σ2 gn,M (s, ω) = M e m 1 , m=1 amj ω (1)ω(m) exp − e and let PN,n,M (A) = µN gn,M (s + imh) ∈ A , PN,n,M (A) = µN gn,M (s + imh, ω) ∈ A ,

A ∈ B(H(D)), A ∈ B(H(D)).

By Lemmas 2.2 and 2.3, both the measures PN,n,M and PN,n,M converge weakly to the same measure Pn,M as N → ∞. Similarly as in Theorem 5.5.2 of [5], we obtain that the family of probability measures {Pn,M } is tight for fixed n. Hence, by the Prokhorov theorem, (see for example, Theorem 1.1.12 of [5]), it is relatively compact. By the definition of gn (s) and gn,M (s), we have lim gn,M (s) = gn (s),

M →∞

and since the series for gn (s) converges absolutely for σ > σ1 , the convergence is uniform on compact subsets of D. Hence, for every ε > 0, limn→∞ lim supN →∞ µN 8(gn,M (s + imh), gn (s + imh)) ≥ ε N

1 ≤ limM →∞ lim supN →∞ (N +1)! 8 gn,M (s + imh), gn (s + imh) = 0. (18) m=0

B(Ω), P) with distribution Let θN be a random variable on a certain probability space (Ω, P(θN = mh) =

1 , N +1

m = 0, 1, ..., N.

We put XN,n,M (s) = gn,M (s + iθN ). D

Denote by −→ the convergence in distribution. Then by the above remark D

XN,n,M (s) N−→ Xn,M , →∞ where Xn,M is an H(D)-valued random element with the distribution Pn,M . Now let XN,n (s) = gn (s + iθN ).


353

Then, in view of Eq. 18, for every ε > 0 lim lim sup P 8(XN,n,M (s), XN,n (s)) ≥ ε = 0.

N →∞ N →∞

(19)

Since the family {Pn,M } is relatively compact, there exists a subsequence Pn,M1 which converges weakly to Pn , say, as M1 → ∞. Then D

Xn,M1 −→

M1 →∞

Pn .

(20)

The space H(D) is separable. Therefore, by relations (18)–(20) and Theorem 1.2.4 of [5], D

XN,n N−→ Pn . →∞

(21)

This means that there is a probability measure Pn such that PN,n converges weakly to Pn as N → ∞. On the other hand, relation (21) shows that the measure Pn is independent of the choice of the subsequence Pn,M1 . This and the relative compactness of {Pn,M } imply the weak convergence of Pn,M to Pn as M → ∞, and also the relation D

Pn . Xn,M M−→ →∞

(22)

Now repeating the same arguments for the random elements N,n,M (s, ω) = gn,M (s + iθ, ω), X N,n (s, ω) = gn (s + iθ, ω), X and taking into account relation (22), we obtain that the measure PN,n also converges weakly to Pn as N → ∞. The lemma is proved.

5

Limit theorems for f2 (s)

In this section, we will consider the weak convergence of probability measures PN,f2 (A) = µN f2 (s + imh) ∈ A , PN,f2 (A) = µN f2 (s + imh, ω) ∈ A ,

A ∈ B(H(D)), A ∈ B(H(D)).

Lemma 5.1. There exists a probability measure P on (H(D), B(H(D))) such that both the measures PN,f2 and PN,f2 converge weakly to P as N → ∞. Proof. The way of the proof is the same as in Lemma 4.1. By this lemma, the measures PN,n and PN,n converge weakly to the same measure Pn as N → ∞. Taking XN,n (s) = gn (s + iθN ), we have that

D

XN,n N−→ Xn , →∞

(23)

354


where Xn is an H(D)-valued random element with the distribution Pn . Also, the family {Pn } is relatively compact. Applying Lemma 3.4, we find that, for every ε > 0, lim lim sup µN 8(f2 (s + imh), gn (s + imh)) ≥ ε n→∞ N →∞

N 1 8(f2 (s + imh), gn (s + imh)) = 0. ≤ lim lim sup n→∞ N →∞ (N + 1)ε m=0

Now we set YN (s) = f2 (s + iθN ). Then we can write lim lim sup P 8(XN,n (s), YN (s)) ≥ ε = 0.

n→∞ N →∞

(24)

From the relative compactness of {Pn } there exists a subsequence {Pn1 } of {Pn } which converges weakly to P as n1 → ∞, i.e., D

P. Xn1 n−→ →∞ 1

(25)

Using (23)–(25) and applying Theorem 1.2.4 of [5] again, we obtain that D

YN n→∞ −→ P. This means that the measure PN,f2 converges weakly to P as N → ∞. The relative compactness of {Pn } and relation (25) show that D

Xn −→ P. n→∞

(26)

Repeating the above arguments, and using (26) and Lemma 3.8, we obtain that the measure PN,f2 also converges weakly to P as N → ∞. The aim of the next lemma is to identify the limit measure in Lemma 5.1. As in Section 3, for s ∈ D, we define f2 (s, ω) =

r ∞

amj ω j (1)ω(m)e−(λm +jλ1 )s .

j=0 m=1

Let Pf2 be the distribution of the random element f2 (s, ω). Lemma 5.2. The measure P in Lemma 5.1 coincides with Pf2 . Proof. Let A ∈ B(H(D)) be a continuity set of P . Then by Lemma 5.1 lim µN f2 (s + imh, ω) ∈ A = P (A) N →∞

(27)

for almost all ω ∈ Ω. Now we fix the set A and define the random variable θ on (Ω, B(Ω)) by the formula 1 if f2 (s, ω) ∈ A, θ(ω) = 0 if f2 (s, ω) ∈ A.


Then

E(θ) = Ω

θdmH = mH ω ∈ Ω : f2 (s, ω) ∈ A = Pf2 (A) < ∞.

355

(28)

By Lemmas 3.5 and 3.6 it follows that N 1 θ(ϕm h (ω)) = E(θ) N →∞ N + 1 m=0

lim

(29)

for almost all ω ∈ Ω. However, the definitions of θ and ϕh give N 1 θ(ϕm h (ω)) = µN f2 (s + imh, ω) ∈ A . N + 1 m=0

(30)

From Eqs. (28)–(30) we obtain lim µN f2 (s + imh, ω) ∈ A = Pf2 (A)

N →∞

for almost all ω ∈ Ω. From this and Eq. (27), we have that P (A) = Pf2 (A) for any continuity set A of P . Since all continuity sets constitute a determining class, hence P (A) = Pf2 (A) for all A ∈ B(H(D)), and the lemma is proved.

6

A limit theorem for f1 (s)

First we observe that f1 (s) =

r

λ1 (sj −s)

1−e

=

r

bm e−λ1 ms

m=0

j=1

is a Dirichlet polynomial with some coefficients bm and exponents mλ1 , and define the H(D)-valued random element f1 (s, ω) by f1 (s, ω) =

r

λ1 (sj −s)

1 − ω(1)e

j=1

=

r

bm ω m (1)e−λ1 ms .

m=0

We define the probability measure PN,f1 (A) = µN f1 (s + imh) ∈ A ,

A ∈ B(H(D)).

Lemma 6.1. The probability measure PN,f1 converges weakly to the distribution Pf1 of the random element f1 (s, ω) as N → ∞. Proof. The lemma is obtained in the same way as Lemmas 2.2 and 5.2.

356

7


A two-dimensional limit theorem

In Sections 5 and 6 we proved discrete limit theorems for the functions f1 (s) and f2 (s) in the space H(D). Now we will prove a joint discrete limit theorem for these functions. Let H 2 (D) = H(D) × H(D). Denote by Pf1 ,f2 the distribution of the H 2 (D)-valued random element F (s, ω) = f1 (s, ω), f2 (s, ω) , ω ∈ Ω, s ∈ D, and let PN,f1 ,f2 (A) = µN (f1 (s + imh), f2 (s + imh)) ∈ A ,

A ∈ B(H 2 (D)).

Lemma 7.1. The probability measure PN,f1 ,f2 converges weakly to Pf1 ,f2 as N → ∞. For the proof of this lemma we need the following results. Lemma 7.2. The family of probability measures {PN,f1 ,f2 } is relatively compact. Proof. In Lemmas 5.1, 5.2 and 6.1 we proved that the probability measures PN,f1 and PN,f2 converge weakly to the measures Pf1 and Pf2 , respectively, as N → ∞. Therefore, the family of probability measures {PN,fj }, j = 1, 2, is relatively compact. The space H(D) is a complete separable space. Hence, by the Prokhorov theorem the family {PN,fj }, j = 1, 2 is tight. Thus we have that for every < > 0 there exists a compact subset Kj ⊂ D such that < PN,fj (H(D)\Kj ) < , j = 1, 2. (31) 2 Let θN be a random variable defined in the proof of Lemma 4.1, and f1,N (s) = f1 (s + iθN ), f2,N (s) = f2 (s + iθN ), FN (s) = f1,N (s), f2,N (s) . Then inequality (31) and the definition of PN,fj yield < P fj,N ∈ H(D)\Kj < , j = 1, 2. 2

(32)

Let K = K1 × K2 . Then K is a compact subset of H 2 (D). In virtue of (32) PN,f1 ,f2 (H 2 (D)\K) = P FN ∈ H 2 (D)\K 2 (fj,N (s) ∈ H(D)\Kj ) =P j=1

≤

2

P fj,N (s) ∈ H(D)\Kj < σ D }.

Moreover, let ujk , j = 1, 2, 1 ≤ k ≤ l, be arbitrary complex numbers, and define a by the formula function v : H 2 (D) → H(D) v(f1 , f2 ) =

2 l

fj ∈ H(D), j = 1, 2. s ∈ D,

ujk fj (sk + s),

(34)

j=1 k=1

Let

Wv (s) = v f1 (s), f2 (s) .

Lemma 7.3. The relation D v F (s, ω) Wv (s + iθN ) N−→ →∞ holds. Proof. By the the definition of the function v, Wv (s) =

2 l

ujk fj (sk + s)

j=1 k=1

for σ > σ1 . In the region σ > σa the f1 (s) and f2 (s) are presented by absolutely convergent Dirichlet series r r λ1 (sj −s) f1 (s) = 1−e = bj e−λ1 js j=1

j=0

and f2 (s) = =

r

1−e

j=1 r ∞

λ1 (sj −s)

∞

am e−λm s

m=1

am,j e−(λm +jλ1 )s .

j=0 m=1

This shows that for s ∈ D Wv (s) = =

l k=1 l k=1

=

r j=0

u1k f1 (sk + s) +

l

u2k f2 (sk + s)

k=1

u1k

r

bj e

−λ1 j(sk +s)

j=0

bj e−λ1 js +

+

l k=1

l k=1

u2k

r ∞ j=0 m=1

u2k

∞ r

amj e−(λm +jλ1 )(sk +s)

j=0 m=1

amjk e−(λm +jλ1 )s = Z1 (s) + Z2 (s),

358


where bj =

l

u1kbj e−λ1 sk ,

bj =

bj , j ≤ r,

k=1

and amjk = amj e−(λm +jλ1 )sk ,

amj =

0,

j>r,

amj , j ≤ r, 0,

j > r.

Z1 (s) is a Dirichlet polynomial, and Z2 (s) is a linear combination of Dirichlet series satisfying conditions (7) and (8). Since the function f2 (s) is regular in the half-plane σ > σ1 , the function Z2 (s) is regular in D. Since the set {λm } is a system of algebraic numbers linearly independent over the field of rational numbers, and the exponents have a property of type (4), we have, that the probability measure µN Wv (s + imh) ∈ A = µN (Z1 (s + imh) + Z2 (s + imh)) ∈ A , A ∈ B(H(D)), (35) converges weakly to the distribution of the H(D)-valued random element Wv (s, ω) =

r

bj ω(1)e−λ1 js +

j=0

l

u2k

r ∞

am,j,k ω j (1)ω(m)e−(λm +jλ1 )s

(36)

j=0 m=1

k=1

as N → ∞. The proof of this is obtained in a similar way like, for example, for the function f2 (s). First we prove a limit theorem for the measure l M r µN (Z1 (s) + u2k amjk e−(λm +jλ1 )s ) ∈ A ,

A ∈ B(H(D)).

j=0 m=1

k=1

After this, applying an approximation of the function Wv (s) in the mean, it remains only to use some elements of the ergodic theory to obtain the explicit form of the limit measure, and it turns out that this limit measure coincides with the distribution of the random element defined by Eq. (36). However, by the definition of v Wv (s, ω) =

l

k=1

u1k

r

j=0

bj ω j (1)e−λ1 j(sk +s) + =

2

r

j=0 k=1

l

k=1

u2k

r

∞

j=0 m=1

am,j,k ω j (1)ω(m)e−(λm +jλ1 )(sk +s)

ujk fj (sk + s, ω) = v f1 (s, ω), f2 (s, ω) .

Therefore, measure (35) converges weakly to the distribution of the random element v f1 (s, ω), f2 (s, ω) as N → ∞. Proof of Lemmas 7.1. By Lemma 7.2 it follows that there exists a sequence N1 → ∞ such that the measure PN1 ,f1 ,f2 converges weakly to some probability measure P on 2 H (D), B(H 2 (D)) as N1 → ∞. Let F1 (s) = f11 (s), f12 (s)


359

be an H 2 (D)-valued random element with distribution P . Then by the choice of N1 we have that D FN1 N−→ F1 . (37) →∞ 1

The function v is continuous. Hence, by Lemma 2.1, we have that D

v(F1 ). v(FN1 ) N−→ →∞ 1

By the definition of Wv (s) D

Wv (s + iθN1 ) N−→ v(F1 ). →∞ 1

(38)

Denoting F (s) = f1 (s), f2 (s) , by Lemma 7.3 we have D

Wv (s + iθN1 ) −→ v(F ). N1 →∞

From this and relation (38) D

v(F ) = v(F1 ).

(39)

→ C be defined by the formula Now let v1 : H(D) f ∈ H(D).

v1 (f ) = f (0),

Then the function v1 is measurable, and (39) gives D

v1 (v(F )) = v1 (v(F1 )), or D

v(F )(0) = v(F1 )(0). By the definition of v we find 2 l

D

ujk fj (sk , ω) =

j=1 k=1

2 l

ujk f1j (sk )

(40)

j=1 k=1

with arbitrary complex numbers ujk . The hyperplanes in the space R4n generates a determining class [1]. Then, the hyperplanes also form a determining class in C2n . Hence, and from (40), we see that C2n -valued random elements fj (sk , ω) and f1j (sk ), j = 1, 2, k = 1, ..., l, have the same distribution. Denote by K a compact subset of D, and let ϕ1 , ϕ2 ∈ H(D). Moreover, let for every < > 0, G = {(g1 , g2 ) ∈ H 2 (D) : sup |gj (s) − ϕj (s)| ≤ 1, construct Wβ using cartesian products.) Notice that for a ∈ Γβ the following conditions are equivalent: 1) a is graphic for Γ, 2) a is graphic for Reg Γ, 3) a is graphic for Wβ (as a subset of W ). The mapping φ|Wβ : Wβ → Rn has generic rank n because Reg Γ ⊂ Wβ . A point a ∈ Wβ is graphic for Wβ as a subset of W iff a is graphic for Wβ as a manifold. The set Eβ of non-graphic points for Wβ as a manifold is analytic in Wβ by the above Theorem 4.1. The K-subanalytic set Eβ = Eβ ∩ Γβ consists of points of Γβ non-graphic for Γ. Now, a point x ∈ ∂U is not in Σ iff all germs Gα at x for a ∈ Γ ∩ φ−1 (x) exist and are equal. If all Ga exist, then they are equal because Ga depends only on a connected component of Γ ∩ φ−1 (x) and fibers of φ|Γ are connected, again by inspecting the proofs in [8]. We have Σ = β φ(Eβ ), hence Σ is K-subanalytic. ✷ Proof of Proposition 3.4. For each x ∈ ∂U \ Σ there is ρ(x) ∈ R such that g extends to an analytic function g˜x defined on U ∪ B(x, ρ(x)). (B(x, r) is the open ball with centre x and radius r.) Define V = U ∪ ∪x∈∂U \Σ B(x, 12 ρ(x)). If z ∈ B(x, 12 ρ(x)) ∩ B(y, 12 ρ(y)) and ρ(x) ≥ ρ(y), then y ∈ B(x, ρ(x)) so g˜x and g˜y agree near y, therefore also at z. This means g extends to an analytic function on V . ✷

Acknowledgements I would like to thank Chris Miller for suggesting the topic and the initial discussion. Much of the expositional work of this paper was done during my stay in the Mathematical Institute in Oxford whom I thank for hospitality.

A. Pi¸ekosz / Central European Journal of Mathematics 2(3) 2004 362–367

367

References [1]

E. Bierstone: ”Control of radii of convergence and extension of subanalytic functions, Proc. of the Amer. Math. Soc., Vol. 132, (2004), pp. 997–1003.

[2]

´ E. Bierstone, P. Milman: ”Semianalytic and subanalytic sets”, Inst. Hautes Etudes Sci. Publ. Math., Vol. 67, (1988), pp. 5–42.

[3]

L. van den Dries: ”A generalization of the Tarski-Seidenberg theorem and some nondefinability results”, Bull. Amer. Math. Soc. (N. S.), Vol. 15, (1986), pp. 189– 193. L. van den Dries, C. Miller: ”Extending Tamm’s theorem”, Ann. Inst. Fourier, Grenoble, Vol. 44, (1994), pp. 1367–1395.

[4] [5]

L. van den Dries, C. Miller: ”Geometric categories and o-minimal structures”, Duke Math. Journal, Vol. 84, (1996), pp. 497–540.

[6]

J.-M. Lion, J.-Ph. Rolin: ”Théorème de préparation pour les fonctions logarithmicoexponentielles”, Ann. Inst. Fourier, Grenoble, Vol. 47, (1997), 859–884.

[7]

C. Miller: ”Expansions of the real field with power functions”, Ann. Pure Appl. Logic, Vol. 68, (1994), pp. 79–84.

[8]

A. Pi¸ekosz: ”K-subanalytic rectilinearization and uniformization”, Central European Journal of Mathematics, Vol. 1, (2003), pp. 441–456.

[9]

J.-Cl. Tougeron: ”Paramétrisations de petit chemins en géométrie analytique réele”, Singularities and differential equations (Warsaw 1993), Banach Center Publications, Vol. 33, (1996), pp. 421–436.

CEJM 2(3) 2004 368–376

On schemes for congruence distributivity I. Chajda∗ , R. Halaˇs† Department of Algebra and Geometry, Palacký University Olomouc, Tomkova 40, 779 00 Olomouc, Czech Republic

Received 14 April 2004; accepted 12 May 2004 Abstract: We present diagrammatic schemes characterizing congruence 3-permutable and distributive algebras. We show that a congruence 3-permutable algebra is congruence meetsemidistributive if and only if it is distributive. We characterize varieties of algebras satisfying the so-called triangular scheme by means of a Maltsev-type condition. c Central European Science Journals. All rights reserved. Keywords: congruence distributivity, congruence 3-permutability, congruence n-permutability, diagrammatic scheme, the triangular scheme MSC (2000): 08A30, 08B10, 08B05

1

Introduction

The first diagrammatic scheme was developed by H.-P.Gumm [6] under the name Shifting Lemma and Shifting Principle in order to characterize congruence modularity. The socalled Triangular scheme was introduced by the first author for a characterization of congruence distributivity in congruence permutable algebras, see [1], [2]. Further, it was modified for congruence semidistributivity [3] and completed in the so-called Trapezoid scheme characterizing congruence distributivity in varieties of algebras, see [4]. In this paper we turn back to congruence distributivity for a single algebra. We will study the case of congruence n-permutable algebra in general and of congruence 3permutable algebra in particular. Several remarks concerning the Triangular scheme and the permutable case are included. ∗ †

E-mail: [email protected] E-mail: [email protected]

I. Chajda, R. Halaˇs / Central European Journal of Mathematics 2(3) 2004 368–376

2

369

The background

Congruence distributive varieties were characterized by B. Jónsson [7] by means of the Maltsev condition. For a single algebra, we did not have an analogous tool to characterize congruence distributivity. The concept of a shift of a lattice identity was introduced by G.Czédli [5]. In particular, the distributive identity β ∧ (α ∨ γ) = (β ∧ α) ∨ (β ∧ γ)

(D)

α ∧ β ≤ γ ⇒ β ∧ (α ∨ γ) ≤ γ.

(S)

has the following shift Hence, (S) is a Horn sentence and a lattice L satisfies (D) if and only if it satisfies (S). Thus (S) is another characterization of distributivity. It turns out that in particular cases it can be more appropriate to verify (S) than (D), see bellow for ConA. Let us remark that, due to self-duality of (D), distributivity is also equivalent to the Horn formula γ ≤ α ∨ β ⇒ γ ≤ β ∨ (α ∧ γ), (DS) which is a dual shift of (D). Up to now, this characterization was not used for ConA. Recall from [1],[2] that a sublattice L of an equivalence lattice EqA satisfies the Triangular scheme if for each α, β, γ ∈ L with α ∩ β ⊆ γ and for x, y, z ∈ A such that x, y ∈ γ, x, z ∈ α, z, y ∈ β we have z, y ∈ γ. This can be visualized as follows

The following assertions are proved in [1],[2]: (a) if ConA is distributive then it satisfies the Triangular scheme; (b) if A is a congruence permutable algebra then ConA is distributive if and only if ConA satisfies the Triangular scheme. Lemma 2.1. Let L be a modular lattice. Then L is distributive if and only if it satisfies the identity (α ∧ (γ ∨ β)) ∨ β = (α ∧ γ) ∨ β (A) Proof. If L is modular it satisfies the identity (α ∨ β) ∧ (γ ∨ β) = (α ∧ (γ ∨ β)) ∨ β.

(M)

370


If L is distributive, it satisfies (α ∨ β) ∧ (γ ∨ β) = (α ∧ γ) ∨ β

(B)

whence (A) follows. Conversely, if L is modular and satisfies (A) then one can easily derive (B).

3

Congruence 3-permutable algebras

A sublattice L of the equivalence lattice EqA is 3-permutable if α◦β◦α=β◦α◦β

for each

α, β ∈ L.

If L = ConA, we say that an algebra A is congruence 3-permutable. We are going to establish a diagrammatic scheme for congruence distributivity in congruence 3-permutable algebras. The scheme is similar to that of H.-P.Gumm [6] for congruence modularity. Definition 3.1. Let L be a sublattice of EqA. We say that L satisfies the 3-scheme if for each α, β, γ ∈ L with α ∩ β ⊆ γ and for x, y, z, w ∈ A such that x, z ∈ α, z, y ∈ β, y, w ∈ α and x, w ∈ γ we have z, y ∈ γ. The 3-scheme can be visualized as follows

Lemma 3.2. If the sublattice L of EqA is distributive then L satisfies the 3-scheme. Proof. Let α, β, γ ∈ L with α ∩ β ⊆ γ and L be distributive. Suppose x, z ∈ α, z, y ∈ β, y, w ∈ α and x, w ∈ γ. Then z, y ∈ β ∩ (α ◦ γ ◦ α) ⊆ β ∩ (α ∨ γ) = (β ∩ α) ∨ (β ∩ γ) ⊆ γ ∨ (β ∩ γ) = γ.

We can introduce a similar diagrammatical scheme: Definition 3.3. A sublattice L of EqA satisfies the weak 3-scheme if for each α, β, γ ∈ L with β ∩ α = β ∩ γ and for x, y, z, w ∈ A such that x, z ∈ α, z, y ∈ β, y, w ∈ α and x, w ∈ γ we have z, y ∈ γ.


371

Also the weak 3-scheme can be drawn:

Remark. If L satisfies the 3-scheme then L also satisfies the weak 3-scheme since β ∩ α = β ∩ γ implies α ∩ β ⊆ γ. However, the converse does not hold in general even for ConA, see the following: Example. Let A = ({a, b, c, d}, {s, t}) where s, t are unary operations defined by the table: s t a d c b c d c b d d a c Then ConA is the N5 as shown in Fig. 1,

Fig. 1

where α = {{a, b}, {c, d}} β = {{b, c}, {a, d}} γ = {{b, c}, {a}, {d}}. To verify the schemes, we need only to check the non-trivial congruences, i.e. α, β, γ. However, the assumption β ∩ α = β ∩ γ is not satisfied, thus we can check only the case α ∩ β = α ∩ γ. It is an easy calculation to show that ConA satisfies the weak 3-scheme. On the contrary, α ∩ β ⊆ γ and c, d ∈ α, d, a ∈ β, a, b ∈ α and c, b ∈ γ but d, a ∈ / γ, thus ConA does not satisfy the 3-scheme.

372


Theorem 3.4. Let A be a congruence 3-permutable algebra. Then ConA is distributive if and only if it satisfies the weak 3-scheme. Proof. If ConA is distributive then it satisfies the 3-scheme by Lemma 3.2 and, by the previous Remark, it satisfies also the weak 3-scheme. Conversely, let A be a congruence 3-permutable algebra and let ConA satisfy the weak 3-scheme. By the way of contradiction, suppose that ConA is not distributive. Since 3-permutability implies congruence modularity, ConA contains a sublattice M3 as shown in Fig. 2. Suppose x, w ∈ γ. Then x, w ∈ α ∨ β = α ◦ β ◦ α due to

Fig. 2

congruence 3-permutability, thus there exist z, y ∈ A such that x, z ∈ α, z, y ∈ β and y, w ∈ α. Of course, α ∩ β = β ∩ γ thus applying the weak 3-scheme, z, y ∈ γ. Hence, z, y ∈ β ∩ γ = β ∩ α ⊆ α. Applying transitivity of α, we conclude x, w ∈ α, i.e. γ ⊆ α, a contradiction. Recall that a lattice L is ∧-semidistributive if β ∧ α = β ∧ γ implies β ∧ (α ∨ γ) = β ∧ γ for all α, β, γ ∈ L. Theorem 3.5. Let A be a congruence 3-permutable algebra. Then ConA is ∧-semidistributive if and only if ConA is distributive. Remark. An analogous result for congruence permutable algebras was proved in [3]. Proof. Let A be congruence 3-permutable algebra and suppose that ConA is ∧-semidistributive. Let α, β, γ ∈ ConA and β ∩ α = β ∩ γ. Suppose x, z ∈ α, z, y ∈ β, y, w ∈ α and x, w ∈ γ. Then z, y ∈ β ∩ (α ◦ γ ◦ α) = β ∩ (α ∨ γ) = β ∩ γ ⊆ γ, i.e. ConA satisfies the weak 3-scheme and, by Theorem 3.4, ConA is distributive. The converse implication is trivial.


4

373

The n-permutable case

A sublattice L of EqA is n-permutable (for n ≥ 2) if α ◦ β ◦ α ◦ · · · = β ◦ α ◦ β ◦ · · · (n factors on each side) for all α, β, γ ∈ L. An algebra A is congruence n-permutable if ConA has this property. Definition 4.1. A sublattice L of EqA satisfies the n-scheme (the weak n-scheme) if for each α, β, γ ∈ L with α∩β ⊆ γ (or β∩α = β∩γ, respectively) and for x, y, z1 , · · · , zn−1 ∈ A such that x, y ∈ β, x, z1 ∈ α, z1 , z2 ∈ γ, z2 , z3 ∈ α, · · · , zn−1 , y ∈ α for n odd and zn−1 , y ∈ γ for n even we have x, y ∈ γ. These schemes can be also visualized but, contrary to the previous cases, classes of the same congruence fail to be parallel:

Remark. Of course, for n = 2 the n-scheme is the Triangular scheme, for n = 3 the (weak) n-scheme is the previous (weak) 3-scheme. Lemma 4.2. Let L be a sublattice of EqA (a)

If L is distributive then it satisfies the n-scheme for each n ≥ 2.

(b) If L is ∧-semidistributive then it satisfies the weak n-scheme for each n ≥ 2. Proof. The proof (a) is analogous to that of Lemma 3.2. For (b), suppose α, β, γ ∈ L with β ∩ α = β ∩ γ and x, y, z1 , · · · , zn−1 ∈ A such that the assumptions of the weak scheme are satisfied. Then x, y ∈ β ∩ (α ◦ γ ◦ α ◦ · · ·) ⊆ β ∩ (α ∨ γ) = β ∩ γ due to ∧-semidistributivity, i.e. it satisfies the weak n-scheme for each n ≥ 2. Theorem 4.3. Let A be a congruence n-permutable algebra. Then (a)

ConA is distributive if and only if it satisfies the n-scheme;

(b) ConA is ∧-semidistributive if and only if it satisfies the weak n-scheme.

374


Proof. Suppose A to be congruence n-permutable. (a) Let ConA satisfy the n-scheme and suppose that it is not distributive. Then ConA contains a sublattice isomorphic to that of Fig. 1 or to that of Fig. 2. We have α ∩ β ⊆ γ in both cases. Suppose x, y ∈ β. Then also x, y ∈ α ∨ γ = α ◦ γ ◦ α ◦ · · · (n times), i.e. there exist z1 , z2 , · · · , zn−1 such that the assumptions of the n-scheme are satisfied. Applying the n-scheme, we have x, y ∈ γ proving β ⊆ γ, a contradiction. Hence, ConA is distributive. The converse follows by (a) of Lemma 4.2. (b) Let ConA satisfy the weak n-scheme, let α, β, γ ∈ ConA and β ∩ α = β ∩ γ. Let x, y ∈ β ∩(α∨γ). Due to congruence n-permutability, we have x, y ∈ β ∩(α◦γ ◦α◦· · ·) with n factors. Thus it is almost evident that the assumptions of the weak n-scheme are satisfied. Applying this scheme, we conclude x, y ∈ β ∩ γ. We have shown β ∩ (α ∨ γ) ⊆ β ∩ γ. The converse inclusion is trivial, i.e. ConA is ∧-semidistributive. The converse implication follows by (b) of Lemma 4.2.

5

A characterization of the Triangular scheme

The following lemma is an immediate observation (and hence we omit the proof). Lemma 5.1. Let L be a sublattice of EqA. (a) L satisfies the Trinagular scheme if and only if it satisfies the Horn sentence α ∩ β ⊆ γ ⇒ β ∩ (α ◦ γ) ⊆ γ. (b) L satisfies the n-scheme if and only if it satisfies the Horn sentence α ∩ β ⊆ γ ⇒ β ∩ (α ◦ γ ◦ α ◦ · · ·) ⊆ γ (with n factors). One can compare the Horn sentences of Lemma 4 with that of (S) to see that it is in fact the shift of distributive identity (D) for the permutable or n-permutable case. Similarly we can rewrite (A) of Lemma 2.1 for the permutable case. However, we can prove a very close identity in a general case: Lemma 5.2. Let L be a sublattice of EqA. Then L satisfies the identity (α ◦ β) ∩ (γ ◦ β) = (α ∩ (γ ◦ β)) ◦ β Proof. Suppose a, b ∈ (α ◦ β) ∩ (γ ◦ β)

(1)

Then a, b ∈ α ◦ β, i.e. there is c ∈ A such that a, c ∈ α

and

c, b ∈ β.

(2)

By (1), a, b ∈ γ ◦ β thus (2) yields a, c ∈ γ ◦ β ◦ β = γ ◦ β.

(3)


375

From (2) and (3) we have a, c ∈ α ∩ (γ ◦ β), i.e. a, b ∈ (α ∩ (γ ◦ β)) ◦ β. The converse inclusion is trivial.

We say that a variety V satisfies the Triangular scheme if ConA satisfies this scheme for each A ∈ V. In what follows, we are going to show that this property can be characterized by a Maltsev type condition. Theorem 5.3. A variety V satisfies the Triangular scheme if and only if there exist n ≥ 0 and ternary terms t0 , · · · , tn such that y = t0 (x, y, z), z = tn (x, y, z) and ti (x, x, y) = ti+1 (x, y, y), ti (x, y, x) = ti+1 (x, y, x) for i even, ti (x, x, y) = ti+1 (x, x, y) for i odd. Proof. Let V satisfy the Triangular scheme, let FV (x, y, z) be the free algebra of V with free generators x, y, z and α = θ(x, z), β = θ(y, z) and γ = (α ∩ β) ∨ θ(x, y). Then α ∩ β ⊆ γ and, by the Triangular scheme, z, y ∈ γ. Hence, there exist n ≥ 0 and ternary terms t0 , · · · , tn such that y = t0 (α ∩ β)t1 θ(x, y)t2 (α ∩ β)t3 · · · tn = z. Using the standard procedure we easily derive the desired Maltsev condition. Conversely, let there exist n ≥ 0 and ternary terms t0 , · · · , tn satisfying the Maltsev type condition of Theorem 4. Let A ∈ V, α, β, γ ∈ ConA and α ∩ β ⊆ γ. Suppose x, y, z ∈ A with x, y ∈ γ, x, z ∈ α, y, z ∈ β. Then ti (x, y, z)αti (x, y, x) = ti+1 (x, y, x)αti+1 (x, y, z) ti (x, y, z)βti (x, y, y) = ti+1 (x, y, y)βti+1 (x, y, z) for i even, i.e. ti (x, y, z), ti+1 (x, y, z) ∈ α ∩ β. For i odd we have ti (x, y, z)γti (x, x, z) = ti+1 (x, x, z)γti+1 (x, y, z) thus y = t0 (x, y, z)(α ∩ β)t1 (x, y, z)γt2 (x, y, z)(α ∩ β) · · · tn (x, y, z) = z. Hence y, z ∈ (α ∩ β) ◦ γ ◦ (α ∩ β) ◦ γ ◦ · · · ⊆ γ ◦ γ ◦ γ ◦ · · · = γ, i.e. A and hence also V satisfies the Triangular scheme.

376


References [1] I. Chajda: ”A note on the triangular scheme”, East-West J. of Mathem., Vol. 3, (2001), pp. 79–80. [2] I. Chajda and E.K. Horváth: ”A triangular scheme for congruence distributivity”, Acta Sci. Math. (Szeged), Vol. 68, (2002), pp. 29–35. [3] I. Chajda and E.K. Horváth: ”A scheme for congruence semidistributivity”, Discuss. Math., General Algebra and Appl., Vol. 23, (2003), pp. 13–18. [4] I. Chajda, E.K. Horváth and G. Czédli: ”Trapezoid Lemma and congruence distributivity”, Math. Slovaca, Vol. 53, (2003), pp. 247–253. [5] I. Chajda, E.K. Horváth and G. Czédli: ”The Shifting Lemma and shifting lattice identities”, Algebra Universalis, Vol. 50, (2003), pp. 51–60. [6] H.-P. Gumm: ”Geometrical methods in congruence modular algebras”, Mem. Amer. Math. Soc., Vol. 45, (1983), pp. viii–79. [7] B. Jónsson: ”Algebras whose congruence lattices are distributive”, Math. Scand., Vol. 21, (1967), pp. 110–121.

CEJM 2(3) 2004 377–381

On log canonical divisors that are log quasi-numerically positive Shigetaka Fukuda∗ Faculty of Education, Gifu Shotoku Gakuen University, Yanaizu-cho, Gifu 501-6194, Japan

Received 26 January 2004; accepted 30 April 2004 Abstract: Let (X, ∆) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, ∆) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisor KX + ∆ is log quasi-numerically positive on (X, ∆) then it is semi-ample. c Central European Science Journals. All rights reserved. Keywords: the log canonical divisor, divisorial log terminal, numerically positive, semi-ample MSC (2000): 14E30

1

Introduction

Throughout the paper every variety is projective over the field of complex numbers. We follow the notation and terminology of the proceedings [7] of “the second Utah seminar”. Definition 1.1. A Q-Cartier Q-divisor L on a projective variety X is numerically positive (nup, for short) if (L, C) > 0 for every curve C on X. A nef Q-divisor L on X is quasinumerically positive (quasi-nup, for short) if there exists a union V of at most countably many Zariski-closed subsets X such that (L, C) > 0 for every curve C not contained in V . A quasi-nup Q-divisor L on X is log quasi-numerically positive (log quasi-nup, for short) on a divisorial log terminal (dlt) variety (X, ∆) if L|B is quasi-nup for every nonKawamata log terminal (non-klt) center B (in other words, for every B ∈ Cnon−klt (X, ∆), under the notation of Section 2). ∗

E-mail: [email protected]

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S. Fukuda / Central European Journal of Mathematics 2(3) 2004 377–381

Of course, the nupness (resp. the log quasi-nupness) implies the log quasi-nupness (resp. the quasi-nupness). In the case where (X, ∆) is Kawamata log terminal (klt), the quasi-nupness is equivalent to the log quasi-nupness. Recently F. Ambro ([1]) reduced the famous log abundance conjecture, which claims that the nef log canonical divisors should be semi-ample, for klt varieties to the log minimal model conjecture and the problem of semi-ampleness of the quasi-nup log canonical divisors: Problem 1.2. Assume that (S, D) is klt and KS +D is quasi-nup. Is KS +D semi-ample? With regard to Problem 1.2, we consider the following Problem 1.3. Assume that (X, ∆) is not klt but dlt and KX + ∆ is log quasi-nup on (X, ∆). Is KX + ∆ semi-ample? We note that the log abundance conjecture (including Problems 1.2 and 1.3) for dlt varieties is known to be true in dimension ≤ 3 ([3], [6]). The subadjunction theory of Kawamata-Shokurov (in Section 2) and a uniruledness theorem of Mori-Miyaoka type (due to Matsuki [9]) enable us to reduce Problem 1.3 (where ∆ = 0) in dimension n to Problem 1.2 (where D = 0) in dimension ≤ n − 1 (see Proposition 3.1) and obtain the following main theorem. Theorem 1.4. Assume that (X, ∆) is not klt but dlt, dim X = 4 and KX + ∆ is log quasi-nup on (X, ∆). Then KX + ∆ is semi-ample. In the case where X is smooth and ∆ is reduced and with only simple normal crossings, the theorem was proved in [4].

2

The subadjunction theory of Kawamata-Shokurov

We recall the subadjunction theory of Kawamata (cf. [5, Lemma 5-1-9]) and Shokurov ([10, Subsection 3.2.3]), clarify the notion of minimal non-klt centers and fix the relevant notation. A log variety (X, ∆) consists of a normal variety X and an effective Q-divisor ∆ on X such that ∆ is reduced. The log variety (X, ∆) is said to be divisorial log terminal (dlt, for short) if KX + ∆ is Q-Cartier and there exists a projective log resolution f : Y → X such that KY + f∗−1 ∆ = f ∗ (KX + ∆) + E with the properties that E ≥ 0, Exc(f ) is a divisor and Supp(f∗−1 ∆) ∪ Exc(f ) is a simply normal crossing divisor. (Moreover if ∆ = 0, the log variety (X, ∆) is said to be Kawamata log terminal (klt, for short).) l We set Di := f∗−1 ∆i , where ∆ = i=1 ∆i and ∆i is a prime divisor. Define Strataf (X, ∆) := {Γ; k ≥ 1, 1 ≤ i1 < i2 < . . . < ik ≤ l, Γ is an irreducible component of Di1 ∩ Di2 ∩ . . . ∩ Dik = ∅}. The set of non-klt centers Cnon−klt (X, ∆) := {f (Γ); Γ ∈ Strataf (X, ∆)} is known not to depend on the choice of f . Note that Exc(f ) Γ for


379

every Γ ∈ Strataf (X, ∆), because Exc(f ) is a divisor and Supp( li=1 Di ) ∪ Exc(f ) is a simply normal crossing divisor. Thus for every B ∈ Cnon−klt (X, ∆), the morphism f is isomorphic over some suitable neighborhood of the generic point of B. Therefore f (Γ1 ) ⊂ f (Γ2 ) if and only if Γ1 ⊂ Γ2 , for Γ1 , Γ2 ∈ Strataf (X, ∆). Consequently the set of minimal non-klt centers MCnon−klt (X, ∆) := {B; B is a minimal element (with respect to inclusion) of Cnon−klt (X, ∆)} coincides with the set {f (Γ); Γ is a minimal element of Strataf (X, ∆)}. Now we focus on the subvariety ∆i of X. From Kollár-Mori ([8, Corollary 5.52]), ∆i is normal and from the subadjunction theorem (Kawamata-Matsuda-Matsuki [5, Lemma 51-9]), Diff ∆i (∆−∆i ) ≥ 0, where (KX +∆)|∆i = K∆i +Diff ∆i (∆−∆i ). Put fi := f |Di and Strataf (X, ∆)|Di := {Γ ∈ Strataf (X, ∆); Γ Di }. Note that KDi + f∗−1 (∆ − ∆i )|Di = fi∗ (K∆i + Diff ∆i (∆ − ∆i )) + E|Di and that (f∗−1 (∆ − ∆i )|Di − E|Di ) = {Dj |Di ; j = i, Dj ∩ Di = ∅} + F (where F is some divisor such that Supp F does not contain any irreducible component of Dj |Di = 0 and that −F is effective and fi -exceptional because Diff ∆i (∆ − ∆i ) ≥ 0). Here Exc(fi ) Γ for any Γ ∈ Strataf (X, ∆)|Di , since Exc(fi ) ⊆ Exc(f ) ∩ Di . Hence, by considering a suitable embedded resolution of Exc(fi ) ⊆ Di , we obtain that (∆i , Diff ∆i (∆ − ∆i )) is dlt (Shokurov [10, Subsection 3.2.3] and Fujino [2, Proof of Theorem 0.1]). Then Cnon−klt (∆i , Diff ∆i (∆−∆i )) = {fi (Γ); Γ ∈ Strataf (X, ∆)|Di } and Cnon−klt (X, ∆) = l j=1 (Cnon−klt (∆j , Diff ∆j (∆ − ∆j )) ∪ {∆j }). Note that f (Γ1 ) ⊂ f (Γ2 ) ⊂ ∆i if and only if Γ1 ⊂ Γ2 ⊂ Di , for Γ1 , Γ2 ∈ Strataf (X, ∆). Therefore MCnon−klt (∆i , Diff ∆i (∆ − ∆i )) = MCnon−klt (X, ∆)∩ Cnon−klt (∆i , Diff ∆i (∆ − ∆i )). We define the maximal dimension of minimal non-klt centers by l(X, ∆) := max{dim B; B ∈ MCnon−klt (X, ∆)} in the case where (X, ∆) is not klt but dlt. Of course, l(X, ∆) ≤ dim X − 1 in this case.

3

Reduction of the non-klt but dlt case, to the klt case in lower dimensions

We reduce Problem 1.3, to Problem 1.2 in lower dimensions. Proposition 3.1. Let (X, ∆) be a log variety that is not klt but dlt and whose log canonical divisor KX + ∆ is log quasi-nup on (X, ∆). Assume that Problem 1.2 has an affirmative answer in dimension ≤ l(X, ∆) (the maximal dimension of minimal non-klt centers ). Then KX + ∆ is semi-ample. Proof 3.1. We shall prove the proposition by induction on n := dim X, heavily relying on the notation introduced in Section 2. Note that (KX + ∆)|∆i = K∆i + Diff ∆i (∆ − ∆i ) is log quasi-nup on the dlt variety (∆i , Diff ∆i (∆ − ∆i )) from the fact that (Cnon−klt (∆i , Diff ∆i (∆ − ∆i )) ∪ {∆i }) ⊆ Cnon−klt (X, ∆). Because MCnon−klt (∆i , Diff ∆i (∆ − ∆i )) = MCnon−klt (X, ∆) ∩ Cnon−klt (∆i , Diff ∆i (∆ − ∆i )), the inequality l(∆i , Diff ∆i (∆ − ∆i )) ≤ l(X, ∆) holds in the case where

380


(∆i , Diff ∆i (∆ − ∆i )) is not klt. Therefore we know that (KX + ∆)|∆i is semi-ample from the induction hypothesis in this case. The Q-divisor (KX + ∆)|∆i is semi-ample also in the case where (∆i , Diff ∆i (∆ − ∆i )) is klt, because the value of l(X, ∆) becomes n − 1 and hence the assumption of the theorem applies. Thus ((KX + ∆)|B )dim B > 0 for every B ∈ Cnon−klt (X, ∆) = lj=1 (Cnon−klt (∆j , Diff ∆j (∆ − ∆j )) ∪ {∆j }). Next we show that (KX + ∆)n > 0. By assuming that (KX + ∆)n = 0, we shall imply the contradiction. Note that −KY f ∗ (KX + ∆)n−1 = (f∗−1 ∆ − E − f ∗ (KX + ∆))f ∗ (KX + ∆)n−1 = (f∗−1 ∆ − E)f ∗ (KX + ∆)n−1 = (f∗−1 ∆)f ∗ (KX + ∆)n−1 ≥ (f∗−1 ∆1 )f ∗ (KX + ∆)n−1 = ((KX + ∆)|∆1 )n−1 = ((KX + ∆)|∆1 )dim ∆1 > 0 because E is f -exceptional and ∆1 ∈ Cnon−klt (X, ∆). Thus from Matsuki (the uniruledness theorem of Mori-Miyaoka type, [9]), Y is covered by f ∗ (KX + ∆)-trivial curves. Therefore also X is covered by (KX + ∆)-trivial curves. This is a contradiction, because KX + ∆ is quasi-nup! So we have that (KX + ∆)n > 0. Consequently KX + ∆ becomes nef and log big on (X, ∆) (i.e. KX + ∆ is nef, (KX + n ∆) > 0 and also ((KX + ∆)|B )dim B > 0 for every B ∈ Cnon−klt (X, ∆)) and thus KX + ∆ is semi-ample, by virtue of the base point free theorem of Reid-type (Fujino [2]).

4

Proof of the main theorem

Proof 4.1 (Proof of Theorem 1.4). Note that l(X, ∆) ≤ dim X − 1 = 3. Thus the assumption of Proposition 3.1 is satisfied, from the log abundance theorem ([3], [6]) for klt varieties in dimensions 2 and 3.

References [1] F. Ambro: The moduli b-divisor of an lc-trivial fibration, math. AG/0308143, August 2003. [2] O. Fujino: “Base point free theorem of Reid-Fukuda type”, J. Math. Sci. Univ. Tokyo, Vol. 7, (2000), pp. 1–5. [3] T. Fujita: “Fractionally logarithmic canonical rings of algebraic surfaces”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Vol. 30, (1984), pp. 685–696. [4] S. Fukuda: A note on the ampleness of numerically positive log canonical and anti-log canonical divisors, math. AG/0305357, May 2003. [5] Y. Kawamata, K. Matsuda and K. Matsuki: “Introduction to the minimal model problem”, In: Algebraic geometry, Sendai (Japan), 1985, North-Holland, Amsterdam, 1987, pp. 283–360. [6] S. Keel, K. Matsuki and J. McKernan: “Log abundance theorem for threefolds”, Duke Math. J., Vol. 75, (1994), pp. 99–119. [7] J. Kollár (Ed.): Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. [8] J. Kollár and S. Mori: Birational geometry of algebraic varieties, Cambridge University Press, Cambridge, 1998.


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[9] K. Matsuki: A correction to the paper “Log abundance theorem for threefolds”, math. AG/0302360, February 2003. [10] V. Shokurov: “3-fold log flips”, Russian Acad. Sci. Izv. Math., Vol. 40, (1993), pp. 95–202.

CEJM 2(3) 2004 382–387

Abstract version of the Cauchy-Kowalewski Problem Oleg Zubelevich∗† Department (#803) of Differential Equations, Moscow State Aviation Institute, Volokolamskoe Shosse 4, 125993 Moscow, Russia

Received 19 February 2004; accepted 15 June 2004 Abstract: We consider an abstract version of the Cauchy-Kowalewski Problem with the right hand side being free from the Lipschitz type conditions and prove the existence theorem. c Central European Science Journals. All rights reserved. Keywords: weighted Banach space, Nishida’s theorem; scales of Banach spaces, fixed point theorems MSC (2000): 35A10

1

Introduction

There are two standard existence theorems in the theory of ODE: the Cauchy-Picard existence and uniqueness theorem and the Peano existence theorem. The Cauchy-Picard theorem states that if the right hand side of ODE satisfies the Lipschitz conditions then initial value problem has unique solution. The proof of this theorem is based on the contraction mapping principle. The Peano theorem states that for existence of a solution it is sufficient to have only continuity of the right hand side. This theorem is proven by means of compactness considerations with the help of the Arzela-Ascoli theorem. The case of initial value problem for PDE in the abstract setup has been studied by many authors and there are existence and uniqueness theorems proved under the assumptions of Lipschitz type conditions. An abstract form of the Cauchy-Kowalewski Problem was first considered by T. Yamanaka in [8] and L. Ovsjannikov [4] in the linear case. Some another aspects of the linear Cauchy-Kowalewski Problem were exposed by J. Treves [7]. ∗ †

Email: [email protected] Partially supported by grants RFBR 02-01-00400

O. Zubelevich / Central European Journal of Mathematics 2(3) 2004 382–387

383

In [2] L. Nirenberg obtained the existence and uniqueness theorem for the abstract nonlinear Cauchy-Kowalewski Problem. The proof of Nirenberg’s theorem uses an iteration procedure of Newtonian type and based on ideas of the KAM theory. In Nirenberg’s theorem it is assumed that the right hand side of the problem is a strong differentiable mapping. T. Nishida [3] simplified the iteration procedure and stated that in Nirenberg’s theorem it is possible to replace strong differentiability with the Lipschitz type conditions. In [5] M. Safonov gave a proof of Nishida’s theorem by constructing a suitable Banach space of functions and then using the contraction mapping principle. In this paper we consider a topological aspect of the abstract nonlinear CauchyKowalewski Problem and prove the Peano type existence theorem. We assume that the right hand side of the equation depends on two arguments: it is bounded and continuous in the first argument (pure Peano’s case) and convex in the second one. Such a setup includes quasilinear PDE as a special case. This theorem is not deduced from Nishida’s result or quasilinear versions of the Cauchy-Kowalewski Problem since the Lipschitz type conditions are not applied. The main tools we use are Browder’s generalization of the Schauder fixed point theorem and a topological construction close to Safonov’s one.

2

Main theorem

Let {(Es , · s )}0<s 0.

(1)

We assume that all embeddings (1) are compact. Such an assumption always holds for the scales of analytic functions. Let Bs (r) = {u ∈ Es | us < r} be an open ball of Es and let B s (r) be its closure. The main object of our study is the following Cauchy-Kowalewski problem: ut = A(t, u, u) + h(t, u),

u |t=0 = 0.

(2)

For some positive constants T, R, M, K the mappings A : [0, T ] × B s+δ (R) × B s+δ (R) → Es , h : [0, T ] × B s+δ (R) → Es ,

δ > 0,

s+δ 0,

s + δ < 1.

(3)

Let the mapping A be convex in the third argument: for all u, v, w ∈ B s+δ (R) and 0 ≤ λ ≤ 1 we have A(t, u, λv + (1 − λ)w)s ≤ λA(t, u, v)s + (1 − λ)A(t, u, w)s .

(4)

384


For example, if the mapping A is linear in the third argument then the above inequality holds. Theorem 2.1. There exists such a large constant a > 0 that problem (2) has a solution C([0, τ ], B s (R)). u(t) ∈ 1−s−τ a>0

Hypothesis 2.2. The condition of convexity (4) is necessary: there must be an example of such mappings A and h that satisfy all the above conditions except (4) and problem (2) does not have the solution. This theorem does not reduce to the Nishida result [3]. Nishida’s theorem uses a type of the Lipschitz condition: f (t, u ) − f (t, u )s ≤

M u − u s+δ , δ

u , u ∈ Bs+δ (R),

s + δ < 1,

where f is the right hand side of the problem. In the case under consideration we separate the arguments of the mapping A. It is bounded in the second argument and unbounded in the third one. Thus it is sufficient to have only continuity in the second argument and linearity or convexity in the third one. If the mapping A equals to zero identically then Theorem 2.1 is a direct generalization from the finite dimensional case to the scale of Banach spaces of the Peano existence theorem. There is no reason to expect uniqueness in Theorem 2.1: even in the case of ordinary differential equations there are systems with continuous (but not Lipschitz) right-hand sides that do not have the uniqueness. Before starting to prove Theorem 2.1 we must build some tools.

3

Preliminary topological construction

Introduce a triangle: ∆ = {(τ, s) ∈ R2 | τ > 0, 0 < s < 1, 1 − s − τ a > 0}. Consider a seminormed space E = (τ,s)∈∆ C([0, τ ], Es ) with a family of norms: uτ,s = max u(t)s . 0≤t≤τ

Obviously, these norms satisfy the following inequalities: · τ,s ≤ · τ +δ,s ,

· τ,s ≤ · τ,s+δ ,

δ > 0.

(5)

The space E is a topological space with a basis of the topology given by the open balls: Bτ,s (R) = {u ∈ E | uτ,s < R}.


385

Definition 3.1. A set G ⊆ E is said to be uniformly continuous if for all ε > 0 and for all (τ, s) ∈ ∆ there is δ = δ(ε, τ, s) > 0 such that if t1 , t2 ∈ [0, τ ] and |t1 − t2 | < δ then sup u(t1 ) − u(t2 )s < ε. u∈G

A set G ⊆ E is said to be bounded if there are such constants Mτ,s that for all u ∈ G we have uτ,s ≤ Mτ,s . Recall the Arzela-Ascoli lemma [6]: Lemma 3.2. Let H ⊂ C([0, T ], X) be a set in the space of continuous functions with values in a Banach space X. Assume that the set H is closed, bounded, uniformly continuous and for every t ∈ [0, T ] the set {u(t) ∈ X} is compact in the space X. Then the set H is compact in the space C([0, T ], X). Obviously there is a similar compactness criteria for the space E. Lemma 3.3. If a closed set G ⊆ E is uniformly continuous and bounded then it is compact. Proof 3.4. Let (τ, s) be an arbitrary point of ∆. Since the set G is bounded and uniformly continuous in the space C([0, τ ], Es+δ ), by the Lemma 3.2 it is compact in the space C([0, τ ], Es ). Thus every sequence {uk }k∈N ⊂ G contains a subsequence that converges with respect to the norm · τ,s . So it remains to prove that there is a subsequence of {uk } that converges by all the norms · τ,s at once. Consider a set ∆Q = ∆ Q2 . This set is countable and let γ : N → ∆Q be a corresponding bijection. Let {u1k } ⊆ {uk } be a subsequence that converges by the norm · γ(1) . By the above arguments there is a subsequence {u2k } ⊆ {u1k } that converges by the norm · γ(2) etc. The diagonal sequence {ukk } converges by the norms { · γ(k) }k∈N . Then due to inequalities (5) it converges in all the norms. ✷ In the conclusion we formulate a generalized version of the Schauder fixed point theorem. Theorem 3.5 ([1]). Let W be a compact and convex subset of the seminormed space E. Then a continuous mapping f : W → W has a fixed point uˆ i.e. f (ˆ u) = uˆ.

4

Proof of Theorem 2.1

Problem (2) is obviously equivalent to the following one: t A(ξ, u(ξ), u(ξ)) + h(ξ, u(ξ)) dξ. u(t) = F (u) = 0

(6)

386


As usual, if a solution of (6) is continuous in the variable t then it is actually C 1 -smooth in t. So, we seek for a fixed point of the mapping F . Let S ⊂ E be a set that consists of such elements v that satisfy the following conditions: v(t)s ≤ R, (7) for all u ∈ E such that u(t)s ≤ R we have A(t, u(t), v(t))s ≤ √ for t1 , t2 ∈ [0, τ ],

1 , 1 − at − s

(8)

(τ, s) ∈ ∆ we have

v(t1 ) − v(t2 )s ≤ K + √

1 |t1 − t2 |. 1 − s − aτ

Note that the set S is nonvoid: 0 ∈ S. The set S is convex by inequality (4) and it is compact by Lemma 3.3. Thus if we show that F (S) ⊆ S

(9)

(10)

then the Proof will be conclude by applying Theorem 3.5 to the mapping F and the set S. So let v ∈ S and we check inclusion (10). First observing that t
1, and the obtained functions would not satisfy the condition: l(fA∪B (x)) + l(gA∪B (x)) ≤ 1. The definition introduced in the next section will overcome this difficulty.

2

Definitions and basic properties

A new definition of an L-valued intuitionistic fuzzy set is introduced in this section. Let L be a complete lattice with the top element T and the bottom element B and α a lattice homomorphism from L to [0,1], such that α(T ) = 1 and α(B) = 0. Recall that the lattice homomorphism is a function α : L → [0, 1] satisfying α(x ∧ y) = min(α(x), α(y)), α(x ∨ y) = max(α(x), α(y)). It is straightforward to prove that such a function is a linearization function as well, so this definition is a special case of LIFS-2. In the rest of the paper by L we will denote a lattice (as well as the underlying set), and by L the lattice together with its homomorphism as above. A lattice valued intuitionistic fuzzy (L-valued intuitionistic fuzzy) set of the type 3 (LIFS-3) is the ordered triple (X, f, g), where X is a nonempty set, f and g are functions from X to L and L a lattice with a homomorphism α as above, such that α(f (x)) + α(g(x)) ≤ 1.

(1)

A lattice valued intuinistic fuzzy set obtained in this way has some advantages when compared with other definitions. The notion is a generalization of the ordinary intuitionistic fuzzy set and its structure is richer. To every LIFS-3 there correspond two families of level subsets, which are lattices under inclusion. An ordinary intuitionistic fuzzy set is obtained by this homomorphism, in a natural way, which is not the case with LIFS-1. However, still for some lattices there is no a convenient lattice homomorphism, mapping the top element to 1 and the bottom element to 0. In comparison with the LIFS-2, in LIFS-3 basic operations can be introduced in a natural way. The following proposition is a straightforward corollary of the analogous proposition in [8]. Proposition 2.1. Let L be a lattice with a homomorphism α : L → [0, 1], and (X, f, g) a LIFS-3. Then, (X, α ◦ f, α ◦ g) is an ordinary intuitionistic fuzzy set.

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T. Gerstenkorn, A. Tepavˇcević / Central European Journal of Mathematics 2(3) 2004 388–398

We define some basic relations and operations over the new introduced fuzzy sets, analogously as for ordinary intuitionistic fuzzy sets. 1. A ⊆ B if and only if for all x ∈ X fA (x) ≤ fB (x) and gA (x) ≥ gB (x). 2. A = B if and only if A ⊆ B and B ⊆ A. 3. A = (X, fA , gA ), where fA (x) = gA (x) and gA (x) = fA (x). 4. A ∩ B = (X, fA∩B , gA∩B ), where for all x ∈ X, fA∩B (x) = fA (x) ∧ fB (x), and gA∩B (x) = gA (x) ∨ gB (x). 5. A ∪ B = (X, fA∪B , gA∪B ), where for all x ∈ X, fA∪B (x) = fA (x) ∨ fB (x), and gA∪B (x) = gA (x) ∧ gB (x). In the following, we prove that by formulas 3-5 the operations complement, union and intersection on the set of all LIFS-3 over the same set is well defined. Proposition 2.2. Let A = (X, fA , gA ) and B = (X, fB , gB ) be LIFS-3, where L is a lattice and α a homomorphism from L to [0, 1]. Then, ordered triples A, A ∩ B and A ∪ B, defined as above are also LIFS-3. Proof. The proof for A is obvious. Further, we have to prove that for every x ∈ X, α(fA∩B (x)) + α(gA∩B (x)) ≤ 1. Indeed, α(fA∩B (x)) + α(gA∩B (x)) = α(fA (x) ∧ fB (x)) + α(gA (x) ∨ gB (x)) = min{α(fA (x)), α(fB (x))} + max{α(gA (x)), α(gB (x))}. Let max{α(gA (x)), α(gB (x))} = α(gA (x)) (the proof is similar in other case). Then min{α(fA (x)), α(fB (x))} ≤ α(fA (x)), and the statement is easily proved. The proof for union is similar.

✷

Now, it is not difficult to prove that the operations ∩ and ∪ satisfy commutative, associative and absorptive laws. Moreover, if L is a distributive lattice, then distributive laws are also satisfied .


393

We already mentioned that a LIFS-3 determines two L-valued fuzzy sets. However, the converse is not true: not every couple of L-valued fuzzy sets corresponds to a LIFS-3. The notions of cut (level) subsets and cut (level) functions for LIFS-3 are introduced similar to those of L-fuzzy sets. For each p ∈ L, there are two cut sets defined by: fp = {x ∈ X | f (x) ≥ p} and gp = {x ∈ X | g(x) ≥ p}. The corresponding cut (level) functions are denoted by f p and g p . Thus, for an L valued intuitionistic fuzzy set (X, f, g), there are two families of cut sets. The following two propositions are consequences of the fact that a LIFS-3 consists of two L-valued fuzzy sets. Proposition 2.3. Every LIFS-3 determines two families of cut sets. Each of these families is closed under intersections and contains set X, and thus it is a lattice under inclusion. Proposition 2.4. Let (X, f, g) be a LIFS-3. Then the following is satisfied. 1. fB = X, and gB = X, where B is the bottom element of L. 2. If p ≤ q, then fq ⊆ fp , and gq ⊆ gp . 3. f (x) = g(x) = 4. If M ⊆ L, then and

3

{p ∈ L | f p (x) = 1};

{p ∈ L | gp (x) = 1}.

(fp | p ∈ M ) = f {p|p∈M }

(gp | p ∈ M ) = g {p|p∈M } .

Theorem of synthesis for the L-valued intuitionistic fuzzy sets LIFS-3

In this section necessary and sufficient conditions under which two families of subsets of a set are families of cut sets of a LIFS-3 are given.

394


Theorem 3.1. Let X be a nonempty set and let F1 and F2 be two families of subsets of X, each of them closed under intersections, containing X, and satisfying the following condition: (∀x ∈ X)(∃A ∈ F1 ∪ F2 )(x ∈ A).

(2)

Then, there is a lattice L with a homomorphism α : L → [0, 1] and two mappings f and g from X to L, such that (X, f, g) is a LIFS-3, and F1 and F2 are its families of cut sets. Proof. The families F1 and F2 are closed under intersections, each of them containing X, thus they are lattices under inclusion. Consider lattices (L1 , ≤1 ) and (L2 , ≤2 ), antiisomorphic to lattices F1 and F2 , under the functions f1 and f2 , respectively, such that L1 ∩ L2 = ∅. We denote by TFi and by BFi , the top and the bottom element, respectively, of lattices Fi , for i = 1, 2. By Ti , Bi , we denote the top and the bottom element, respectively, of lattices Li , i = 1, 2. Let (L, ≤) be a linear sum of L1 , L2 and one element lattice ({T }, ≤), for T ∈ L1 and T ∈ L2 . Namely, L = L1 ∪ L2 ∪ {T }, and     x ∈ L1 and y ∈ L2 or         x ≤1 y and x, y ∈ L1 or

x ≤ y if and only if   

x ≤2 y and x, y ∈ L2 or         y = T.

Obviously T is the top element of the lattice L. Now, we define functions f : X → L and g : X → L, as follows.    

f (x) =

T,

if x ∈ BF1 ;

   f1 ( {X ∈ F1 | x ∈ X}) otherwise.    

g(x) = 

T,

if x ∈ BF2 ;

  f2 ( {X ∈ F2 | x ∈ X}) otherwise.

We have to prove that the families of cut sets of the LIFS-3 (X, f, g) are F1 and F2 . First we will consider all p ∈ L and prove that every cut set fp coincides with a set from F1 .


395

Case 1. p = T . Since f (x) = T for x ∈ BF1 , and for all other x ∈ X, f (x) < T , we have that fT = BF1 . Case 2. p ∈ L2 . Since f (x) ∈ L1 or f (x) = T , we have that fp = fT , for all p ∈ L2 . Case 3. p = T1 . x ∈ fT1 if and only if f (x) ≥ T1 if and only if f (x) = T or f (x) = T1 if and only if f1 ( {X ∈ F1 | x ∈ X}) = T1 or x ∈ BF1 if and only if {X ∈ F1 | x ∈ X} = BF1 or x ∈ BF1 if and only if x ∈ BF1 . Case 4. p ∈ L1 \ {T1 }. We prove that fp = f2−1 (p). If x ∈ BF1 , then obviously x ∈ fp and x ∈ f2−1 (p) , hence x ∈ fp if and only if x ∈ f2−1 (p). Now, suppose that x ∈ BF1 . x ∈ fp if and only if f (x) ≥ p if and only if f1−1 (f (x)) ⊆ f1−1 (p) if and only if {X ∈ F1 | x ∈ X} ⊆ f1−1 (p) if f1−1 (f1 ( {X ∈ F1 | x ∈ X})) ⊆ f1−1 (p) if and only if and only if x ∈ f1−1 (p). Thereby we also proved that every element from family F1 coincides with a cut set of f . This proves that the family of cut sets of f coincides with family F1 . The proof that the family of cut sets of g coincides with family F2 is similar. Now, we have to define a homomorphism α from L to [0, 1], such that (1) is satisfied. We define a function satisfying (1), as follows: α(T ) = 1, and α(x) = 0, for all T = x ∈ L. This function is obviously a homomorphism, satisfying the condition (1). Indeed, the condition (1) is not satisfied only in the case when there is an x ∈ X, such that f (x) = T , and g(x) = T . In this case, x ∈ fT , and x ∈ gT , and by Proposition 2.4(2), x ∈ fp , and x ∈ gp for all p ∈ L, which is in contrary to (2). It is evident that α need not be the only homomorphism satisfying (1), but here it is enough to prove that there exists such a function. ✷ Theorem 3.2. Necessary and sufficient conditions under which two families F1 and F2 of subsets of X are families of cut sets of a LIFS-3 are that both are closed under intersections, X belongs to both of them, and that condition (2) is satisfied. Proof. The first part follows by Theorem 3.1. Suppose that F1 and F2 are families of cut sets of a LIFS-3 (X, f, g). They are closed under intersections and contain X, by Proposition 2.3. Now, suppose that condition (2) is not satisfied, i.e., that there is x ∈ X, such that x belongs to all elements of F1 and F2 . Now, f (x) = T and g(x) = T , by Proposition 2.4 (3). Since α(T ) = 1, α(f (x)) + α(g(x)) = 2, a contradiction. ✷ Example 3.3. Let X = {a, b, c, d}, F1 = {{a, b, c, d}, {a, c}, {b, d}, {c}, ∅} and F2 = {{a, b, c, d}, {a, b, c}, {a, b, d}, {a, b}, {b, c}, {b}}. F1 and F2 satisfy conditions of Theo-

396


rem 3.1, i.e., both families are closed under intersections, contain X, and the condition (2) is also satisfied. By the construction in Theorem 3.1, we determine lattices L1 and L2 , antiisomorphic to (F1 , ⊆) and (F2 , ⊆), respectively. The isomorphisms f1 and f2 are given by the following tables and corresponding lattices are presented in Figure 2a. Lattice L in Figure 2b is a linear sum of L1 , L2 and {T }. 



 {a, b, c, d} {a, c} {b, d} {c} ∅   

f1 =  

r

B1

t

s T1





 {a, b, c, d} {a, b, c} {a, b, d} {a, b} {b, c}{b}

  

f2 =  

B2

u

v

q

p

T2

Now, by the mentioned construction, we have that: 



a b c d  

f = 

r ts t

and 



a b c d . 

g= 

qT pv

One possibility for the homomorphism is the one from Theorem 3.1. We present here another homomorphism satisfying the conditions:    



x

T p T2 u

q B 2 v T1 s

r t B1 

α(x) 1 0.9 0.9 0.8 0.8 0.5 0.5 0.1 0.1 0.1 0 0

 


397

T❝

T❝1

❏ s ❝ ❏ ❏❝ t ❝ r ❅ ✡✡ ❅ ❝✡

B1

❝T2 ❅ ❅ ❝q p❝ ❅ ❅ ❅❝ ❅❝ v ❅ u❅ ❝

T❝2

❅ ❅ ❝q p❝ ❅ ❅ ❅ ❝v ❅❝ u❅❅ ❝

B2

L1

L2 Fig. 2a

B2

❝T 1 ❏ s ❝ ❏ ❏❝ t ❝ r ❅ ✡✡ ❅ ❝✡

B1 L

Fig. 2b

The families of level sets of the obtained LIFS-3 (X, f, g) coincide with F1 and F2 , which is straightfoward to check.

References [1] K. Atanassov: “Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, Vol. 20, (1986), pp. 87–96. [2] K. Atanassov, S. Stoeva: “Intuitionistic L-fuzzy sets”, Cybernetics and Systems Research, Vol. 2, R. Trappl (ed.) Elsevier Science Publishers B.V., North-Holland, (1984), pp. 539–540. [3] K. Atanassov: Intuitionistic fuzzy sets, Theory and Applications, Physica-Verlag, Springer Company. Heilderberg, New York, 1999. [4] B.A. Davey, H.A. Priestly: Introduction to lattices and order, Cambridge University Press, 1990. [5] T. Gerstenkorn, J. Ma´ nko: “Bifuzzy probabilistic sets”, Fuzzy Sets and Systems, Vol. 71, (1995), pp. 207–214. [6] T. Gerstenkorn, J. Ma´ nko: “Bifuzzy probability of intuitionistic fuzzy sets”, Notes on Intuitionistic Fuzzy Sets, Vol. 4, (1998), pp. 8–14. [7] T. Gerstenkorn, J. Ma´ nko: “On probability and independence in intuitionistic fuzzy set theory”, Notes on Intuitionistic Fuzzy Sets, Vol. 1, (1995), pp. 36–39. [8] T. Gerstenkorn, A. Tepavˇcević: “Lattice valued bifuzzy sets, New Logic for the New Economy”, VIII SIGEF Congress Proceedings, ed. by G. Zollo, pp. 65–68. ˇ selja, A. Tepavˇcević: “Representation of lattices by fuzzy sets”, Information [9] B. Seˇ Sciences, Vol. 79, (1993), pp. 171–180. ˇ selja, A. Tepavˇcević, G. Vojvodić: “L-fuzzy sets and codes”, Fuzzy sets and [10] B. Seˇ systems, Vol. 53, (1993), pp. 217–222.

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ˇ selja, A. Tepavˇcević: “Completion of ordered structures by cuts of fuzzy sets, [11] B. Seˇ an overview”, Fuzzy Sets and Systems, Vol. 136, (2003) pp. 1–19. ˇ selja, A. Tepavˇcević: “Representing ordered structures by fuzzy sets, an [12] B. Seˇ overview”, Fuzzy Sets and Systems, Vol. 136, (2003), pp. 21–39.

CEJM 2(3) 2004 399–419

On the computation of scaling coefficients of Daubechies’ wavelets ˇ a1 , Václav Finˇek2∗ Dana Cern´ 1

Institute of Mathematics and Didactics of Mathematics, Technical University of Liberec, H´ alkova 6, Liberec 461 17, Czech Republic 2 Institute of Numerical Mathematics , Dresden University of Technology, Willersbau, Zellescher Weg 12 - 14, D - 01062 Dresden, Germany

Received 11 March 2004; accepted 9 June 2004 Abstract: In the present paper, Daubechies’ wavelets and the computation of their scaling coefficients are briefly reviewed. Then a new method of computation is proposed. This method is based on the work [7] concerning a new orthonormality condition and relations among scaling moments, respectively. For filter lengths up to 16, the arising system can be explicitly solved with algebraic methods like Gr¨ obner bases. Its simple structure allows one to find quickly all possible solutions. c Central European Science Journals. All rights reserved. Keywords: Daubechies’ wavelets, computation of scaling coefficients MSC (2000): 65T60

1

Introduction

The approach which leads up to wavelets appeared at the beginning of the twentieth century in works conceived by Haar, Franklin, and Littlewood and Paley. The purpose was to deal with problemmas that were not resolved by the Fourier transform, such as the study of the regularity and other local properties of functions. One of the sequels of these efforts is the theory of orthonormal wavelets that was developed by Yves Meyer beginning in 1985 and the multiresolution analysis that was introduced by Stéphane Mallat and Yves Meyer in 1987. By definition, a multiresolution analysis is a sequence {Vj }j∈Z of closed subspaces of ∗

E-mail: fi[email protected]

400

ˇ D. Cern´ a, V. Finˇek / Central European Journal of Mathematics 2(3) 2004 399–419

L2 (R) that satisfy the following properties: • . . . ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ . . . (embedding), • j Vj is dense in L2 (R) (density), • j Vj = {0} (separation), • f (.) ∈ Vj ⇔ f (2−j .) ∈ V0 (scaling), • {φ(. − k)}k∈Z forms an orthonormal basis for V0 (orthonormality). From ’scaling’ and ’orthonormality’ it follows that {φj,k }k∈Z forms an orthonormal basis of Vj , where φj,k (.) := 2j/2 φ(2j . − k).

(1)

The function φ is called the scaling function or sometimes the ’father wavelet’ and plays an essential role in wavelet theory. The orthonormal wavelets can be introduced in the following way: Let Wj be the orthogonal complemmaent of Vj in Vj+1 . Then Wj represents the additional information necessary to come over from the approximation at the scale 2j to the approximation at the finer scale 2j+1 . Furthermore L2 (R) =

Wj

j∈Z

and suppose, as in the case of V0 , that there is a function ψ from W0 such that the family ψ(x − k)k∈Z is an orthonormal basis of W0 . Then ’scaling’ implies Wj = span {2j/2 ψ(2j x − k); k ∈ Z}. Now, the wavelet ψ can be defined: Definition 1.1. Any function ψ(.) ∈ L2 (R) which generates an orthonormal basis of the space L2 (R) by the system of translations and dilations {ψj,k (.)}j,k∈Z = {2j/2 ψ(2j . − k)}j,k∈Z is called an orthonormal wavelet. From V0 W0 = V1 , it immediately follows that the scaling function φ satisfies the scaling equation (identity) hk φ(2. − k), (2) φ(.) = 2 k∈Z

where hk are the corresponding scaling coefficients, and the wavelet ψ satisfies the wavelet equation gk φ(2. − k); (3) ψ(.) = 2 k∈Z

gk are called wavelets coefficients. Remark 1.2. Strictly speaking the wavelet ψ can be constructed directly from the scaling function φ. For the scaling function with support in [0, 2N − 1], the coefficients gk can


401

be chosen so that gk := (−1)k h2N −k−1 . Thus the support of the corresponding wavelet is also contained in [0, 2N − 1] and this wavelet generates an orthonormal basis of the space L2 (R) by the system of its translations and dilations. Example 1.3. Let V0 be the space of functions that are constant on the intervals [k, k+1], k ∈ Z. These are the splines of order N = 0. In this case, the scaling function is defined by φ(.) := 1[0,1) , and the wavelet ψ is defined by ψ(.) := 1[0, 1 ) − 1[ 1 ,1) , respectively. This 2 2 function generates the Haar system, which was introduced by Alfred Haar (1910).

2

Daubechies’ wavelets

This section is about an important family of wavelets - Daubechies’ wavelets. The importance of this special construction consists in its combination of three key properties: • compact support, • orthogonality, • approximation properties. To construct orthonormal exampleamples in which ψ is compactly supported, it pays to start from scaling coefficients {hk ; k ∈ Z} rather than from φ of Vj . The orthonormal wavelet bases thus obtained cannot, in general, be written in a closed analytic form. However, their graphs can be computed with arbitrarily high precision, via an algorithm which is called the ’cascade algorithm’. Before introducing Daubechies’ wavelets, some definitions, orthogonality conditions and conditions ensuring certain approximation properties will be briefly recalled. First of all from the orthogonality, it follows that the number of the scaling coefficients is even. By contradiction – let the number of the scaling parameters be odd and at the same time h0 = 0 and h2N = 0. Applying the scaling equation and using the orthogonality of translations of the scaling function give 2N k=0 hk hk−2l = 0 ∀ l ∈ Z. Since 2N is even, h2N h0 = 0 is obtained and this is the contradiction. Then we can assume that / {0, 1, . . . , 2N − 1}. hk = 0 for ∀k ∈

(4)

As for orthogonality, recall that the 2π-periodic function HN , defined by HN (ξ) =

2N −1

hk e−ikξ ,

(5)

k=0

is the discrete Fourier transform of the sequence of {hk } and that these numbers are coordinates of φ(x/2). The function HN is usually called the ’quadrature mirror filter’ or the ’conjugate quadrature filter’. Then the following Theorem holds:

402


Theorem 2.1. Let φ(.) = 2 k∈Z hk φ(2. − k) then ∞ φ(x − m)φ(x)dx = δm,0 for m ∈ Z

(6)

−∞

implies: 1. |HN (ξ)|2 + |HN (ξ + π)|2 = 1, 1 for m ∈ Z. 2. j hj h2m+j = 2 δm,0 For the proof of this Theorem and more details on the subject of orthonormality see for instance [4, 9, 13]. However the implications from this Theorem are only necessary conditions on orthonormality and if we use them for the derivation of scaling coefficients we have to check orthonormality afterwards. A first necessary and sufficient condition on HN was identified by Cohen (1990). His condition involves the structure of the zero-set of HN . A very different approach to the derivation of necessary and sufficient conditions was initiated by Lawton (1990). Define αl = φ(x)φ(x − l) dx. Now, we apply the scaling equation:

αl = 4

2N −1

hn hm

φ(2x − n)φ(2x − 2l − m) dx

n,m=0

=2

2N −1

hn hm α2l+m−n = 2

n,m=0

2N −2 k=2−2N

2N −1

hn hk−2l+n α2l+m−n .

n=0

Then we define the (4N − 3) × (4N − 3) matrix A by Alk :=

2N −1

−2N + 2 ≤ k, l ≤ 2N − 2.

hn hk−2l+n ,

(7)

n=0

and thus we have a very simple sufficient criterion for orthonormality of translations of φ. Theorem 2.2. (Lawton (1990)) Let HN (0) = 1 be a 2π-periodic function satisfying |HN (ξ)|2 + |HN (ξ + π)|2 = 1, and let φ be defined by (8). Then the eigenvalue 1 of matrix A defined by (7) is nondegenerate, if and only if the translations of φ are orthonormal. For the proof and more details on this subject see [4, 6, 8]. Furthermore, the scaling function corresponding to (4) has the compact support in [0, 2N − 1]. It is well-known that the function φ can be examplepressed, according to ˆ = φ(ξ)

∞ k=1

HN (2−k ξ),

(8)


403

by the convolution product φ = ()∞ k=1

2N −1 n=0

hk δ(2−k n)

= ()∞ k=1 uk ,

which converges in the sense of distributions. The support of the distribution uk is [0, 2−k (2N − 1)] and consequently, φ is a distribution with support in the interval [0, 2N − −1 1]. Since 2N k=1 |hk ||k| < ∞ and HN (0) = 1 the infinite product (8) makes sense and even converges absolutely and uniformly on compact sets. Then we can use the following Lemma proved by Mallat (1989): Lemma 2.3. If HN (0) = 1 is a 2π-periodic function satisfying |HN (ξ)|2 + |HN (ξ + π)|2 = 1, ˆ and if (8) converges pointwise a.e., then its limit φ(ξ) is in L2 (R), and ||φ||L2 < 1. For more details see [4, 6]. Further important topics in wavelet theory are conditions ensuring certain approximation properties. The order of accuracy is determined by computing with polynomials. Mainly, our attention is paid to the conditions which ensure that combinations of φ(x−k) exampleactly reproduce locally the polynomials 1, x, ...xn−1 - so the order of accuracy is n. The test for accuracy of order n is Condition An which is recognized in the time domain, the frequency domain, and also in the eigenvalue domain. Theorem 2.4. The order of accuracy is n if the scaling coefficients hk satisfy one of these three forms of Condition An : k j for j = 0, 1, ..., n − 1, 1. n sum conditions on the coefficients: k∈Z (−1) k hk = 0

2. H(ω) =

1+eiω 2

n

Q(ω) possess a zero of order n at π, n−1 , where M := {h2k−l }. 3. among eigenvalues of the matrix M are 1, 12 , ..., 12

Example 2.5. The matrix M for scaling the function corresponding to this scaling equation φ(.) = 3k=0 hk φ(2. − k) has this form: .

. . h1 h0 0 0 0 0 0 0 0

 0 h h 3 2   M = 0 0 0  0 0 0  0 0 0



 h1 h0 0 0 0 0 0    h3 h2 h1 h0 0 0 0  .  0 0 h3 h2 h1 h0 0   ... 0 0 0 0 h3 h2

Now it remarkains to answer the question whether Condition An is satisfactory for reproducing polynomials by combinations of scaling functions. The answer provides the following Theorem:


404

n−1 Theorem 2.6. Let the values 1, 12 , ..., 12 be among eigenvalues of the matrix M and k let yk denote left eigenvector of the matrix M related to the eigenvalue 12 . Then yk (l)φ(x + l) = xk for k = 0, 1, ..., n − 1, l∈Z

where yk (l) is the l-component of the vector yk . For proofs of the previous two Theorems and more details on this subject see [5, 9, 12]. Now, the filters of Daubechies’ scaling functions can be defined in the following way: Definition 2.7.

ω 1 sin2N −1 t dt, |HN (ω)| = qN (ω) := 1 − cN 0 π Γ(1/2)Γ(N ) . cN := sin2N −1 t dt = Γ(N + 1/2) 0 2

Then it is well-known that there exampleists the factorization N 1 + eiω HN (ω) = FN (ω), 2

(9)

where FN is a trigonometric polynomial and the condition |HN (ξ)|2 + |HN (ξ + π)|2 = 1

(10)

is also satisfied. For proofs, the proof of exampleistence of filters HN , and more details see for instance [9]. There is also the nice book written by Daubechies [6]. In this book an explicit solution |HN (ξ)|2 of (10) satisfying the condition (9) was found by computing with polynomials and it was shown that it is the unique lowest degree solution. This solution corresponds to the above definition. Note that HN is not unique! To conclude: From the filters HN can be constructed orthonormal scaling functions with support in [0, 2N − 1] and any polynomial up to degree N − 1 can be reproduced by linear combinations of φ(x − k). These wavelets introduced by Daubechies possess a minimal support for the prescribed approximation properties by maintenance of orthogonality.

3

On the computation of the scaling coefficients

1. Method So, there are some conditions on the scaling coefficients, but how to compute them? The first possibility consists in exampleploiting the Definition 2.7 and to compute the square root of |HN (ξ)|2 . A small illustration: Example 3.1. For N = 2, the following is obtained ω 1 1 9 1 2 sin2N −1 t dt = + cos(ω) − cos(3ω). |H2 (ω)| = q2 (ω) := 1 − cN 0 2 16 16


Ranging from H2 (ξ) =

3

k=0

2

|H2 (ξ)| =

405

hk e−ikξ , hk ∈ R 3

dk e−ikξ

with

dk =

k=−3

hk+l hl .

l

The coefficients dk are symmetric: dk = d−k . Thus |H2 (ξ)|2 is a cosine polynomial of the third degree |H2 (ω)|2 = d0 + 2d1 cos(ω) + 2d2 cos(2ω) + 2d3 cos(3ω) and comparison of coefficient yields 1 , 2 9 h0 h1 + h1 h2 + h2 h3 = , 32 h0 h2 + h1 h3 = 0, 1 h0 h3 = − . 32 h20 + h21 + h22 + h23 =

Another possibility for exampletracting the square root is presented in a Lemma by Riesz (see [10]). Lemma 3.2. Let A be a positive trigonometric polynomial invariant under the substitution ξ → −ξ; A is necessarily of the form A(ξ) =

M

am cos mξ,

with am ∈ R.

m=0

Then there exampleists a trigonometric polynomial B of order M, i.e., B(ξ) =

M

bm eimξ ,

with bm ∈ R,

m=0

such that |B(ξ)|2 = A(ξ). Note that this proof is constructive. It uses roots of |HN (ξ)|2 for the construction of roots of HN (ξ). This approach was initiated by Daubechies [6]. 2. Method The second possibility consists in using the equivalent statements from Theorem 2.1 and Theorem 2.4. Consequently a slightly simpler system of equations can be obtained. Let N be some positive integer thus: i) hk = 0 for ∀k ∈ / {0, 1, . . . , 2N − 1}, 2N −1 −1 ii) δm,0 = 2 for 1 − N ≤ m ≤ N − 1, j=0 hj h2m+j 2N −1 iii) hk = 1, k=0 2N −1 k n iv) k=0 (−1) hk k = 0 for 0 ≤ n ≤ N − 1.


406

Example 3.3. For N = 2, it results: h0 + h1 + h2 + h3 = 1, h0 − h1 + h2 − h3 = 0, −h1 + 2h2 − 3h3 = 0, h0 h2 + h1 h3 = 0. 3. Method And at the end a new approach based on relations among scaling moments will be introduced. First, the orthogonality conditions have to be replaced. In [7], the following Theorem was proved: Theorem 3.4. Let φ(.) = 2

2N −1 0

hk φ(2. − k) then

|HN (ξ)|2 + |HN (ξ + π)|2 = 1 is equivalent to 2n 1 2n (−1)i (ai a2n−i + bi b2n−i ) δ0,n = i 2 i=0

for N > n ≥ 0, n ∈ N,

where N −1

(2k)i h2k

ai =

and

bi =

k=0

N −1

(2k + 1)i h2k+1 .

(11)

k=0

Consequently, applying the third condition from Theorem 2.4 which ensures some approximation properties 2N −1

(−1)k hk k n = 0 for 0 ≤ n ≤ N − 1,

k=0

implies ai = bi for 0 ≤ i ≤ N − 1 and the following Theorem holds: −1 hk φ(2. − k) and let φ(.) be an orthonormal scaling Theorem 3.5. Let φ(.) = 2 2N 0 function with the order of accuracy m. Then the following formulas hold: δ0,n = n j=0

where Mn :=

n n i=0

i

(−1)i mi mn−i

for n = 0, ..., 2m − 1.

(12)

n (−1)j+1 Mj Mn−j = 0 j xn φ(x) dx

and

for n = 1, ..., 2m − 1,

mn :=

k

hk k n ,

respectively.

(13)


407

For proofs and more details see [7]. Similar results appeared also in [1, 2, 3]. Consequently, the coefficients ai for i > N − 1 are a linear combination of ai for i < N and the same holds also for the coefficients bi . It can be seen by considering that both the system of a0 , . . . , aN −1 and the system of b0 , . . . , bN −1 contain only N variables which are from their definition (see (11)) linearly independent. Together with ai = bi for 0 ≤ i ≤ N − 1, it implies that the discrete scaling moments mi = ai +bi for i > N −1 can be examplepressed by linear combinations of those with indexample smaller than N and that the same holds also for the continuous scaling moments. Then the following system is equivalent to the previous one corresponding to the second method: i) Find {m1 , . . . , mN −1 }, ii) δ0,n = ni=0 ni (−1)i mi mn−i for 1 ≤ m ≤ N − 1, iii) m0 = 1. The previous approach can be further enhanced. As mentioned above, mi for i > N −1 can be examplepressed by linear combinations of previous ones (i < N ), but these linear combinations depend on the displacement of the scaling function or equivalently on the displacement of scaling coefficients. It means that shifted scaling moments mi

=

2N −1 k=0

hk

2N − 1 k− 2

i (14)

−1 i can be treated instead of the originally defined moments mi = 2N k=0 hk k . The main advantage of this approach consists in the symmetry of ai and bi (defined similar to the original ones) for even i and antisymmetry of odd ones, respectively. For exampleample, i i the first term by ai is − 2N2−1 h0 and the last one by bi is 2N2−1 h2N −1 and so on. Thus the even moments for i > N − 1 are examplepressed only by linear combinations of even moments (for i < N ) and by analogy for the odd moments. This useful trick enables one to substantially reduce the complexampleity of arising system. Example 3.6. For N = 3, the following system will be obtained m 0 = 1

m 2 = m 1

2

2

m 4 − 4m 1 m 3 + 3m 2 = 0

and further

13 m1 4 after substitution and elimination m 3 =

m 4 =

4

11 45 m2− 2 16

2

16m 1 − 40m 1 − 15 = 0. Now, we will study further properties of m1 . Are solutions in variable m 1 symmetrical according to point 0? Is there any estimate for the localization of m 1 ? We start with the first problemma. First of all, we notice some symmetry in the second method for the computation of scaling coefficients. Lemma 3.7. If


408

{h0 , . . . , h2N −1 } satisfies the conditions for the scaling coefficients in the second method then {h2N −1 , . . . , h0 } also satisfies these conditions. Proof. First we check the condition ensuring approximation properties. 2N −1

(−1)k hk k n = 0 for 0 ≤ n ≤ N − 1.

k=0

can be replaced by 2N −1

(−1)k hk (k − l)n = 0 for 0 ≤ n ≤ N − 1.

k=0

for any fixed integer l. And the choice l = 2N − 1 brings n

(−1)

2N −1

(−1)k hk (2N − 1 − k)n = 0 for 0 ≤ n ≤ N − 1.

k=0

At last the symmetry of the orthonormality conditions δm,0 = 2−1 finishes the proof.

2N −1 j=0

hj h2m+j ✷

Thus, the symmetrical scaling function can be defined as follows φs (x) := φ(2N − 1 − x) and the computation of the first continuous scaling moments gives

M1s

=

s

x φ (x) dx =

x φ(2N − 1 − x) dx =

(2N − 1 − y) φ(y) dy = 2N − 1 − M1 .

Furthermore m0 = 1, M0 = 1 and M1 = 1 = 2

1

x φ(x) dx = 2

k

1 xhk φ(2x − k) dx = 2

(y + k) hk φ(y) dy

k

1 (m0 y + m1 ) φ(y) dy = (m0 M1 + m1 M0 ), 2

imply M1 = m1 and consequently ms1 = 2N − 1 − m1 , where ms1 is symmetrical discrete moments defined analogical to the continuous one. It means that all solutions in variable m1 are symmetrical according to point 2N2−1 . We formulate it in the nexamplet Lemma.


409

Lemma 3.8. The solutions in variable m1 coming from the above definition of Daubechies’ filters possess pairs of roots symmetrical according to point 2N2−1 or equivalently the solutions in variable m1 possess pairs of roots symmetrical according to point 0. Now, we will study the second question concerning the localization of m1 . We are mainly concerned with the real solution. −1 k Lemma 3.9. Let HN (z) = 2N k=0 hk z be any nonconstant polynomial with real coefficients and h0 h2N −1 = 0. Further let HN (1) = 1 and sup|z|=1 |HN (z)| ≤ 1, then HN (1) belongs to the interval (0, 2N − 1). Proof. We use the following version of the Bernstein inequality for trigonometric polynomials: Let p be any polynomial with complexample coefficients and at most degree 2N − 1. Then max|z|=1 |p (z)| ≤ (2N − 1) max|z|=1 |p(z)|. Equality holds iff there exampleists a constant c such that p(z) = cz 2N −1 . We apply this inequality to the polynomials HN (z) an z 2N −1 HN (z) and we obtain |HN (1)| < 2N − 1 and |2N − 1 − HN (1)| < 2N − 1, respectively. Since HN has real coefficients, HN (1) is a real number and the proof is finished. ✷ To conclude: The defining system in the second method is partly linear and partly quadratic. Unlike the second method the third one enables one to simply eliminate not only linear equations but also some quadratic one - even variables can be namely immediately examplepressed by combinations of odd ones. Furthermore, solving these systems for shifted moments defined by (14) can again reduce the complexampleity. For filter lengths up to 16 this system can be exampleplicitly solved via algebraic methods like Gröbner bases. Its particularly simple structure allows one to find all possible solutions. At the end, the two Theorems mentioned above imply that m1 belongs to the interval ( 1−2N , 2N2−1 ) and due to the symmetry, we can restrict it only to the interval [0, 2N2−1 ). 2 The obtained results are presented in Tables 1-5 and in Figures 1,2,3 and 4. We should also note that exampleact examplepressions in closed forms of Daubechies orthogonal filters for N = 4 and 5 are given in [11]. For each filter the first collection of coefficients corresponds to the ”minimum phase filter” (or with the extremal phase choice) scaling function and the last one corresponds to the ”least asymmetric” scaling function, respectively both constructed by Daubechies (see [6]). At last, we examplepress one hypothesis - the number N −2 of solutions is equal to 2[ 2 ] for N > 1 using the notation [x] for the integer part of x.

Acknowledgment This work is partially supported by Grant Agency of the Czech Republic - Grant No 201/01/1200 and by Ministry of Education of the Czech Republic - Grant No MSM 113200007.

410


References [1] B. Han: “Analysis and construction of optimal multivariate biortogonal wavelets with compact support”, SIAM J. Math. Anal., Vol. 31, (2000), pp. 274–304. [2] B. Han: ”Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets”, J. Approx. Theory, Vol. 110, (2001), pp. 18–53. [3] B. Han: ”Symmetric multivariate orthogonal refinable functions”, Appl. Comput. Harmon. Anal., to appear. [4] A. Cohen and R.D. Ryan: Wavelets and Multiscale Signal Processing, Chapman & Hall, London, 1995. [5] A. Cohen: “Wavelet methods in numerical analysis“, In: P.G. Ciarlet et al. (Eds.): Handbook of numerical analysis, Vol. VII, Amsterdam: North-Holland/Elsevier, 2000, pp. 417–711. [6] I. Daubechies: Ten Lectures on Wavelets. Society for industrial and applied mathematics, Philadelphia, 1992. [7] V. Finˇek: Approximation properties of wavelets and relations among scaling moments II, Preprint, TU Dresden, 2003. [8] W. Lawton: “Necessary and sufficient conditions for constructing orthonormal wavelet bases“, J. Math. Phys., Vol. 32, (1991), pp. 57–61. [9] A.K. Louis, P. Maass and A. Rieder: Wavelets: Theorie und Anwendungen, B.G. Teubner, Stuttgart, 1994. [10] G. Polya and G. Szegö: Aufgaben und Lehrs¨ atze aus der Analysis, Vol. II, SpringerVerlag, Berlin, 1971. [11] W.C. Shann and C.C. Yen: “On the exampleact values of the orthogonal scaling coefficients of lenghts 8 and 10“, Appl. Comput. Harmon. Anal., Vol. 6, (1999), pp 109–112. [12] G. Strang and T. Nguyen: Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996. [13] P. Wojtaszczyk: A Mathematical introduction to wavelets, Cambridge University Press, 1997.


N=1

N=2

N=3

n

hN (n)

0

0.5

1

hN (n)

0

0.1132094912917792

0.5

1

0.4269717713525142

√

2

0.5121634721295985

3

0.0978834806739047

4

-0.1713283576914674

5

-0.0228005659417736

6

0.0548513293210668

7

-0.0044134000541791

8

-0.0088959350509771

9

0.0023587139695339

0

0.0193273979773840

1

0.0208734322107417

2

-0.0276720930583110

0

(1 −

1

(3 −

2

(3 +

3

(1 +

√ √ √

N=5

3)/8 3)/8 3)/8 3)/8

5

√ √ (1 + 10 + 5 + 2 10)/32 √ √ (5 + 10 + 3 5 + 2 10)/32 √ √ (5 − 10 + 5 + 2 10)/16 √ √ (5 − 10 − 5 + 2 10)/16 √ √ (5 + 10 − 3 5 + 2 10)/32 √ √ (1 + 10 − 5 + 2 10)/32

0

0.1629017140256492

3

0.1409953484269096

1

0.5054728575459144

4

0.5115264834471958

2

0.4461000691233798

5

0.4482908241899161

3

-0.0197875131178223

6

0.0117394615680619

4

-0.1322535836845199

7

-0.1239756813064712

5

0.0218081502370886

8

-0.0149212499343306

6

0.0232518005354909

9

0.0138160764789039

7

-0.0074934946651807

0

-0.0535744507091029

1

-0.0209554825625298

2

0.3518695343281499

3

0.5683291217038204

4

0.2106172671017885

5

-0.0701588120892717

6

-0.0089123507208356

7

0.0227851729479811

0 1 2 3 4

N=4

n

Table 1

411

412


N=6

n

hN (n)

n

hN (n)

0

0.0788712160014507

0

0.0156461310906403

1

0.3497519070376178

1

0.0353192853999830

2

0.5311318799408690

2

0.0117281274817440

3

0.2229156614650177

3

0.1071338130028424

4

-0.1599932994460614

4

0.4346881891482093

5

-0.0917590320301476

5

0.5038492536130889

6

0.0689440464873723

6

0.0883403719454407

7

0.0194616048541647

7

-0.1773998458661412

8

-0.0223318741650945

8

-0.0590712131095395

9

0.0003916255761486

9

0.0349375154878786

10

0.0033780311814639

10

0.0086683934435052

11

-0.0007617669028013

11

-0.0038400216376517

0

-0.0278054939010449

0

0.0108923501632923

1

-0.0542339970448559

1

0.0024683061859233

2

0.1433852735866131

2

-0.0834316077060730

3

0.5030939996188508

3

-0.0341615607932860

4

0.4566824370358502

4

0.3472289864792177

5

0.0333558930263133

5

0.5569463919628689

6

-0.1043053858735158

6

0.2389521856664347

7

0.0253108621610915

7

-0.0513624849308392

8

0.0362577173208787

8

-0.0148918756492697

9

-0.0096875351296210

9

0.0316252813300217

10

-0.0042145481687812

10

0.0012499610463979

11

0.0021607773682213

11

-0.0055159337546888

Table 2


N=7

n

hN (n)

n

hN (n)

0

0.0550497153728118

0

0.0084961844541972

1

0.2803956418127626

1

0.0121716951094147

2

0.5155742458180987

2

-0.0458968894614975

3

0.3321862411055397

3

-0.0453476699090422

4

-0.1017569112133463

4

0.2547129164153979

5

-0.1584175056403329

5

0.5529020609727706

6

0.0504232325046941

6

0.3419645579347671

7

0.0570017225798716

7

-0.0401668308104380

8

-0.0268912262948454

8

-0.0714255071199577

9

-0.0117199707821033

9

0.0316376187152903

10

0.0088748961896808

10

0.0144703799496919

11

0.0003037574977011

11

-0.0128174454082603

12

-0.0012739523590969

12

-0.0023216421725990

13

0.0002501134265612

13

0.0016205713302650

0

0.0122869726957853

0

0.0072606973809772

1

0.0401825838182185

1

0.0028356713428721

2

0.0415476463363710

2

-0.0762319359477674

3

0.0972532493036416

3

-0.0990283534036613

4

0.3605202757013507

4

0.2040919698622592

5

0.5172831726067475

5

0.5428913549072109

6

0.1829932437451860

6

0.3790813009818498

7

-0.1917050162867341

7

0.0123328297443059

8

-0.1267065470940570

8

-0.0350391456110007

9

0.0448861205161673

9

0.0480073839677329

10

0.0330231232026814

10

0.0215777262909796

11

-0.0090207011145457

11

-0.0089352158255621

12

-0.0036647145873175

12

-0.0007406129572978

13

0.0011205911565050

13

0.0018963292671016

Table 3

413

414


N=8

n

hN (n)

n

hN (n)

0

0.0384778110540762

0

-0.0140599273346535

1

0.2212336235761249

1

-0.0474992790529680

2

0.4777430752138737

2

0.0293082221104462

3

0.4139082662111959

3

0.3372088679700805

4

-0.0111928676668802

4

0.5412154843748935

5

-0.2008293163904890

5

0.2599529093929467

6

0.0003340970462201

6

-0.1059636902982449

7

0.0910381784236577

7

-0.0716510778673998

8

-0.0122819505228484

8

0.0693444907236722

9

-0.0311751033251394

9

0.0244482622756116

10

0.0098860796483508

10

-0.0258540306350766

11

0.0061844224098159

11

-0.0018208836477897

12

-0.0034438596284418

12

0.0067774635347850

13

-0.0002770022744794

13

-0.0008661330388585

14

0.0004776148556496

14

-0.0007680124758218

15

-0.0000830686306866

15

0.0002273339683771

0

0.0094498497971799

0

0.0065454468010230

1

0.0395337686865075

1

0.0165703003941643

2

0.0608499587635737

2

-0.0163698122406393

3

0.1006302254096596

3

-0.0371090122732223

4

0.3003296986863678

4

0.1811173956086418

5

0.4986973760508475

5

0.5129122499471088

6

0.2717739958371097

6

0.4288623480595546

7

-0.1625575734146116

7

0.0100776356718280

8

-0.1987199044539242

8

-0.1281131292342110

9

0.0282520262435560

9

0.0076145068166549

10

0.0692481011989093

10

0.0367799913713535

11

-0.0063298770937416

11

-0.0136782361297209

12

-0.0143467303751760

12

-0.0100584701307134

13

0.0021122922003640

13

0.0041008796209177

14

0.0014150305459597

14

0.0012362297649909

15

-0.0003382380825814

15

-0.0004883240477305

Table 4


N=8

n

hN (n)

n

hN (n)

0

0.0054415277153865

0

0.0016075105997628

1

0.0076498515596055

1

0.0015520018691811

2

-0.0475454550534975

2

-0.0055265028067328

3

-0.0971496984757495

3

0.0126037598652244

4

0.1172294763649463

4

0.0548137999395381

5

0.4911302349154492

5

0.0213688058172216

6

0.4673628458051815

6

0.0223745704952175

7

0.0767420208091848

7

0.3028973863573692

8

-0.0777259100294995

8

0.5411936062172783

9

0.0289523449374786

9

0.2576993351871367

10

0.0431284462760154

10

-0.1137359201143157

11

-0.0106105081339174

11

-0.1409528956890003

12

-0.0087167003659729

12

-0.0026467581147138

13

0.0038731444084098

13

0.0257245907196667

14

0.0008257692874402

14

0.0019196937839655

15

-0.0005873900204611

15

-0.0019883533437054

0

-0.0034530083139349

0

-0.0023917292557492

1

-0.0062576695346180

1

-0.0003833454481163

2

0.0150124055521158

2

0.0224118115218331

3

0.0295649066056758

3

0.0053793058752817

4

0.0016918191659450

4

-0.1013243276431463

5

0.1187221221820791

5

-0.0433268077029274

6

0.4427716837343578

6

0.3403726735948683

7

0.4929634329673349

7

0.5495533152683713

8

0.0901859480155989

8

0.2576993351871367

9

-0.1760890318598142

9

-0.0367312543805018

10

-0.0613950240642198

10

-0.0192467606317034

11

0.0491271030478224

11

0.0347452329556765

12

0.0168636846274549

12

0.0026931943768827

13

-0.0089565197595468

13

-0.0105728432641898

14

-0.0016775087173178

14

-0.0002141971501220

15

0.0009256563510668

15

0.0013363966964058

Table 5

415

416


1

0,8

0,6

0,4

0,2

0

-0,2

-0,4 0

2

4

6

8

10

Fig. 1 φ for N = 6 (corresponding to the 1st collection of coefficient presented in the 1st column of Table 2).


1

0,8

0,6

0,4

0,2

0

-0,2 0

2

4

6

8

10

Fig. 2 φ for N = 6 (corresponding to the 2nd collection of coefficients presented in the 1st column of Table 2).

417

418


1

0,8

0,6

0,4

0,2

0

-0,2

-0,4 0

2

4

6

8

10

Fig. 3 φ for N = 6 (corresponding to the 1st collection of coefficients presented in the 2nd column of Table 2).


1

0,8

0,6

0,4

0,2

0

-0,2 0

2

4

6

8

10

Fig. 4 φ for N = 6 (corresponding to the 2nd collection of coefficients presented in the 2nd column of Table 2).

419

CEJM 2(3) 2004 420–447

Review article

Matrix problems and stable homotopy types of polyhedraa Yuriy A. Drozd∗ Department of Mechanics and Mathematics, Kyiv Taras Shevchenko University, 01033 Kyiv, Ukraine

Received 4 March 2004; accepted 6 June 2004 Abstract: This is a survey of the results on stable homotopy types of polyhedra of small dimensions, mainly obtained by H.-J. Baues and the author [3, 5, 6]. The proofs are based on the technique of matrix problems (bimodule categories). c Central European Science Journals. All rights reserved. Keywords: polyhedra, homotopy type, matrix problems, tame and wild problems MSC (2000): 55P12, 15A36, 16G60 a

Dedicated to C. M. Ringel.

1

Introduction

This paper is a survey of some recent results on stable homotopy types of polyhedra. The common feature of these results is that their proofs use the technique of the so called matrix problems, which was mainly elaborated within framework of representation theory. I think that this technique is essential in homotopy theory too, and perhaps even in much more general setting of triangulated categories. I hope that the considerations of Section 3 are persuasive enough. Certainly, I could not cover all such results in an expository work, thus I have restricted to the stable homotopy classification of polyhedra of small dimensions obtained in [3, 5, 6, 7]. I tried to present these results in a homogeneous way and also to replace references to rather sophisticated topological sources by simpler ones. The latter mainly concerns with some basic facts about homotopy groups of spheres, which can be found in [18] or [21]. I also used the book [20] as a standard source of references; maybe some readers will prefer [19] or [10]. Most of these references are ∗


Yu. Drozd / Central European Journal of Mathematics 2(3) 2004 420–447

421

collected in Section 1. For the matrix problems I have chosen the language of bimodule categories explained in Section 2, since it seems to be the simplest one as well as the most appropriate for applications. Note that almost the same arguments that are used in Sections 5 and 6 can be applied to the classification of polyhedra with only 2 non-trivial homology groups [6], while the dual arguments were applied to the spaces with only 2 non-trivial homotopy groups in [4]. Rather similar are also calculations in [17] (see also the Appendix by Baues and Henn to [3]). I hope that any diligent reader of this survey will be able to comprehend the arguments of these papers too. I am extremely indebted to H.-J. Baues, who was my co-author and my guide to the topological problems, and to C. M. Ringel, whose wonderful organising activity had made such a pleasant and fruitful collaboration possible. H.-J. Baues and I obtained most of our joint results during my visits to the Max-Plank-Institut f¨ ur Mathematik, and I highly acknowledge its support.

2

Generalities on stable homotopy types

All considered spaces are supposed pathwise connected and punctured ; we denote by ∗X (or by ∗ if there can be no ambiguity) the marked point of the space X. B n and S n−1 denote respectively the n-dimensional ball { x ∈ Rn | ||x|| ≤ 1 } and the (n − 1)dimensional sphere { x ∈ Rn | ||x|| = 1 }, both with the marked point (1, 0, . . . , 0) . As usually, we denote by X ∨ Y the bouquet (or one point union) of X and Y , i.e. the factor space X Y by the relation ∗X = ∗Y , and identify it with ∗X × Y ∪ X × ∗Y ⊂ X × Y ; we denote by X ∧Y the factor space X ×Y /X ∨Y . In particular, we denote by ΣX = S 1 ∧X the suspension of X and by Σn X = Σ . . . Σ X its n-th suspension. The word “polyhedron” n times

is used as a synonym of “finite CW-complex.” One can also consider bouquets of several spaces si=1 Xi ; if all of them are copies of a fixed space X, we denote such a bouquet by sX. We recall several facts on stable homotopy category of CW-complexes. We denote by Hot(X, Y ) the set of homotopy classes of continuous maps X → Y and by CW the homotopy category of polyhedra, i.e. the category whose objects are polyhedra and morphisms are homotopy classes of continuous maps. The suspension functor defines a natural map Hot(X, Y ) → Hot(ΣX, ΣY ). Moreover, the Whitehead theorem [20, Theorem 10.28 and Corollary 10.29] shows that the suspension functor reflects isomorphisms of simply connected polyhedra. It means that if f ∈ Hot(X, Y ), where X and Y are simply connected, f is an isomorphism (i.e. a homotopy equivalence) if and only if so is Σf . We set Hos(X, Y ) = limn Hot(Σn X, Σn Y ). If α ∈ Hot(Σn X, Σn Y ), β ∈ Hot(Σm Y, Σm Z), −→ one can consider the class Σn β ◦ Σm α ∈ Hot(Σm+n X, Σn+m Z), whose stabilization is, by definition, the product βα of the classes of α and β in Hos(X, Z). Thus we obtain the stable homotopy category of polyhedra CWS. Actually, if we only deal with finite CW-complexes, we need not go too far, since the Freudenthal theorem [20, Theorem 6.26]

422


implies the following fact. Proposition 2.1. If X, Y are of dimensions at most d and (n − 1)-connected, where d < 2n − 1, then the map Hot(X, Y ) → Hot(ΣX, ΣY ) is bijective. If d = 2n − 1, this map is surjective. In particular, the map Hot(Σm X, Σm Y ) → Hos(X, Y ) is bijective if m > d − 2n + 1 and surjective if m = d − 2n + 1. Here (n − 1)-connected means, as usually, that πk (X), the k-th homotopy group of X, is trivial for k ≤ n − 1. Thus for all polyhedra of dimension at most d the map Hot(Σm X, Σm Y ) → Hos(X, Y ) is bijective if m ≥ d and surjective if m = d − 1. Note also that the natural functor CW → CWS reflects isomorphisms of simply connected polyhedra. Since we are only interested in stable homotopy classification, we identify, in what follows, polyhedra and continuous maps with their images in CWS. We denote by CWF the full subcategory of CWS consisting of all spaces X with torsion free homology groups Hi (X) = Hi (X, Z) for all i. Recall that any suspension Σn X is an H-cogroup [20, Chapter 2], commutative if n ≥ 2, so the category CWS is an additive category. Moreover, one can deduce from the Adams’ theorem [20, Theorem 9.21] that this category is actually fully additive, i.e. every idempotent e ∈ Hos(X, X) splits. In our case it means that there is a decomposition Σm X Y ∨ Z for some m, such that e comes from the map ε : Y ∨ Z → Y ∨ Z with ε(y) = y for y ∈ Y and ε(z) = ∗Y ∨Z for z ∈ Z. We call a polyhedron X indecomposable if X Y ∨ Z implies that either Y or Z are contractible (i.e. isomorphic in CW to the 1point space). Actually, the category CWS is a triangulated category [16]. The suspension f

plays the role of shift, while the triangles are the cone sequences X −→ Y → Cf → ΣX (and isomorphic ones), where Cf = CX ∪f Y is the cone of the map f , i.e the factor space CX Y by the relation (x, 0) ∼ f (x); CX = X × I/X × 1 is the cone over the space X. Note that cone sequences coincide with cofibration sequences in the category CWS [20, Proposition 8.30]. Recall that a cofibration sequence is a such one f

g

Σf

h

X −→ Y −→ Z −→ ΣX −→ ΣY

(1)

that for every polyhedron P the induced sequences f∗

g∗

Σf∗

h

∗ Hos(P, X) −→ Hos(P, Y ) −→ Hos(P, Z) −→ Hos(P, ΣX) −→ Hos(P, ΣY ),

Σf ∗

g∗

h∗

f∗

Hos(ΣY, P ) −→ Hos(ΣX, P ) −→ Hos(Z, P ) −→ Hos(Y, P ) −→ Hos(X, P )

(2)

are exact. In particular, we have an exact sequence of stable homotopy groups f∗

g∗

h

Σf∗

∗ S S πk−1 (X) −→ πk−1 (Y ), πkS (X) −→ πkS (Y ) −→ πkS (Z) −→

(3)

where πkS (X) = limm πk+m (Σm X) = Hos(S k , X). Certainly, one can prolong the se−→ quences (2) and (3) into infinite exact sequences just taking further suspensions. Every CW-complex is obtained by attaching cells. Namely, if X n is the n-th skeleton of X, then there is a bouquet of balls B = mB n+1 and a map f : mS n → X n such


423

that X n+1 is isomorphic to the cone of f , i.e. to the space X n ∪f B. It gives cofibration sequences like (1) and exact sequences like (2) and (3). We denote by CWkn the full subcategory of CW formed by (n − 1)-connected (n + k)dimensional polyhedra and by CWFkn the full subcategory of CWkn formed by the polyhedra X with torsion free homology groups Hi (X) for all i. Proposition 2.1 together with the fact that every map of CW-complexes is homotopic to a cell map, also implies the following result. Proposition 2.2. The suspension functor Σ induces equivalences CWkn → CWkn+1 for all n > k + 1. Moreover, if n = k + 1, the suspension functor Σ : CWkn → CWkn+1 is a full representation equivalence, i.e. it is full, dense and reflects isomorphisms. (Dense means that every object from CWkn+1 is isomorphic (i.e. homotopy equivalent) to ΣX for some X ∈ CWkn .) Therefore, setting CWk = CWkk+2 CWkn for n > k + 1, we can consider it as a full subcategory of CWS. The same is valid for CWFk = CWFkk+2 . Note also that CWkn naturally embeds into CWkn+1 . It leads to the following notion [2]. Definition 2.3. An atom is an indecomposable polyhedron X ∈ CWkk+1 not belonging to the image of CWkk . A suspended atom is a polyhedron Σm X, where X is an atom. Then we have an obvious corollary. Corollary 2.4. Every object from CWkn with n ≥ k + 1 is isomorphic (i.e. homotopy equivalent) to a bouquet si=1 Xi , where Xi are suspended atoms. Moreover, any suspended atom is indecomposable (thus indecomposable objects are just suspended atoms). Note that the decomposition in Corollary 2.4 is, in general, not unique [14]. That is why an important question is the structure of the Grothendieck group K0 (CWk ). By definition, it is the group generated by the isomorphism classes [X] of polyhedra from CWk subject to the relations [X ∨ Y ] = [X] + [Y ] for all possible X, Y . The following results of Freyd [14, 10] describe the structure of this group. Definition 2.5. (1) Two polyhedra X, Y ∈ CWk are said to be congruent if there is a polyhedron Z ∈ CWk such that X ∨ Z Y ∨ Z (in CWk ). (2) A polyhedron X ∈ CWk is said to be p-primary for some prime number p if there is a bouquet of spheres B such that the map pm 1X : X → X can be factored through B, i.e. there is a commutative diagram X AA

pm 1X

AA AA A

B

/X }> } } }} }}

424


Theorem 2.6 (Freyd). The group K0 (CWk ) (respectively K0 (CWFk ) ) is a free abelian group with a basis formed by the congruence classes of p-primary suspended atoms from CWk (respectively from CWFk ) for all prime numbers p ∈ N. Therefore, if we know the “place” of every atom class [X] in K0 (CWk ) or K0 (CWFk ), i.e. its presentation as a linear combination of classes of p-primary suspended atoms, we can deduce therefrom all decomposition rules for CWk or CWFk .

3

Bimodule categories

We also recall main notions concerning bimodule categories [11, 13]. Let A, B be two fully additive categories. An A-B-bimodule is, by definition, a biadditive bifunctor U : A◦ ×B → Ab. As usual, given an element u ∈ U(A, B) and morphisms α ∈ A(A , A), β ∈ B(B, B ), we write βuα instead of U(α, β)u. Given such a functor, we define the bimodule category El(U) (or the category of elements of the bimodule U, or the category of matrices over U) as follows. • The set of objects of El(U) is the disjoint union ob El(U) = U(A, B). A∈ob A B∈ob B

• A morphism from u ∈ U(A, B) to u ∈ U(A , B ) is a pair (α, β) of morphisms α ∈ A(A, A ), β ∈ B(B, B ) such that u α = βu in U(A, B ). • The product (α , β )(α, β) is defined as the pair (α α, β β). Obviously, El(U) is again a fully additive category. Suppose that ob A ⊃ { A1 , A2 , . . . , An } , ob B ⊃ { B1 , B2 , . . . , Bm } such that every object A ∈ ob A (B ∈ ob B) decomposes as A ni=1 ki Ai (respectively, B m i=1 li Bi ). ◦ Then A (respectively, B) is equivalent to the category of finitely generated projective right (left) modules over the ring of matrices (aij )n×n with aij ∈ A(Aj , Ai ) (respectively, (bij )m×m with bij ∈ B(Bj , Bi )). We denote these rings respectively by |A| and |B|. We also denote by |U| the |A|-|B|-bimodule consisting of matrices (uij )m×n , where uij ∈ U(Aj , Bi ). Then U(A, B), where A, B are, respectively, a projective right |A|-module and a projective left |B|-module, can be identified with A ⊗|A| |U| ⊗|B| B. Elements from this set are usually considered as block matrices (Uij )m×n , where the block Uij is of size li × kj with entries from U(Aj , Bi ). To form a direct sum of such elements, one has to write direct sums of the corresponding blocks at each place. Certainly, some of these blocks can be “empty,” if kj = 0 or li = 0. An empty block is indecomposable if and only if it is of size 0 × 1 (in U(Aj , 0)) or 1 × 0 (in U(0, Bi ) ); we denote it respectively by ∅j or by ∅i . In many cases the rings |A| and |B| can be identified with tiled subrings of rings of integer matrices. Here a tiled subring in Mat(n, Z) is given by an integer matrix (dij )n×n such that dii = 1 and dik |dij djk for all i, j, k; the corresponding ring consists of all matrices (aij ) such that dij |aij for all i, j (especially aij = 0 if dij = 0).


425

Example 3.1. Let A ⊂ Mat(2, Z) be the tiled ring given by the matrix 

 1 12  , 0 1 U be the set of 2 × 2-matrices (uij ) with uij ∈ Z/24 if i = 1, j = 2, uij ∈ Z/2 otherwise. We define U as an A-A-bimodule setting      a 12b u1 u2  au1 + bu3 au2 + 12bu4   = ;   u3 u4 cu3 cu4 0 c      u1 u2  a 12b au1 cu2 + 12bu1   =   . 0 c u3 u4 au3 cu4 + bu3 If we need to indicate this action, we write     ∗ 1 12   Z/2 Z/24    and   ∗ 0 1 Z/2 Z/2 for the matrix defining the ring A and for the bimodule U. Thus the multiplications of the elements marked by stars is given by the ∗-rule: (12a∗ ) · (u mod 2∗ ) = au mod 2.

(4)

Example 3.2. In the classification of torsion free atoms below the following bimodule plays the crucial role. We consider the tiled rings A2 ⊂ Mat(2, Z) and B2 ⊂ Mat(7, Z) given respectively by the matrices 

 1   1   1    ∗  1 12      and 0  0 1  0    0  0

2 2 12 24 12 24 1 1 12 24 6 24 2 1 12 24 12 24 0 0 1 2 12∗ 12 0 0 1 1 12 6 00 0 0

1

1

00 0 0

0

1

          .        

426


The A2 -B2 -bimodule U2 is defined as the set of matrices of the form   Z/24 0      Z/12  0      Z/12 0         Z/2 Z/24  .      0 Z/12        ∗ Z/2   Z/2   0 Z/2 The multiplication in U2 is given by the natural matrix multiplication, but taking into account the ∗-rule (4). We shall use the following description of indecomposable elements in El(U2 ). Set I1 = { 1, 2, 3, 4, 6 } , I2 = { 4, 5, 6, 7 }, V = { v ∈ N | 1 ≤ v ≤ 6 }, V1 = { v ∈ N | 1 ≤ v ≤ 12 }, V2 = { 1, 2, 3 }. Theorem 3.3. A complete list L2 of non-isomorphic indecomposable objects from El(U2 ) consists of • empty objects ∅j (j = 1, 2) and ∅i (1 ≤ i ≤ 7); • objects vij ∈ U(Aj , Bi ) (j = 1, 2; i ∈ Ij ; v ∈ V1 if i = 1; v = 1 if i = 6, 7 or (ij) = (14); v ∈ V otherwise); j vi j (j = 1, 2; i = 1, 2, 3, l = 4, 6 if j = 1; i = 4, 5, l = 6, 7 if j = 2; • objects vil = 1jl if (il) = (26) or (57) then v ∈ V2 ; otherwise v ∈ V ); • objects v44 =  (114 v42 ) with v ∈ V ; 1

2

14 v4  • objects v4l =   with l = 6, 7 and v ∈ V ; 0 12l   1 vi 0  • objects vi w44 =   with i = 1, 2, 3 and v, w ∈ V ; 114 w42   1 vi 0    1 2 • objects vi w4l =  14 w4  with i = 1, 2, 3, l = 6, 7 and v, w ∈ V .   2 0 1l Here the indices define the block containing the corresponding element. Proof. Decompose U into 2-primary and 3-primary parts. Since for every two matrices M2 , M3 ∈ GL(n, Z) there is a matrix M ∈ GL(n, Z) such that M ≡ M2 mod 2 and


427

M ≡ M3 mod 3, we can consider the 2-primary part and the 3-primary part separately. Note that in the 3-primary part the blocks u14 , u16 , u26 and u27 vanish, while the other nonzero blocks of u ∈ ob(U2 ) are with entries from Z/3 and there are no restrictions on elementary transformation of the matrix u. Thus every element in the 3-primary part is a direct sum of elements 1ji with j = 1, i = 1, 2, 3 or j = 2, i = 4, 5. For elements u, u of the 2-primary part write u < u if u = ua for some non-invertible a ∈ A2 . Then we have the following relations: 111 < 113 < 112 < 211 < 213 < 212 < 411 , 116 < 114 < 411 and 116 < 212 ;

124 < 125 < 224 < 225 < 424 ,

127 < 126 < 424 and 127 < 225 .

Using them, one can easily decompose the parts     1  u1  2    u4  2 1  and u u˜1 =  ˜ =    u2  2   u5 u13 into a direct sum of empty and 1 × 1 matrices. Now we obtain a column splitting of the remaining matrices, and with respect to the transformation that do not change u˜1 and u˜2 , these columns are linearly ordered. Therefore, we can also split them into empty and 1 × 1 blocks. Together with u˜1 and u˜2 , it splits the whole matrix u into a direct sum of matrices of the forms from the list L2 , where v, w are powers of 2. Adding 3-primary parts, we get the result. Example 3.4. Consider the idempotents e = i∈I1 eii ∈ A2 and e = e11 ∈ B. Set A1 = eAe, B1 = e B2 e Z and U1 = e U2 e. Then U1 is an A1 -B1 -bimodule; elements from El(U1 ) can be identified with those from El(U2 ) having no second column and fifth row. Hence we get the following result. Corollary 3.5. A complete list L1 of non-isomorphic indecomposable objects from El(U1 ) consists of • empty objects ∅i (i ∈ I1 ); ; v ∈ V1 if i = 1, v ∈ V if i = 2, 3, v = 1 if i = 4, 6); • objects vi (i ∈I1 vi (i = 1, 2, 3, l = 4, 6; if (il) = (26) then v ∈ V2 , otherwise v ∈ V ). • objects vil = 1l Here the indices show the blocks where the corresponding elements are placed.

4

Bimodules and homotopy types

Bimodule categories arise in the following situation. Let A and B be two fully additive subcategories of the category Hos. We denote by A † B the full subcategory of Hos

428


consisting of all objects X isomorphic (in Hos) to the cones of morphisms f : A → B with A ∈ A, B ∈ B, or, the same, such that there is a cofibration sequence f

g

h

A −→ B −→ X −→ ΣA,

(5)

where A ∈ A, B ∈ B. Consider the A-B-bimodule H, which is the restriction on A◦ × B of the “regular” Hos-Hos-bimodule Hos. If f ∈ Hos(A, B) is an element of H, it gives rise to an exact sequence like (5) with X = Cf . Moreover, since this sequence is a cofibration one, for every morphism (α, β) : f → f , where f ∈ Hos(A , B ), there is a morphism γ : X → X , where X = Cf , such that the diagram f

g

A −−−→ B −−−→     α β

h

X −−−→  γ

Σf

ΣA −−−→ ΣB    Σβ Σα

(6)

A −−−→ B −−−→ X −−−→ ΣA −−−→ ΣB f

g

h

Σf

commutes. In what follows we suppose that the categories A and B satisfy the following condition: Hos(B, ΣA) = 0 for all A ∈ A, B ∈ B. (7) In this situation, given a morphism γ : X → X , we have that h γg = 0, hence γg = g β for some β : B → B . Moreover, since the sequence g

h

Σf

B −→ X −→ ΣA −→ ΣB is cofibration as well, and Σ : Hos(A, B) → Hos(ΣA, ΣB) is a bijection, there is a morphism α : A → A , which makes the diagram (6) commutative. Note that neither γ is uniquely determined by (α, β), nor (α, β) is uniquely restored from γ. Nevertheless, we can control this non-uniqueness. Namely, if both γ and γ fit the diagram (6) for given (α, β), their difference γ = γ − γ fits an analogous diagram with α = β = 0. The equality γg = 0 implies that γ = σh for some σ : ΣA → X , and the equality h γ = 0 implies that γ = g τ for some τ : X → B. On the contrary, if γ¯ = σσ = τ τ for some morphisms σ

σ

X −−−→ ΣY −−−→ X

τ

τ

and X −−−→ Z −−−→ X ,

where Y ∈ A, Z ∈ B, the condition (7) implies that γg = h γ = 0, so γ fits the diagram (6) with α = β = 0. Fix now γ, and let both (α, β) and (α , β ) fit (6) for this choice of γ. Then the pair (α, β), where α = α − α , β = β − β , fits (6) for γ = 0. The equality g β = 0 implies that β = f σ for some σ : B → A , and the equality (Σα)h = 0 implies that Σα = Στ Σf , or α = τ f for some τ : B → A . On the contrary, if (α, β) : f → f is such that β = f σ and α = τ f with σ, τ : B → A , then g β = (Σα)h = 0, hence this pair fits (6) with γ = 0. Summarizing these considerations, we get the following statement.


429

Theorem 4.1. Let A, B be fully additive subcategories of Hos satisfying the condition (7), A † B be the full subcategory of Hos consisting of all spaces such that there is a cofibration (5) with A ∈ A, B ∈ B. Denote by H the bimodule Hos considered as A-Bbimodule, by I the ideal in A† B consisting of all morphisms γ : X → X that factor both through an object from ΣA and through an object from B, and by J the ideal in El(H) consisting of all morphisms (α, β) : f → f such that β factors through f and α factors through f . Then the factor categories El(H)/J and A† B/I are equivalent; an equivalence is induced by the maps f → Cf and (α, β) → γ, where γ fits a commutative diagram (6). Moreover, I 2 = 0, thus the functor A† B → A† B/I reflects isomorphisms. τ

Proof. We only have to check the last statement. But if γ : X → X factors as X −→ h

g

σ

B −→ X and γ : X → X factors as X −→ ΣA −→ X , where A ∈ A, B ∈ B, then γ γ = 0, since h g : B → ΣA and Hos(B, ΣA) = 0. Corollary 4.2. In the situation of Theorem 4.1, suppose that Hos(B, A) = 0 for each A ∈ A, B ∈ B. Then El(H) A † B/I. Moreover, the functor A † B → El(H) is a representation equivalence, i.e. it is dense, preserves indecomposable and reflects isomorphisms. ∼

Note also that any isomorphism f : A −→ B is a zero object in El(H)/J , since its identity map (1A , 1B ) can be presented as (f −1 f, f f −1 ). Obviously, the corresponding object from A† B is zero (i.e. contractible) too.

5

Small dimensions

We now use Theorem 4.1 to describe stable homotopy types of atoms of dimensions at most 5, or, the same, indecomposable objects in the categories CW12 and CW23 . Example 5.1. It is well known that πn (S n ) = Z (freely generated by the identity map). It allows easily to describe atoms in CW12 . Such an atom X is (stably!) of the form Cf for some map f : mS 2 → nS 2 . Since Hos(S n , S n+1 ) = 0, Theorem 4.1 can be applied. The map f is given by an integer matrix. Using automorphisms of mS 2 and nS 2 , we can transform it to a diagonal form. Hence, indecomposable gluings can only be if m = n = 1; thus f = q1S 2 . One can see that such a gluing is indecomposable if and only if q is a power of a prime number. The corresponding atom S 2 ∪q B 3 will be denoted by M (q) and called Moore atom. It occurs in a cofibration sequence q

g(q)

h(q)

q

S 2 −→ S 2 −−→ M (q) −−→ S 3 −→ S 3 . For the next section we need more information about 2-primary Moore atoms. We denote Mt = M (2t ) and write gt , ht instead of g(2t ), h(2t ). These atoms can be included

430


into the following commutative “octahedral” diagram [16], where t = r + s: gr

9 S2

/ Mr

LLLhr LLL s s s % s 2 ktr 2 S KKK 9 S3 r r r KKK rrrht % 2t / 2 M S LLL t gt L rrr gs LL% xrrrkst 2r sss

(8)

Ms

Moreover, in this diagram hs kst = 2r ht . The exact sequence (3) is here of the form q

q

πkS (S 2 ) −→ πkS (S 2 ) −→ πkS (M (q)) −→ πkS (S 3 ) −→ πkS (S 3 ), which gives the values of stable homotopy groups of the spaces M (q) shown in Table 1 below. (By the way, this table implies that all Moore atoms are pairwise non-isomorphic.) k

2

3

4

πkS (M (q)), q odd

Z/q

0

0

πkS (Mt ), t > 1

Z/q

Z/2

Z/2 ⊕ Z/2

πkS (M1 )

Z/2

Z/2

Z/4

Table 1

Actually, the only non-trivial case is the group π4S (M1 ). It can be obtained as π6 (Σ2 M1 ), which is isomorphic to the 2-primary component of π6 (S 3 ) = Z/12 (cf. [18, Lemma XI.10.2]). To prove that the sequence 0 −→ π4S (S 2 ) = Z/2 −→ π4S (Mt ) −→ π4S (S 3 ) = Z/2 −→ 0 splits if t > 1, it is enough to consider the commutative diagram 0

/ π S (S 2 ) 4

0

0

/ π S (S 2 ) 4

/ π S (M ) 1 4

/ π S (M ) t 4

/ π S (S ) 3 4

/0

(9) / π S (S ) 3 4

/0

arising from the diagram (8) with r = 1. It shows that the second row of this diagram is the pushdown of the first one along the zero map; thus it splits. Example 5.2. Now we are able to describe atoms in CW23 . They are cones Cf for some f : mS 4 → Y with 2-connected Y of dimension 4. Again Hos(Y, S 5 ) = 0, so Theorem 4.1 can be applied. Example 5.1 shows that Y is a bouquet of spheres S 3 , S 4 and suspended Moore atoms ΣM (q). Note that π4S (Y ) = π4 (Y ) for every Y ; in particular π4 (S 4 ) = Z, π4 (S 3 ) = Z/2 (generated by the suspended Hopf map η1 = Ση; η : S 3 → S 2 CP1


431

which is given by the rule η(a, b) = (a : b), where (a, b) ∈ C2 are such that |a|2 + |b|2 = 1) and Z/2 if q = 2r π4 (ΣM (q)) = π3S (M (q)) = 0 otherwise. The Hopf map η2 = Σ2 η : S 4 → S 3 and the inclusion j : S 2 → M (q) give rise to an epimorphism η∗ : π4 (S4 ) → π4 (S 3 ) and to an isomorphism j∗ : π4 (S 3 ) → π4 (ΣMr ), where Mr = M (2r ). Moreover, if t > r, there is a map M (2r ) → M (2t ) that induces an 4 isomorphism π4 (Mr ) → π4 (Mt ). If Y = s4 S 4 ∨ s3 S 3 ∨ ( ∞ r=1 mr Mr ), a map f : mS → Y can be given by a matrix of the form F4 F3 G1 G2 . . . , where Fi is of size m × si with entries from π4 (S i ); Gr is of size m × mr with entries from π4 (ΣMr ) (some of these matrices can be “empty,” containing no columns). Using automorphisms of Y and B, one can easily transform this matrix to the shape where there is at most two non-zero elements in every row (if two, one of them necessarily in the matrix F4 and even) and at most one non-zero element in every column, as shown below: F4

F3

Gr

q η 2t

η η

2t

η

Thus X decomposes into a bouquet of the spaces Σ2 M (q) (which are not atoms, but suspended atoms), spheres and the spaces C(η), C(η2t ), C(2r η) and C(2r η2t ), which are gluings of the following forms: 5 4 3

•

• •

•

•

C(η)

C(η2t )

• •

•

C(2r η)

• • •

•

C(2r η2t )

Here, following Baues, we denote the cells by bullets and the attaching maps by lines; the word in brackets shows which maps are chosen to attach bigger cells to smaller ones. We do not show the fixed point, which coincide here with X 2 (since X is 2-connected); thus the lowest bullets actually describe spheres, not balls. These polyhedra are called Chang atoms. Again one can check that all of them are pairwise non-isomorphic.

432


Thus we have proved the following classical result. Theorem 5.3 (Whitehead [23], Chang [9]). The atoms of dimension at most 5 are: • sphere S 1 (of dimension 1); • Moore atoms M (q), where q = pr , p is a prime number (of dimension 3); • Chang atoms C(η), C(η2r ), C(2r η2t ), C(2r η) (of dimension 5). In what follows, we often use suspended Moore and Chang atoms. We shall denote them by the same symbols but indicating the dimension. Thus M d (q) = Σd−3 M (q) and C d (w) = Σd−5 C(w) for w ∈ { η, 2r η, η2r , 2r η2t }; in particular, M (q) = M 3 (q) and C(w) = C 5 (w). The same agreement will also be used for other atoms constructed below.

6

Dimension 7

We shall now consider the category CW3 . Its objects actually come from CW34 , so we have to classify atoms of dimension 7. Such an atom X is 3-connected, so we may suppose that X 3 = ∗. Set B = X 5 , then X/B only has cells of dimensions 6 and 7. Therefore X ∈ Σ3 CW1 † Σ2 CW1 ΣCW1 † CW1 . Consider the bifunctor W(A, B) = Hos(ΣA, B) restricted to the category CW1 . Since, obviously, Hos(B, Σ2 A) = 0 for A, B ∈ CW1 , we can apply Theorem 4.1. So we first classify indecomposable elements of the bimodule category El(W). Indecomposable objects of the category CW1 are spheres S 2 , S 3 and Moore atoms M (q) (q = pr , r prime). If q is odd, one easily sees that W(A, M (q)) = 0 for all A, so we may only consider the spaces Mr = M (2r ). From the cofibration sequence gr

h

r S 2 −→ S 2 → Mr −→ S3 → S3

and the diagram (5), we get the values of the Hos-groups shown in Table 2. S2

S3

M1

Mr (r > 1)

S2

Z

Z/2

Z/2

Z/2

S3

0

Z

Z/2

Z/2r

M1

Z/2

Z/2

Z/4

T1r

Mt (t > 1)

Z/2t

Z/2

T1t

Ttr

Table 2

ab Here Ttr denotes the set of matrices with a ∈ 2m Z/2m , b ∈ Z/2, where m = 0a min(r, t). The equality Hos(M1 , M1 ) = Z/4 follows from the fact that this ring acts on π4S (M1 ) = Z/4, so 2 Hos(M1 , M1 ) = 0. The diagram (8) implies that the sequence 2r

0 −→ Hos(S 3 , Mt ) −→ Hos(Mr , Mt ) −→ Ker{Hos(S 2 , Mt ) −→ Hos(S 2 , Mt )} −→ 0


433

splits if min(r, t) > 1. The generator of the subgroup of diagonal matrices in Ttr is ktr , 01 while the matrix corresponds to the morphism gt ηhr . 00 Analogous calculations, using Table 1 of the preceding section and the diagram (8), produce the following table for the values of the functor Hos(ΣA, B): S2

S3

M1

Mr (r > 1)

S2

Z/2

Z/2

Z/4

Z/2 ⊕ Z/2

S3

Z

Z/2

Z/2

Z/2

M1

Z/2

Z/4

Z/2 ⊕ Z/2

Z/2 ⊕ Z/4

Mt (t > 1)

Z/2

Z/2 ⊕ Z/2

Z/4 ⊕ Z/2

Z/2 ⊕ Z/2 ⊕ Z/2

Table 3

It is convenient to organize this result in the form of Table 4 below, as in [5]. 1⊗

2⊗

3⊗

⊗1

Z/2

Z/2

Z/2 . . . Z/2

⊗2

Z/4

Z/2

Z/2 . . . Z/2

...

∞⊗

⊗3 Z/4 Z/2 Z/2 . . . Z/2 .. . ........................... ⊗∞

Z/4

Z/2

∞∗

0

0

.. .

∗∞ . . .

∗3

∗2

∗1

...

0

0

0

Z/2 . . . Z/2

Z/2

0

Z/2 . . . Z/2

Z/2

0

0

..........................

Z/2 . . . Z/2

Z/2 . . . Z/2

Z/2

...

Z/2 . . . Z/2

Z/2 Z/2

0

Z

...........................

0

..........................

3∗

0

0

0

...

0

Z/2 . . . Z/2

Z/2 Z/2

2∗

0

0

0

...

0

Z/2 . . . Z/2

Z/2 Z/2

1∗

0

0

0

...

0

Z/4 . . . Z/4

Z/4 Z/2

Table 4

In this table the row marked by ⊗t (respectively, t ∗) shows the part of the group Hos(ΣMr , Mt ) that comes from Hos(ΣMr , S 2 ) (respectively, from Hos(ΣMr , S 3 ) ). In the same way, the column marked by r ⊗ (respectively, ∗r ) shows the part of this group that comes from Hos(S 3 , Mt ) (respectively, from Hos(S 4 , Mt ) ). The columns ∞ ⊗ and ∗∞ correspond, respectively, to Hos(S 4 , ) and Hos(S 3 , ); the rows ⊗∞ and ∞ ∗ correspond, respectively, to Hos( , S 2 ) and Hos( , S 3 ). Therefore we consider the elements from El(W) as block matrices (Wyx ), where x ∈ { r ⊗, ∗r }, y ∈ { ⊗t ,t ∗ } and the block Wyx is with entries from the corresponding cell of Table 4. Moreover, morphisms between Moore spaces induce the following transforma-

434


tions of vertical stripes W x and horizontal stripes Wy of such a matrix, which we call admissible transformation: r r r r (a) replacing the stripes M ⊗ and M ∗ by M ⊗ X and M ∗ X; (a ) replacing the stripes M⊗t and Mt ∗ by XM⊗t and XMt ∗ ; r r r s (b) replacing M ∗ by M ∗ + M ∗ X + M ⊗ Y , where r > r, s arbitrary; (b ) replacing M⊗t by M⊗t + XM⊗t + Y Ms ∗ , where t > t, s arbitrary; r r r r r r (c) replacing M ⊗ and M ∗ by M ⊗ + M ⊗ X and M ∗ + 2r−r M ∗ X, where r < r; (c ) replacing Mt ∗ and M⊗t by Mt ∗ + XMt ∗ and M⊗t + 2t−t XM⊗t , where t < t; 1 1 r s (d) replacing M ⊗ by M ⊗ + 2M ⊗ X + 2M ∗ Y ; r, s arbitrary; (d ) replacing M1 ∗ by M1 ∗ + 2XMt ∗ + 2Y M⊗s ; r, s arbitrary; s r r (e) replacing M1∗∗ by M1∗∗ + 2M⊗⊗1 X; s arbitrary; 1 1 1 (e ) replacing M⊗⊗t by M⊗⊗t + 2XMs∗∗ ; s arbitrary. Here X, Y denote arbitrary integer matrices of the appropriate size; in the transformations of types (a) and (a ) the matrix X must be invertible. Two matrices W, W are isomorphic in El(W) if and only if W can be transformed to W using admissible transformations. ∞ It is convenient first to reduce the block W∞ ∗⊗ to a diagonal form D = diag(a1 , a2 , . . . , am ) ∞k with a1 |a2 | . . . |am . Let ak = 2dk bk with odd bk . Denote by W ⊗ and W∞k ∗ the parts of ∞ the stripes W ⊗ and W∞ ∗ corresponding to the columns and rows with dk = d (k = ∞ if dk = 0). Since all other matrices of these stripes are with entries from Z/2, we can ∞0 make the parts W ⊗ and W∞0 ∗ zero. Moreover, using admissible transformations that ∞k ∞k ∞l do not change the block D, we can replace W ⊗ by W ⊗ + W ⊗ X and W∞k ∗ by W∞k ∗ + Y W∞l ∗ for any l < k. In what follows we always suppose that W is already in this form. Call two matrices of this form W, W 2-equivalent, if there is a matrix W W such that W ≡ W mod 2. One can easily see that the problem of 2-equivalence of matrices from El(W) is actually a sort of bunch of chains in the sense of [8, 12]. We use the paper [12] as the source for the further discussion. Namely, we have the chain E = { ⊗t ,t ∗, ∞k ∗ } for the rows and the chain F = r ⊗, ∗r , ∞k ⊗ for the columns, where 1∗ 1

< 2 ∗ < 3 ∗ < · · · < ∞∞ < · · · < ∞3 ∗ < ∞2 ∗ < ∞1 ∗ < ⊗∞ < · · · < ⊗3 < ⊗2 < ⊗1 ,

⊗ < 2 ⊗ < 3 ⊗ < · · · < ∞∞ ⊗ < . . . ∞3 ⊗ < ∞2 ⊗ < ∞1 ⊗ < ∗∞ < · · · < ∗3 < ∗2 < ∗1 .

The equivalence relation ∼ on X = E ∪ F is given by the rule ⊗t ∼ t ∗ (t = ∞), r ⊗ ∼ ∗r (r = ∞),

∞k

⊗ ∼ ∞k ∗

for all possible values of t, r and k = ∞. Thus we can get a classification of our matrices up to 2-equivalence from [12]. Namely, we write x − y if either x ∈ E, y ∈ F or vice versa, at least one of them belongs to { ⊗t } ∪ { ∗r }, moreover, { x, y } = { ⊗t , ∗1 } and { x, y } = { ⊗1 , ∗r }. We call an X -word a sequence w = x1 ρ2 x2 ρ3 . . . ρn xn , where xi ∈ X , ρi ∈ { ∼, − }, ρi = ρi+1 (i = 2, . . . , n − 1) and xi−1 ρi xi holds in X for all i = 2, . . . , n. Such a word is called full if the following conditions hold: • either ρ2 =∼ or x1 ∼ y for all y ∈ X , y = x1 ; • either ρn =∼ or xn ∼ y for all y ∈ X , y = xn .


435

w is called a cycle if ρ2 = ρn = − and xn ∼ x1 in X . If, moreover, w cannot be written in the form v ∼ v ∼ · · · ∼ v for a shorter word v, it is called aperiodic. We call a polynomial f (t) ∈ Z/2[t] primitive if it is a power of an irreducible polynomial with the leading coefficient 1. We shall identify any word w with its inverse and any cycle w with any of its cyclic shifts. Then the set of indecomposable representations of this bunch of chains is in 1-1 correspondence with the set S ∪ B, where S is the set of full words (up to inversion) and B is the set of pairs (w, f ), where w is an aperiodic cycle (up to a cyclic shift) and f = td is a primitive polynomial. We call representations corresponding to S strings and those corresponding to B bands. Note that an X -word can contain at most one element ∞k ⊗, at most one element ∗∞k and at most one subword of the form ⊗t − ∗r or its inverse. Replacing w by its inverse, we shall suppose that there are no words of the form ∗r − ⊗t or ∞k ⊗ ∼∞k ∗. It is convenient to rewrite this answer in a modified form. Namely, we replace the subword ∞k ∗ ∼∞k ⊗, if it occurs, by k εk , also omit x1 if ρ2 =∼, omit xn if ρn =∼ and omit all remaining symbols ∼. Then we replace every subword r ⊗ −⊗t by r ⊗t , ⊗t −r ⊗ by t ⊗r , t ∗ −∗r by r r r r r t ∗ , ∗ −t ∗ by ∗t and ⊗t − ∗ by t θ . Note that in the last case r = 1 and t = 1. We also omit all signs ∼, replace any double superscript rr by r and any double subscript tt by t . Certainly, the original word can be easily restored from such a shortened form. Now, any full word or its inverse can be written as a subword of one of the following words: r1

⊗t1 ∗r2 ⊗t2 ∗ . . .rn ⊗tn

t−m t−m

⊗ ...

r−2

⊗ ...

r−2

r−1

∗t−2 ⊗

r−1

∗t−2 ⊗

(“usual word”),

∗t−1 θr1 ⊗t1 ∗r2 ⊗t2 · · · ∗rn k

r2

rn

∗k ε ⊗t1 ∗ ⊗t2 · · · ∗

(“theta-word”), (“epsilon-word”),

Moreover, • ∞ can only occur at the ends of a word, not in a theta-word or epsilon-word. • In any theta-word t−1 = 1 and r1 = 1. Any cycle or its shift can be written as r1

⊗t1 ∗r2 ⊗t2 ∗ . . .rn ⊗tn ∗r1 .

The description of the representations in [12] also implies the following properties. Proposition 6.1. (1) Any row (column) of a string contains at most 1 non-zero element. (2) There are at most 2 zero rows or columns in a string, namely, they are in the following stripes: (a) Mt ∗ if w has an end ⊗t (or t ⊗), t = ∞; r (b) M ∗ if w has an end r ⊗, r = ∞; (c) M⊗t if w has an end t ∗; r (d) M ⊗ if w has an end ∗r (or r ∗); (e) M∞k ∗ if the left end of w is k εk ; ∞k (f) M ⊗ if the right end of w is k εk . We call each end occurring in this list a distinguished end.

436


(3) The horizontal and vertical stripes of a band can be subdivided in such a way that every new horizontal or vertical band has exactly 1 non-zero block, which is invertible. Recall that elements modulo 4 only occur in the stripes W

1⊗

and W1 ⊗ .

∞

Corollary 6.2. Let W ∈ El(W) (with diagonal W∞ ∗⊗ ). Denote by W its reduction the matrix obtained from W by replacing all invertible entries with modulo 2 and by W are even). Suppose that W = m W i , where all W i are strings 0 (thus all entries of W i=1 or bands. Then W W , where W = W and the only non-zero rows and columns of W can be those corresponding to the distinguished ends of types 2(a-d) of Proposition 6.1. In particular, if some of W i is a band, a theta-string or an epsilon-string, W has a direct i = 0. summand Wi such that W i ≡ Wi mod 2 and W Thus we only have now to consider the case, when W = W and every W i is a usual string. Suppose that Wi corresponds to a string wi . It is easy to verify that if wi and wj have a common distinguished end, there is a sequence of distinguished transformations, which does not change W and adds the row (or column) corresponding to this end in i to the row (or column) corresponding to this end in W j or vice versa. Hence, such W rows (columns) are in some sense linearly ordered. As a consequence, we can transform to a matrix having at most one non-zero element in every row and every column W (without changing W ). It gives us the following description of indecomposable matrices = 0. from El(W) with W Corollary 6.3. Suppose that W is an indecomposable matrix from El(W), such that = 0 for every matrix W W . Let W = m W i , where each W i is a usual string. W i=1 There are, up to isomorphism, the following possibilities: has a unique non-zero element in the (1) m = 1, W corresponds to a word w and W block Wba for the following choices: w = t1 ∗r2 ⊗t2 . . . , a = 1 ⊗, b = ⊗t1 (t1 == 1); r1

r2

r1

w = ⊗t1 ∗ ⊗ . . . , a = ∗ , b = 1 ∗ (r1 = 1); , r1

r2

r

w = 1 ∗ ⊗t1 ∗ . . . , a = ∗ , b = 1 ∗; , 1

r2

1

w = ⊗t1 ∗ ⊗t2 . . . , a = ⊗, b = ⊗t ; .

(a) (b) (c) (d)

has a unique non-zero (2) m = 2, W i (i = 1, 2) correspond to the words wi and W a element in the block Wb , where w1 = 1 ∗r−1 ⊗t−1 ∗r−2 ⊗ . . . , w2 = r1 ⊗t1 ∗r2 ⊗t2 . . . , a = ∗r1 , b = 1 ∗,

(e)

w1 = 1 ⊗t1 ∗r1 ⊗t2 ∗ . . . , w2 = t−1 ∗r−1 ⊗t−2 ∗r−2 ⊗ . . . , a = 1 ⊗, b = ⊗t−1 .

(f)


437

We encode these matrices by the following words w: w = . . . ⊗r2 ∗t2 ⊗r1 ∗t1 θ1 r1

in case (a),

r2

w = 1 θ ⊗t1 ∗ ⊗t2 ∗ . . . r2

r1

r2

r3

in case (b),

w = · · · ∗t2 ⊗ ∗t1 ⊗ ∗1 θ 1

r

w = t θ ⊗t1 ∗ ∗t2 ⊗ ∗ . . . r−2

w = ··· ∗

r−1

⊗t−2 ∗ r−1

w = · · · ∗ t−2 ⊗

in case (c), in case (d),

r1

∗1 θ ⊗t1 ∗r2 ⊗ . . . 1

r2

∗t−1 θ ⊗t1 ∗ ⊗t2 ∗ . . .

in case (e), in case (f),

We call these words “theta-words” as well. Obviously, cases (a-d) always give indecomposable matrices. On the other hand, one can check that in case (e) W is indecomposable if and only if (r−1 + 1, t1 , r−2 , t2 , . . . ) < (r1 , t−2 , r2 , t−3 , . . . ) with respect to the lexicographical order [5]. In case (f) W is indecomposable if and only if (t1 + 1, r−1 , t2 , r−2 , . . . ) < (t−1 , r2 , t−2 , r3 , . . . ) lexicographically. Thus we obtain a complete list of non-isomorphic indecomposable matrices from El(W). Moreover, it is easy to verify that they remain pairwise non-isomorphic and indecomposable in El(W)/I as well. Thus, using Theorem 4.1, we get the following result. Theorem 6.4 (Baues–Hennes [7]). Indecomposable polyhedra from CW34 are in 1-1 correspondence with usual words, theta-words, epsilon-words and bands defined above, with the only restriction that in a theta-word w = . . .r−2 ∗t−2 ⊗r−1 ∗t−1 θr1 ⊗t1 ∗r2 ⊗t2 . . . the following conditions hold: if t−1 = 1, then (r−1 + 1, t1 , r−2 , t2 , . . . ) < (r1 , t−2 , r2 , t−3 , . . . ) lexicographically, if r1 = 1, then (t1 + 1, r−1 , t2 , r−2 , . . . ) < (t−1 , r2 , t−2 , r3 , . . . ) lexicographically. The gluings of spheres corresponding to these words can be described as follows: 7 6 5 4

• • • •88 •88 •GGG 8 8 GG 88 88 GG ··· 88 88 GG GG 8 8 • • • GG 88 88 GG 88 88 ··· GG 8 8 G

•

•

for a usual word

•

438


7 6 5

•888

···

•888

• • 88 88 8 G 88 • • GGG •88 8 GG 88 88 GG 88 8 G 8 • • 8 8 88 ··· 8

88 88 88 ww• w 88 w w 88 ww w w •

•

4

•

•

for a theta-word 7 6 5 4

···

•888

• • 88 88 G GG 88 • • •888 • GGG GG 8 GG 88 GG 88 GG GG 88 8 G 88 • • • • 88 ··· 88

•

•

•

for an epsilon-word In these diagrams vertical segments present the suspended atoms Mr , slanted lines correspond to the gluings arising from Hopf maps S d+1 → S d , while the long slanted line in a theta-word shows the gluing arising from the doubled Hopf map S 6 → S 4 . Note that all atoms from CW34 are p-primary (2-primary, except M (q) with odd q). Therefore, we have the uniqueness of decomposition of spaces from CW3 into bouquets of suspended atoms.

7

Bigger dimensions. Wildness

Unfortunately, if we pass to bigger dimensions, the calculations as above become extremely complicated. In the representations theory the arising problems are usually called “wild.” Non-formally it means that the classification problem for a given category contains the classification of representations of arbitrary (finitely generated) algebras over a field. It is well-known, since at least 1969 [15], that it is enough to show that this problem contains the classification of pairs of linear mappings (up to simultaneous conjugacy), or, equivalently, the classification of triples of linear mappings V1

/

/ V2 /

(10)

On the other hand, problems like the one considered in the preceding section, where indecomposable objects can be parameterised by several “discrete,” or combinatorial parameters (as X -words above) and at most one “continuous” parameter (as a primitive polynomial in the description of bands), are called “tame.” The problems, where the answer is purely combinatorial, like the classification of atoms of dimensions d ≤ 5, are called “finite.” I shall not precise these notions formally. The reader can consult, for


439

instance, the survey [13], where it is done within the framework of representation theory. An important question in the representation theory is to distinguish finite, tame and wild cases. The following result accomplishes such an investigation for stable homotopy types. Proposition 7.1 (Baues [5]). The classification problem for the category CW4 is wild. Proof. Let B be the category of bouquets of Moore atoms M = M1 , A = Σ2 B. Then CW4 contains the subcategory Σ3 (A† B) A† B. Corollary 4.2 shows that the category A† B is representation equivalent to El(H), where H is the restriction of Hos onto A◦ × B. We know that Hos(M, M ) = Z/4. Therefore, we only have to show that Hos(Σ2 M, M ) Z/2 ⊕ Z/2 ⊕ Z/2. Indeed, it implies the category El(H) is representation equivalent to the category of diagrams of the shape (10). 2 2 The cofibration sequence S 2 −→ S 2 → M → S 3 −→ S 3 and the Hopf map η : S 5 → S 4 produce the following commutative diagram: 0 −−−→ Z/2 −−−→ π4S (M ) −−−→     η∗

Z/2 −−−→ 0  

0 −−−→ Z/2 −−−→ π5S (M ) −−−→ Z/2 −−−→ 0, Since η 3 = 4ν, where ν is the element of order 8 in π5S (S 2 ) = Z/24 [21], actually η ∗ = 0, so the lower row splits and π5S (M ) = Z/2 ⊕ Z/2. Just in the same way we show that Hos(Σ2 M, S 2 ) = Z/2 ⊕ Z/2. Now, applying the functors Hos( , S 2 ) and Hos( , M ) to the same cofibration sequence, we get the commutative diagram 0 −−−→

Z/2  

−−−→ Hos(Σ2 M, S 2 ) −−−→  

Z/2 −−−→ 0  

0 −−−→ Z/2 ⊕ Z/2 −−−→ Hos(Σ2 M, M ) −−−→ Z/2 −−−→ 0. Since the upper row of this diagram splits, the lower one splits as well, hence Hos(Σ2 M, M ) = Z/2 ⊕ Z/2 ⊕ Z/2. It accomplishes the proof. We can summarize the obtained results in the following theorem. Theorem 7.2. The category CWk is of finite type for k ≤ 2, tame for k = 3 and wild for k ≥ 4.

8

Torsion free atoms. Dimension 9

Nevertheless, if we consider torsion free atoms, the situation becomes much simpler. Namely, in this case neither sphere of dimension d can be attached to the spheres of dimension d − 1, thus in the picture describing the gluing of spheres there is no fragments of the sort • d d−1 •

440


Therefore, a calculation of atoms from CWFkk+1 can be organized as follows. Denote by Bk the full subcategory of CW consisting of bouquets of torsion free suspended atoms of dimension 2k and by Sk the category of bouquets of spheres S 2k . Let Γm (X) denote the S S S subgroup Im{πm (X m−1 ) → πm (X)} of πm (X). When X runs through Bk , Γ2k can be considered as an Sk -Bk -bimodule; we denote this bimodule by Gk . Then the following analogue of Theorem 4.1 holds (with essentially the same proof). Proposition 8.1. Denote by I the ideal of the category CWFkk+1 consisting of all morphisms X → X that factor through an object from Bk , and by J the ideal of the category El(Gk ) consisting of such morphisms (α, β) : f → f that α factors through f and β factors through f . Then CWFkk+1 /I El(Γk )/J . Moreover, both I 2 = 0 and J 2 = 0, hence the categories CWFkk+1 and El(Gk ) are representation equivalent. Proof. The only new claim here is that J 2 = 0. But this equality immediately follows from the fact that if a morphism X → S m factors through X m−1 , it is zero. Thus a torsion free atom of dimension 7 can be obtained as a cone of a map f : mS 6 → Y , where Y is a bouquet of spheres S 4 , S 5 and suspended Chang atoms C 6 (η), while f ∈ Γ6 (Y ). Easy calculations, like above, give the following values of Γ6 :

X

S4

S5

Γ6

Z/2 Z/2

C 6 (η) 0

(The last 0 is due to the fact that the map η∗ : π6 (S 5 ) → π6 (S 4 ) is an epimorphism [21]). The Hopf map η : S 5 → S 4 induces an isomorphism Γ6 (S 5 ) → Γ6 (S4 ). Therefore, the only indecomposable torsion free atom of dimension 7 is the gluing C(η 2 ) = C 7 (η 2 ) = S 4 ∪η2 B 7 . (Note that such an atom must contain at least one 4-dimensional cell.) Moreover, all torsion free atoms of dimensions d ≤ 7 are 2-primary. A torsion free atom of dimension 9 is a cone of some map f : mS 8 → Y , where Y is a bouquet of spheres S i (5 ≤ i ≤ 7), suspended Chang atoms C 7 (η), C 8 (η) and suspended atoms C 8 (η 2 ). One can calculate the following table of the groups Γ8 for these spaces: X Γ8

S5

S6

S7

Z/24 Z/2 Z/2

C 7 (η) C 8 (η) C 8 (η 2 ) Z/12

0

Z/12

Morphisms between these spaces induce epimorphims Γ8 (S 5 ) → Γ8 (C 7 (η 2 )) → Γ8 (C 7 (η)), Γ8 (S 7 ) → Γ8 (S6 ) and monomorphisms Γ8 (S 7 ) → Γ8 (C 7 (η) → Γ8 (S 5 ), Γ8 (S 6 ) → Γ8 (S 5 ). It can be deduced either from [21] or, perhaps easier, from the results of [22], cf. [3]. (The only non-trivial one is the monomorphism Γ8 (S 7 ) → Γ8 (C 7 (η)) ). Again we consider the


441

map f as a block matrix F =

F1 F2 F3 F4 F6

.

Here Fi is of size mi × m with entries from Γ8 (Yi ), where

  S5     7    C (η) Yi = C 8 (η 2 )     S6     5 S

if i = 1, if i = 2, if i = 3, if i = 4, if i = 6.

We have written F6 , not F5 , in order to match the notations of the Example 3.2; so we set I1 = { 1, 2, 3, 4, 6 }. Using the automorphisms of mS 7 and of Y , one can replace the matrix F by P F Q, where P ∈ GL(m, Z) and Q = (Qij )i,j∈I1 is an invertible integer block matrix, where the block Qij is of size mi × mj with the following restrictions for the entries a ∈ Qij :

a =0

for i ∈ { 4, 6 } , j < i,

a ≡0 mod 2 for (ij) ∈ { (12), (13), (23) } , a ≡0 mod 6 for (ij) = (26), a ≡0 mod 12 for j ∈ 4, 6, i ∈ { 1, 2, 3 } , (ij) = (26).

Thus we have come to the bimodule category El(U1 ) considered in Example 3.4, so we can use Corollary 3.5, which describes all indecomposable objects of this category. Certainly, we are not interested in the “empty” objects ∅i , since they correspond to the spaces with no 9-dimensional cells. Note also that the matrices (14 ), (16 ) correspond not to atoms, but to suspended atoms C 9 (η 2 ) and C 9 (η). We use the following notation for the atoms corresponding to other indecomposable matrices F :

442


A(v)

for

(v1 ),

A(ηv)

for

(v2 ),

A(η 2 v)

for

A(vη)

for

A(vη 2 )

for

A(ηvη)

for

(v3 ), v1 , 16 v1 , 14 v2 , 16 v2 , 14 v3 , 16 v3 . 14

A(ηvη 2 ) for A(η 2 vη) for A(η 2 vη 2 ) for So we have proved

Theorem 8.2 (Baues–Drozd [3]). Every torsion free atom of dimension 9 is isomorphic to one of the atoms A(w) with w ∈ { v, ηv, η 2 v, vη, vη 2 , ηvη, ηvη 2 , η 2 vη, η 2 vη 2 }. Using the gluing diagrams, these atoms can be described as in Table 5 below. •

9 8 7 6 5

• A(v)

9 8 7 6 5

• • •

•

A(η 2 v)

A(ηv)

•

•

•

•

•

• •

•

•

•

A(η 2 vη)

A(η 2 vη 2 )

• • • A(vη) • • •

• •

•

A(vη 2 ) •

•

•

•

•

A(ηvη)

A(ηvη 2 )

Table 5

One can also check that the 2-primary atoms in this list are those with v divisible by 3, while the only 3-primary one is A(8). Thus there are altogether 29 primary suspended atoms of dimension at most 9. The congruent ones are only A(3) and A(9). Indeed,


443

3 A(3)∨S corresponds to the matrix mod 24. But the latter can be easily transformed 0 9 to mod 24, which corresponds to A(9) ∨ S 5 : 0 3 3 −9 9 9 → → → → . 0 12 12 12 0 5

(At the last step we add the first row multiplied by 4 to the second one; all other transformations are obvious.) One can verify that all other 2-primary atoms are pairwise non-congruent. Corollary 8.3. The Grothendieck group K0 (CWF4 ) is a free abelian group of rank 29. Note that the matrix presentations allows easily to find the images in K0 (CWF4 ) of all atoms. For instance, the equivalence of matrices     81 0  11 0   ∼  0 31 0 01 implies that A(8) ∨ A(3) A(1) ∨ S 5 ∨ S 9 , thus in K0 (CWF4 ) we have [A(1)] = [A(8)] + [A(3)] − [S 5 ] − [S 9 ]. The reader can easily make analogous calculations for all atoms of Table 5.

9

Torsion free atoms. Dimension 11

For torsion free atoms of dimension 11 analogous calculations have been done in [6]. Nevertheless, they are a bit cumbersome, so we propose here another, though rather similar, approach. Namely, denote by Sk the category of bouquets of spheres S 2k−1 and S 2k , by Bk the category of bouquets of suspended atoms of dimension 2k − 1 and by Gk the Sk -Bk -bimodule such that Gk (S 2k−1 , B) = Γ2k−1 (B) and Gk (S 2k , B) = Γ2k (B). Proposition 9.1. Denote by I the ideal of the category CWFkk+1 consisting of all morphisms X → X that factors through an object from Bk , and by J the ideal of the category El(Gk ) consisting of such morphisms (α, β) : f → f that α factors through f and β factors through f . Then CWFkk+1 /I El(Γk )/J . Moreover, both (I )2 = 0 and (J )2 = 0, hence the categories CWFkk+1 and El(Γk ) are representation equivalent. Thus we obtain torsion free atoms if dimension 11 as cones of maps S → Y , where S is a bouquet of spheres of dimensions 9 and 10, while Y is a bouquet of 5-connected

444


suspended atoms of dimensions 6 ≤ d ≤ 9. Note that at least one of these atoms must have a cell of dimension 6 in order that such a cone be an atom. Just as above, we have the following values of Γ9 and Γ10 for such atoms:

X

S6

C 8 (η) C 9 (η 2 )

S7

C 9 (η)

S8

S9

Γ9

Z/24

Z/12

Z/12

Z/2

0

Z/2

0

Γ10

0

0

0

Z/24 Z/12

Z/2 Z/2

(We have arranged this table taking into account the known maps between these groups, as above.) The Hopf map S 10 → S 9 induces monomorphisms in the 4th and the 6th columns of this table, while the maps between suspended atoms induce homomorphisms analogous to those of the preceding section. Thus a morphism f : S → Y can be described by a matrix   F1 F2 F3 F4 0 F6 0  F =  , 0 0 0 G4 G5 G6 G7 where the matrix Fi (Gi ) has entries from the first row (respectively, second row) and the i-th column of the table above. Two matrices, F and F , define homotopic polyhedra if F = P F Q, where P, Q are matrices over the tiled orders, respectively,                   

 1 2 2 12 24 12 24 1 1 1 12 24 6 1 2 1 12 24 12 0 0 0 1 2 12∗ 0 0 0 1 1 12 000 0 0

1

000 0 0

0

  24    24     12    6     1  1



 ∗

and

 1 12  .  0 1

Here 12∗ shows that the corresponding element obeys the ∗-rule (4), i.e. induces a nonzero map Z/2 → Z/2 and acts as usual multiplication by 12 in all other cases. Thus we have obtained the bimodule category El(U2 ) from Example 3.2, so we can use the list of indecomposable objects from Theorem 3.3. Moreover, we only have to consider the matrices having non-empty G-column and one of the parts F1 , F2 , F3 (otherwise we have no 11-dimensional or no 6-dimensional cells). Therefore, a complete list of atoms


arises from the following matrices: 



 vi 0  ,  14 w4









 vi 0    1 w  ,  4 4   0 16

445

 vi 0    1 w  ,  4 4   0 17

where i ∈ { 1, 2, 3 }, v, w ∈ { 1, 2, 3, 4, 5, 6 }. We omit the upper indices of Theorem 3.3, since here they coincide with the column number; the lower indices show to which horizontal stripe of the matrix F the corresponding elements belong. It gives the following list of 11-dimensional torsion free atoms. Theorem 9.2 (Baues–Drozd [6]). Every torsion free atom of dimension 11 is isomorphic to one of the atoms of Table 6 below.

11 10 9 8 7 6

•

•

• •

A(vη 2 w)) 11 10 9 8 7 6

•

•

• • • •

A(ηvη 2 wη)

• • • • •

A(vη 2 wη 2 )

•

•

• •

•

• A(η 2 vη 2 wη 2 )

• • • • •

A(η 2 vη 2 w)

•

•

• • •

A(vη 2 wη)

• • • • • •

A(η 2 vη 2 wη)

•

•

• • •

A(ηvη 2 w)

• • • • • •

A(ηvη 2 wη 2 )

Table 6

Again 2-primary atoms are those with v, w ∈ { 3, 6 } and there are no 3-primary spaces in this table. Moreover, the new 2-primary atoms are pairwise non-congruent, therefrom we obtain the following result. Corollary 9.3. The Grothendieck group K0 (CWF5 ) is a free abelian group of rank 85. We end up with the following statements about the higher dimensional torsion free spaces. Proposition 9.4. (1) There are infinitely many non-isomorphic (even non-congruent) 2-primary atoms of dimension 13. Hence the Grothendieck group K0 (CWFk ) is of infinite rank for k ≥ 6.

446


(2) If k ≥ 11, the classification problem for the category CWFk is wild. S S (A11 (η 2 v), or, the same, π10 (A(η 2 v) equals Z/2 ⊕ Z/2. Proof. We shall show first that π12 We consider the cofibration sequences f

g

Σf

h

S 8 −→ ΣC −→ A −→ S 9 −→ Σ2 C,

(a)

S 6 −→ S 4 −→ C −→ S 7 −→ S 5 ,

(b)

where A = A(η 2 v), C = C(η 2 ). Note that the map f factors through S 5 . From the sequence (b) we get π9S (C) π9S (S 7 ) Z/2 and π9S (ΣC) π8S (C) = 0. The second equality follows from the fact that the induced map π8S (S 7 ) → π8S (S 5 ) is known to be S S injective [21]. Since π10 (S 5 ) = π10 (S 6 ) = 0, the sequence (a) gives then an exact sequence S S S (ΣC) Z/2 −→ π10 (A) −→ π10 (S 9 ) Z/2 −→ 0. 0 −→ π10 S To show that this sequence splits, we have to check that 2α = 0 for every α ∈ π10 (A). In any case, 2α factors through ΣC, which gives rise to a commutative diagram φ

M 10 (2) −−−→   γ S8

2

S 10 −−−→  β

S 10  α

−−−→ ΣC −−−→ A f

g

S for some β, γ (we have used the cofibration sequence for M (2) ). Since π9S (S 5 ) = π10 (S 5 ) = 0, also Hos(M 10 (q), S 5 ) = 0. But the map βφ = f γ factors through S 5 , so βφ = 0 and S β = 2σ for some σ ∈ π10 (ΣC) Z/2. Hence β = 0 and 2α = 0. Analogous calculations show that any endomorphism of A acts as a homothety on S π10 (A). Since, obviously, Hos(A, S 10 ) = 0, Corollary 4.2 shows that the category of spaces arising as cones of mappings mS 12 → nA11 (η 2 v) is equivalent to the category 1 , or, the same, of diagrams of Z/2-vector of representations of the Kronecker quiver A spaces of the shape V1 ⇒ V2 . But it is well-known that this quiver is of infinite type, i.e. has infinitely many non-isomorphic indecomposable representations. Obviously, all corresponding spaces are 2-primary and non-congruent, which proves the claim (1). S The claim (2) follows from the equality π20 (S 11 ) (Z/2)3 . It implies that the category of spaces, which are cones of mappings mS 20 → mS 11 , is equivalent to that of diagrams / / V2 . The latter is well-known to be wild. V1 /

Perhaps, the estimate 11 in the claim (2) of Proposition 9.4 is too big, but at the moment I do not know a better one. On the other hand, there is some evidence that the classification problem for CWF6 is still tame.

References [1] H.J. Baues: Homotopy Type and Homology, Oxford University Press, 1996.


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[2] H.J. Baues: “Atoms of Topology“, Jahresber. Dtsch. Math.-Ver., Vol. 104, (2002), pp. 147–164 . [3] H.J. Baues and Yu.A. Drozd: “The homotopy classification of (n − 1)-connected (n + 4)-dimensional polyhedra with torsion free homology“, Expo. Math., Vol. 17, (1999), pp. 161–179. [4] H.J. Baues and Yu.A. Drozd: “ Representation theory of homotopy types with at most two non-trivial homotopy groups“, Math. Proc. Cambridge Phil. Soc., Vol. 128, (2000), pp. 283–300. [5] H.J. Baues and Yu.A. Drozd: “Indecomposable homotopy types with at most two non-trivial homology groups, in: Groups of Homotopy Self-Equivalences and Related Topics“, Contemporary Mathematics, Vol. 274, (2001), pp. 39–56. [6] H.J. Baues and Yu.A. Drozd: “Classification of stable homotopy types with torsionfree homology“, Topology, Vol. 40, (2001), pp. 789–821. [7] H.J. Baues and Hennes: “The homotopy classification of (n − 1)-connected (n + 3)dimensional polyhedra, n ≥ 4“, Topology, Vol. 30, (1991), pp. 373–408. [8] V.M. Bondarenko: “Representations of bundles of semichained sets and their applications“, St. Petersburg Math. J., Vol. 3, (1992), pp. 973–996. [9] S.C. Chang: “Homotopy invariants and continuous mappings“, Proc. R. Soc. London, Vol. 202, (1950), pp. 253–263. [10] J.M. Cohen: Stable Homotopy, Lecture Notes in Math., Springer-Verlag, 1970. [11] Yu.A. Drozd: “Matrix problems and categories of matrices“, Zapiski Nauch. Semin. LOMI, Vol. 28, (1972), pp. 144–153. [12] Yu.A. Drozd: “Finitely generated quadratic modules“, Manus. Math., Vol. 104, (2001), pp. 239–256. [13] Yu.A. Drozd: “Reduction algorithm and representations of boxes and algebras“, Comptes Rendues Math. Acad. Sci. Canada, Vol. 23, (2001), pp. 97–125. [14] P. Freyd: “Stable homotopy II. Applications of Categorical Algebra“, Proc. Symp. Pure Math., Vol. 17, (1970), pp. 161–191. [15] I.M. Gelfand and V.A. Ponomarev: “ Remarks on the classification of a pair of commuting linear transformations in a finite-dimensional space“, Funk. Anal. Prilozh., Vol. 3:4, (1969), pp. 81-82. [16] S.I. Gelfand and Yu.I. Manin: Methods of Homological Algebra, Springer–Verlag, 1996. [17] H.W. Henn: “Classification of p-local low dimensiona; spectra“, J. Pure and Appl. Algebra, Vol. 45, (1987), pp. 45–71. [18] Hu Sze-Tsen: Homotopy Theory, Academic Press, 1959. [19] E. Spanier: Algebraic Topology, McGraw-Hill, 1966. [20] R.M. Switzer: Algebraic Topology — Homotopy and Homology, Springer-Verlag, 1975. [21] H. Toda: Composition Methods in the Homotopy Groups of Spheres, Ann. Math. Studies, Vol. 49, Princeton, 1962. [22] H.M. Unsöld: “A4n -Polyhedra with free homology“, Manus. Math., Vol. 65, (1989), pp. 123–145. [23] J.H.C. Whitehead: “The homotopy type of a special kind of polyhedron“, Ann. Soc. Polon. Math., Vol. 21, (1948), pp. 176–186.

CEJM 2(3) 2004 448–477

Review article

A survey of certain results on strong approximation by orthogonal series László Leindler∗ Bolyai Institute, University of Szeged, Aradi vértan´ uk tere 1, H-6720 Szeged, Hungary

Received 3 September 2003; accepted 12 June 2004 Abstract: This is a survey of results in a particular direction of the theory of strong approximation by orthogonal series, related mostly with author’s contributions to the subject. c Central European Science Journals. All rights reserved. Keywords: Strong approximation, orthogonal series, summability methods, Fourier series MSC (2000): 40C05, 40C15, 40F05, 41-02, 41A25, 42A10, 42A24, 42C05, 42C15

1

The origin of the problem

Let {ϕn (x)} be an orthonormal system on the finite interval (a, b). We consider the orthogonal series ∞ n=0

cn ϕn (x) with

∞

c2n < ∞.

(1)

n=0

By the Riesz-Fischer theorem the series (1) converges in L2 to a square-integrable function f . Let us denote the partial sums and the (C, α)-means of (1) by sn (x) and σnα (x), respectively. Before recalling some results on strong approximation by orthogonal series, the first of them due to G. Sunouchi [51], we have to mention that the study of strong approximation for Fourier series was initiated by G. Alexits [2], [3] at the beginning of the 1960s, linked to the research concerning the strong summability of Fourier series which had been started by Hardy and Littlewood [11] in the 1910s. ∗


L. Leindler / Central European Journal of Mathematics 2(3) 2004 448–477

449

Consequently, in order to give some short historical background, we have to consider the case when {ϕn } is the trigonometric system. Then, by the famous result of Fejér [8] the (C, 1)-means of (1) converge to f (x) if f is a continuous and 2π-periodic function, in other words, the series (1) is (C, 1)-summable to f (x), i.e. σn1 (x) :=

1 sk (x) → f (x). n + 1 k=0 n

It was a natural question that if the Fourier series of a continuous function is not convergent then its summability is a consequence of the cancellations of the various deviations summed in the means σn1 (x)

1 − f (x) = (sk (x) − f (x)). n + 1 k=0 n

This question and the negative answer was given by Hardy and Littlewood [11] in 1913, namely they proved that if f is an integrable function and continuous at a given point x0 then the strong means 1 |sk (x0 ) − f (x0 )| n + 1 k=0 n

(2)

also converge to zero. This fact focuses interest to the investigations of strong summability by Fourier series, whose literature, initiated by the previous result, has grown very rapidly shortly after the appearance of the Hardy-Littlewood’s paper. Here we recall an important result of Zygmund [59], whose special case p = 2 was proved by Marcinkiewicz [39] as one of the most important results in this subject. It states as follows: If p > 0 and f is an integrable function then the means 1 |sk (x) − f (x)|p n + 1 k=0 n

(3)

converge to zero almost everywhere, specially at any point of continuity of f . In [18] we generalized (3) to strong (C, γ)-summability, i.e. we verified that if f is integrable, then n 1 γ−1 n+γ p A |sk (x) − f (x)| → 0 An = (4) n Aγn k=0 n−k almost everywhere for any positive p and γ. As we have already mentioned the problem of strong approximation is due to G. Alexits, who in a joint paper with D. Králik [3] sharpened a classical result of Bernstein [5] proving, among others, that if f ∈ Lip α, 0 < α < 1, then 1 |sk (x) − f (x)| = O(n−α ). n + 1 k=0 n

450


This result has many generalizations, a survey of the results can be found in my monograph [27]. Another monograph dealing with strong approximation of Fourier series was published by V.V. Zhuk [56] in 1989. It contains several interesting theorems, but his work and mine are mutually disjoint. We also mention that F. Schipp [47] proved a certain analogue of the theorem of Alexits-Králik for Walsh-Fourier series. In [4] G. Alexits and D. Králik proved a theorem of strong approximation pertaining to polynomial-like, constant preserving orthonormal system. Now we turn to the question of strong approximation by general orthogonal series, but we would like to emphasize that the literature of strong summability by orthogonal series is much more abundant that of the strong approximation. To see this it is enough only to consider the recently published proficient monograph by O.A. Ziza [57], entitled ”Summability of Orthogonal Series” (in Russian with English Summary). The first result of strong approximation by orthogonal series was proved by G. Sunouchi in 1967, who generalized one of my results from the ordinary approximation to the strong one. In [16] we proved that if 0 < γ < 1 and ∞

c2n n2γ < ∞

(5)

n=1

then f (x) − σn1 (x) = σx (n−γ ) almost everywhere in (a, b). G. Sunouchi [51] generalized this result and proved the first theorem pertaining to strong approximation. His theorem states as follows: Theorem 1.1. If 0 < γ < 1 and (5) is satisfied, then

n 1 α−1 A |sk (x) − f (x)|p Aαn k=0 n−k

1/p = ox (n−γ )

(6)

holds almost everywhere in (a, b) for any α > 0 and 0 < p < γ −1 . In the next section of this survey we will collect some results on strong approximation. Most of them are certain generalizations of the previous Sunouchi’s result. At the end of this point let me emphasize that this survey will be restrained just to the point-wise approximation. However there are several results treating similar problems in different spaces. E.g. H.-J. Schmeisser and W. Sickel [48] consider the Lp -case; further references can be found in their paper.


2

451

Results (mainly) on classical means

Continuing the generalizations, in [19] we showed that the assumptions of Sunouchi’s theorem imply, for any increasing sequence {νk } of the natural numbers, that 1/p n 1 α−1 A |sν (x) − f (x)|p = ox (n−γ ) (7) Cn (f ; α, p, {νk }; x) := Aαn k=0 n−k k also holds almost everywhere in (a, b). But if we consider an arbitrary sequence {µk } of the natural numbers which is not necessarily increasing then (7) does not hold with sµk (x) instead of sνk (x). For such mixed sequences we [19] could prove only the following weaker result: If 0 < γ ≤ 1/2, 0 < p ≤ 2, pγ < 1 and ∞

c2n n2γ (log log n)2 < ∞,

(8)

n=4

then

1 |sµk (x) − f (x)|p n + 1 k=0 n

1/p = ox (n−γ )

(9)

holds almost everywhere in (a, b) for any fixed (not necessarily monotonic) subsequence of the natural numbers. Later on we shall speak on strong, very strong and extra strong (or mixed) approximation according as in the investigated means the following partial sums sk (x), sνk (x) (νk < νk+1 ) or sµk (x) (where {µk } is a permutation of a subsequence of the natural numbers, or briefly a mixed sequence), appear, respectively. The previous results, in other words, state that the conditions of Sunouchi’s theorem imply the very strong approximation with the same order, but in the case of extra strong approximation we need some additional assumptions. In [19] we also proved that for the strong de la Vallée Poussin means the restriction γ < 1 can be omitted, furthermore that the partial sums in (6) can be replaced by (C, δ)means, where δ would also take negative values. More precisely the following theorems were proved: Theorem 2.1. Suppose that 0 0, and that (5) is satisfied. Then we have 1/p 2n 1 |sk (x) − f (x)|p = ox (n−γ ) (10) n k=n+1 almost everywhere in (a, b). Theorem 2.2. If α > 0, 0 < γ < 1, 0 (1 − δ)p, (11) 2

452


then

n 1 α−1 δ−1 An−k |σk (x) − f (x)|p α An k=0

1/p = ox (n−γ )

(12)

almost everywhere in (a, b). The next step in the generalizations was made in [20], where the (C, α)-means were changed by a wide class of regular summation methods Tdetermined by a certain positive n αnk . But the main goal remained triangular matrix αnk /An αnk ≥ 0 and An = k=0

to use our general result to get easily consequences for the classical strong means. As sample results we recall the following theorems, where in the sequel K, K1 , . . . will denote positive constants, not necessarily the same on any two occurrences. Theorem 2.3. Suppose that 0 < γ < 1, 0 1 such that pq(q − 1)−1 ≥ 2 and with this q for any 0 < ρ < 1 and 2m < n ≤ 2m+1  1/q n +1 m min(2  ,n) q αnk (k + 1)q(1−ρ)−1 ≤K αnk n−ρ .   =0

k=0

k=2 −1

Then for arbitrary δ > 1 − (q − 1)/pq we have 1/p n 1 αnk |σkδ−1 (x) − f (x)|p = ox (n−γ ) An k=0

(13)

almost everywhere in (a, b). We call the attention of the reader to two papers of I. Szalay [52, 53] where he proved some theorems having conclusion like (13), however his conditions have form not like condition (5) but ”block-type” form, e.g. that 2m+1 κ ∞ c2n n2γ < ∞. m=0

n=2m +1

Corollary 2.4. Suppose that 0 < γ < 1, 0 1 − min(1/2, 1/p) almost everywhere in (a, b). Theorem 2.5. Suppose that 0 < γ < 1/2, 0 < p ≤ 2, and that (8) is satisfied. Then, if (2−p)/2 n n (αnk )2/(2−p) ≤K αnk n−p/2 , k=0

k=0


we have

n 1 αnk |sµk (x) − f (x)|p An k=0

453

1/p = ox (n−γ )

almost everywhere in (a, b) for any (not necessarily monotonic) sequence {µk } of distinct natural numbers. Corollary 2.6. If 0 < γ < 1/2, 0 p/2, furthermore (8) is satisfied, then 1/p n 1 α−1 A |sµ (x) − f (x)|p = ox (n−γ ) (14) Aαn k=0 n−k k almost everywhere in (a, b). By (11) we see that the parameter δ − 1 of the means σkδ−1 (x) in (12) can be negative,

but if α is small the range of this parameter is very narrow, it is only the interval − αp , 0 . This is one of the motives of proving of the following theorem [22]. A theorem of similar type for strong summability was proved by K. Endl [6]. Theorem 2.7. If 0 < α < 1, δ > 1/2 and 0 < γ < α/2, furthermore ∞

c2n n2γ+1−α < ∞,

(15)

n=1

then

n 1 α−1 δ−1 A |σ (x) − f (x)|2 Aαn k=0 n−k k

1/2 = ox (n−γ )

almost everywhere in (a, b). W. Henrich [12] generalized Theorem 2.7 as follows: Theorem 2.7 (*). Let 0 < α < 1, p > 2, 0 < γ < α/2 and δ > 1/2 + α(1/2 + 1/p). If (15) holds, then 1/p n 1 α−1 δ−1 A |σ (x) − f (x)|p = ox (n−γ ) Aαn k=0 n−k almost everywhere in (a, b). The following two theorems [7] also have similar character with large p (≥ 2). Theorem 2.8. Let us suppose that 0 < α ≤ 1, p ≥ 2, 0 ≤ γ < αp , and that the series (1) is (C, α)-summable almost everywhere in (a, b). Then the following pairs of the conditions δ>

∞ 1 c2n n2γ+1−2α/p < ∞, and 2 n=1

454


or

∞ 1 c2n n2γ+1−2α/p log n < ∞, δ = and 2 n=1

imply that

Cnα,δ (x)

:=

n 1 α−1 δ−1 An−k |σk (x) − f (x)|p α An k=0

1/p = ox (n−γ )

holds almost everywhere in (a, b). Theorem 2.9. If 0 < α ≤ 1 and p ≥ 2, then the following pairs of condition δ> or

∞ 1 c2n n < ∞, and 2 n=1

∞ 1 c2n n log n < ∞, δ = and 2 n=1

imply that

α Cnα,δ (x) = Ox n− p

holds almost everywhere in (a, b). A similar problem in connection with the extra strong approximation also appears. Since we cannot apply Corollary 2.6 if α ≤ p/2, this was the reason of proving [22]: Theorem 2.10. If 0 < α < 1, 0 ≤ γ < α/2 and (15) holds then 1/2 n 1 α−1 A |sµ (x) − f (x)|2 = ox (n−γ ) Aαn k=0 n−k k almost everywhere for any sequence {µk } of distinct natural numbers. Let me mention some other problems appearing in connection with Corollary 2.6 which initiated my investigations. If we compare the restrictions on the parameters in the previous results, the assumption γ < 1/2 in Corollary 2.6 seems to be the most artificial, and it differs from all of the other conditions given for γ. Secondly, the factor (log log n)2 in (8) also seems to be superfluous. In [24] we extended the validity of (14) for any positive γ; naturally, hereby the range of the remaining parameters changed. Simultaneously we showed that the mentioned factor (log log n)2 being in (8) can be omitted, but then, for large γ, a new additional condition on α appeared. Theorem 2.11. Suppose that γ > 0, 0 p max α ≥ 1 if p = 2; and that (5) holds. Then (14) holds almost everywhere in (a, b).

1 2

, γ or

We want to emphasize that Theorem 2.11 and (7) in the special case α = 1 include the following corollary.


Corollary 2.12. If γ > 0, 0 < p < γ −1 , and (5) is satisfied, then 1/p n 1 p |sνk (x) − f (x)| = ox (n−γ ) n + 1 k=0

455

(16)

holds almost everywhere in (a, b) for any increasing sequence {νk }. This corollary shows that the restriction γ < 1 from the assumptions of Sunouchi’s theorem in the special case α = 1 can be omitted. This observation has become the source of many further research papers. On the other hand, using Theorem 2.11 with α = 1, we can easily obtain, by an appropriate application of the Hölder inequality, the same order of extra strong approximation for a larger class of regular summation methods. This class includes the method given by αnk = k β−1 , which has been investigated very frequently in connection with problems of strong approximation. In [21] we proved: Theorem 2.13. Suppose that γ > 0, 0 < pγ < β and that (5) holds. Moreover, if β ≤ 2, or if β > 2 but at least either γ < 1 or p ≤ 2, then 1/p n 1 β−1 k |sk (x) − f (x)|p = ox (n−γ ) (17) hn (f, β, p; x) := nβ k=1 holds almost everywhere in (a, b). This is the place where we can mention that P.S. Kantawala, S.R. Agrawal and C.M. Patel [13] proved some similar results for strong Nörlund and Euler means. E.g. in the case of Nörlund means their result holds only if γ < 1/2, that is, the rate of approximation is not so good as in (17). We also would like to underline that some of the results cited previously have been extended to multiple orthogonal series by F. Móricz (see e.g. [43, 44, 45]). Let us denote by A(β) the range of the positive parameters γ and p determined by the condition γp < β, i.e. A(β) := {γ, p : γ > 0, p > 0, and γp < β}, moreover, let B(β) := {γ, p : γ ≥ 1, p > 2, and γp < β}. Using these notations, Theorem 2.13 states that if (γ, p) ∈ A(β)\B(β), then (17) holds. Already in 1980 we had the conjecture that (5) implies (17) for any (γ, p) ∈ A(β), but then we were not able to prove this assertion. Then we could prove (17) only under the stronger condition ∞ n=1

c2n n2γ+1−2/p < ∞ (p ≥ 2, 0 < pγ < β).

(18)

456


In the same paper we investigated the order of approximation of the means hn (f, β, p; x) (defined in (17)) under the assumptions β = γp and (18). We showed that then hn (f, β, p; x) = Ox (n−β/p ), almost everywhere or equivalently ∞

nβ−1 |sn (x) − f (x)|p = Ox (1)

(19)

n=1

almost everywhere. The last formulation of the result in the limit case β = γp offers a natural generalization of this problem: What is a sufficient condition of the almost everywhere convergence of the series ∞ µn |sn (x) − f (x)|p (20) n=1

for any positive p and a monotone sequence {µn }? A possible answer is given by the next theorem [25]. Theorem 2.14. Let {µn } be a monotone sequence with the property 0 < K1 ≤ µ2n /µn ≤ K2 . Then the following pairs of conditions 0 < p ≤ 2 and

∞

µn Enp < ∞,

(21)

n=1

or p ≥ 2 and

∞

2 µ2/p n En < ∞,

(22)

n=1

imply that series (20) converges almost everywhere in (a, b), where En denotes the best approximation of f in the metric L2 (a, b). In this section we also present a result concerning the strong and very strong approximation by generalized Abel method. Let us introduce the following strong mean: 1/q ∞ q + k rk |sνk (x) − f (x)|p , A(p, r, q, {νk }; x) := (1 − r)q+1 q k=0 where 0 < r < 1, p > 0, q is a natural number, and {νk } is an arbitrary increasing sequence of the natural numbers. Theorem 2.15. Let γ and p be positive numbers, furthermore let q be an arbitrary natural number, {νk } be an increasing sequence of the natural numbers. Suppose that the coefficients (1) satisfy (5). Then we have that in the case pγ < 1 A(p, r, q, {νk }; x) = ox ((1 − r)γ );

(23)


and if pγ ≥ 1 but 0 pγ, q + 1 = pγ, q + 1 < pγ,

hold almost everywhere in (a, b) as r → 1 − . The special case p = 1 of Theorem 2.15 includes and generalizes a theorem of L. Rempulska [46]. In a joint paper with H. Schwinn [38] we proved the following four theorems. Theorem 2.16. If α and γ are positive numbers, 0 < pγ < 1, and {νk } is an increasing sequence, then condition (5) implies (7) almost everywhere in (a, b). Theorem 2.17. If γ > 0 and 0 < pγ < β then condition (5) implies 1/p n (k + 1)β−1 |sνk (x) − f (x)|p = ox (n−γ ) hn (f, β, p, {νk }; x) := (n + 1)β

(24)

k=0

almost everywhere in (a, b) for any increasing sequence {νk }. Theorem 2.18. If γ > 0 and 0 < pγ < min(α, 1) then (5) implies (14) almost everywhere in (a, b) for any sequence {µk } of distinct positive integers. Theorem 2.19. If γ > 0 and 0 < pγ < min(β, 1) then (5) implies hn (f, β, p, {µk }; x) = ox (n−γ )

(25)

almost everywhere in (a, b) for any sequence {µk } of distinct positive integers. In the light of Theorem 2.16 we would like to focus your attention on the fact that the restriction γ < 1 in the Sunouchi’s theorem is superfluous, consequently the assumption γ < 1 can be omitted here and in many other generalizations. Comparing the assumptions of Theorem 2.16 and 2.17 we see that the restriction pγ < 1 appears only in connection with the means Cn (f, α, . . .). This raised the next question: Can we replace the restriction pγ < 1 among the assumptions of Theorem 2.16 if α > 1 by pγ < α? In [22] we showed that the answer for this question is negative in general. We would like to make one more remark. Theorem 2.17 shows that if β > 1 then the restriction γp < 1 is not required to estimation (24) for the strong and very strong approximation, i.e. if the sequence {νk } is either the sequence of the natural numbers (νk ≡ k) or an arbitrary increasing subsequence of the natural numbers. But we mention that it can also be proved that statement (25) or Theorem 2.19 requires the assumption pγ < 1 for β > 1, i.e. for extra strong approximation the condition pγ < 1 is necessary.

458


Finally, by comparing of Theorem 2.16 and 2.17, we mention that if pγ < 1 then the means Cn (f, α, . . .) for any small α behave like the means hn (f, 1, . . .), i.e. then the strong Cesàro-means are more effective than the strong Riesz-means; or in other words, the strong Cesàro-means for any positive α behave like the strong (C, 1)-means. We [26] also investigated the order of strong approximation e.g. under the assumption β = pγ; i.e. we considered the limit case of the restrictions of the parameters. The results are as follows: Theorem 2.20. If α and p are positive numbers then for any increasing sequence {νk } the condition ∞ c2n n2/p < ∞ (γ = 1/p) n=1

implies that Cn (f, α, p, {νk }; x) = ox (n−1/p (log n)1/p ) holds almost everywhere in (a, b). Theorem 2.21. If β and p are positive numbers then for any increasing sequence {νk } the condition ∞ c2n n2β/p < ∞ (γ = β/p) n=1

implies that hn (f, β, p, {νk }; x) = ox (n−β/p (log n)1/p )

(26)

holds almost everywhere in (a, b). Theorem 2.22. If α and p are positive numbers, and α = min(α, 1), then for any sequence {µk } of distinct positive integers the condition ∞

c2n n2α/p < ∞ (γ = α/p)

n=1

implies that Cn (f, α, p, {µk }; x) = ox (n−α/p (log n)1/p ) holds almost everywhere in (a, b). Theorem 2.23. If β and p are positive numbers, and β = min(β, 1), then for any sequence {µk } of distinct positive integers the condition ∞

c2n n2β/p < ∞ (γ = β/p)

n=1

implies that hn (f, β, p, {µk }; x) = ox (n−β/p (log n)1/p )


459

holds almost everywhere in (a, b). In [25] we proved that in the special case p = 2 the statement of Theorem 2.21 can be improved, instead of (26) hn (f, β, 2, {νk }; x) = Ox (n−β/2 ) (γ = β/2)

(27)

also holds. The theorems that will be next formulated show that in the special case p = 2 similar improvements can be achieved for Theorems 2.20, 2.22 and 2.23, too. But if p = 2 then an estimate like (27) requires new conditions instead of the conditions of the previous theorems. The following theorems were proved in [25] and [26]. Theorem 2.24. For any positive α and for any increasing sequence {νk } the following pairs of conditions ∞ p/2 ∞ 0 < p ≤ 2 and c2k 0, α = min(α, 1) and {µk } is an arbitrary permutation of some subsequence of the natural numbers, then each of the conditions (28) and (29) implies that Cn (f, α, p, {µk }; x) = Ox (n−α/p ) holds almost everywhere in (a, b). Theorem 2.27. If β > 0, β = min(β, 1) and {µk } is a sequence given in Theorem 2.26, then each of the conditions (31) and (32) with β instead of β implies that hn (f, β, p, {µk }; x) = Ox (n−β/p ) holds almost everywhere in (a, b).

3

Theorems on de la Vall´ ee Poussin means

As we have mentioned the strong approximation investigations started in the 1960s. Till the 1990s it has become more and more clear that most of the results concerning any property of ordinary approximation have an analogue in strong sense. In other words, we have the same rate of approximation for strong means as for ordinary ones if we consider any one of the most frequently used means. This is true in spite of the facts that, in general, neither strong summability nor strong approximation follow from the suitable general ordinary summability and approximation (see e.g. F. M´ oricz [42] and [34]). Some sample theorems showing the great analogy between the ordinary and strong approximation results have been mentioned in this work previously and further examples can be found e.g. in the works [1, 15, 17, 19, 28, 38]. In 1991 [28, 29] we discovered some lacks in this compatibility even in the case of the classical de la Vallée Poussin approximation, and in the general one as well. In this point we present some theorems filling up these gaps. In order to formulate our assertion precisely we recall some definitions. In the following let p > 0, λ := {λn } be a nondecreasing sequence of natural numbers such that λ0 = 1 and λn+1 ≤ λn + 1 hold, furthermore let ν := {νk } denote an arbitrary increasing sequence of positive integers. Then the ordinary, the strong, the very strong and the generalized very strong de la Vallée Poussin means are defined as follows: 2n 1 sν (x), n ν=n+1 1/p 2n 1 Vn |p; x| := |sν (x) − f (x)|p , n ν=n+1

Vn (x) :=


Vn |p, ν; x| := and

Vn |λ, p, ν; x| :=

2n 1 |sνk (x) − f (x)|p n k=n+1

1 λn

n

461

1/p ,

1/p p

|sνk (x) − f (x)|

.

k=n−λn +1

We note that if λn = n then the generalized means reduce to the (C, 1)-means, if λn ≡ 1 then to the partial sums, and if λn = n2 , where [β] denotes the integral part of β, then we get the classical de la Vallée Poussin means with 2n in place of n. Now we recall some theorems relevant to this theme. In [15] we proved: Theorem 3.A. Let {ρn } be a monotone sequence of positive numbers such that m

ρ22k ≤ K ρ22m .

(33)

k=0

If

∞

c2n ρ2n < ∞

(34)

n=0

then Vn (x) − f (x) = ox (ρ−1 n ) holds almost everywhere in (a, b). A similar result for the very strong approximation was proved in [29] and it states as follows. Theorem 3.1. Under the assumptions of Theorem 3.A Vn |p, ν; x| = ox (ρ−1 n ) also holds almost everywhere in (a, b) for any 0 < p ≤ 2 and for any increasing sequence ν := {νk } of positive integers. We would like to mention that R. Taberski [54] proved some interesting results pertaining to the strong de la Vallée Poussin means in connection with continuous functions and Fourier-Chebyshev series. In [28] we proved two theorems in connection with the generalized very strong de la Vallée Poussin means. Theorem 3.2. Let {ρn } and {(n } be monotone nondecreasing sequences. If the condition (34) implies that the means Vn (x) of (1) converge for any {ϕn (x)} and {cn } almost

462


everywhere on a set E of positive measure, then the conditions ∞

c2n ρ2n (2n < ∞ and (µm+1 ≤ K (µn with 1 ≤ K
1, ω > 0 and m = 0, 1, . . . we define ρm (ω, r) :=

1 µm+1

1/r

µm+1 −1

(αk (ω))r

.

k=µm

K(.) will denote a constant depending only on those parameters as indicated. In terms of the quantities introduced above we can recall the theorem proved in [36]: Theorem 4.1. Let p > 0 and g(t) be a nondecreasing positive function on [0, ∞). Suppose that there exist r > 1 and a constant K(r, µ, γ) such that for every ω > 0 ∞

µm ρm (ω, r)γ(µm )−p ≤ K(r, µ, γ)(g(ω)/γ(ω))p .

(36)

m=0

If

∞

c2n γ(n)2 < ∞,

(37)

n=0

then Aω (f, p, ν; x) :=

∞

1/p αk (ω)|sµk (x) − f (x)|p

= Ox (g(ω)/γ(ω))

(38)

k=0

holds almost everywhere in (a, b) for any increasing sequence ν := {νk } of positive integers. If, in addition, for every fixed m, ρm (ω, r) = o((g(ω)/γ(ω))p ), as ω → ∞,

(39)

then the Ox in (38) can be replaced by ox . We note that the most important special case of Theorem 4.1 is when both (36) and (39) are satisfied with g(ω) ≡ 1. In this case we get that Aω (f, p, ν; x) = ox (γ(ω)−1 ) holds almost everywhere in (a, b). Next we recall some results which can be derived from Theorem 4.1 in the special case when g(ω) ≡ 1 and both (36) and (39) are satisfied. For precise reasoning see [36]. (i) If n k pnk (t) := t (1 − t)n−k , k = 0, 1, . . . , n; n = 1, 2, . . . k

464


and φ(t) ∈ L(0, 1) is a nonnegative function with φ1 = 1, then the matrix {αk (n)} defined by 1 pnk (t)φ(t)dt, (40) αk (n) := 0

yields the coefficients of a regular Hausdorff transformation. For these transformations, using Theorem 4.1, we have the following result. Theorem 4.2. Let γ > 0. Suppose that {αk (n)} is given by (40), where φ(t) ∈ Lr (0, 1) with some r > 1. If (37) with γ(n) = nγ holds and 0 < pγ < 1 − r−1 , then n

1/p αk (n)|sνk (x) − f (x)|p

= ox (n−γ )

(41)

k=0

almost everywhere in (a, b) for any increasing sequence {νk } of positive integers. Hence it immediately follows: Corollary 4.3. If {αk (n)} is the matrix of a Cesàro (C, δ) or a Hölder (H, δ) transformation, then (41) holds whenever o < pγ < min(1, δ). (ii) If λk (ω, t) :=

(ω log(1/t))k ω t , k!

k = 0, 1, . . .

and φ(t) ∈ L(0, 1) is a nonnegative function with φ1 = 1, then the sequence {αk (ω)} defined by 1 λk (ω, t)φ(t)dt (42) αk (ω) := 0

yields the coefficients of a regular [J, f ]-transformation. For this transformation we have Theorem 4.4. Let γ > 0. Suppose that {αk (ω)} is given by (42), where φ(t) ∈ Lr (0, 1) with r > 1. If (37) with γ(n) = nγ holds and 0 < pγ < 1 − r−1 then ∞

1/p αk (ω)|sνk (x) − f (x)|p

= ox (ω −γ )

(43)

k=0

almost everywhere in (a, b) for any increasing sequence {νk } of positive integers. This implies: Corollary 4.5. If {αk (ω)} is the coefficient-sequence of the Abel transformation, then (43) holds whenever 0 < pγ < 1. This clearly gives (23) with ω = (1 − r)−1 if q = 0.


465

(iii) If the function φ(t) in (40) satisfies 0 ≤ φ(t) ≤ K(β)tβ−1 with β > 0, then αk (n) ≤ K(β)

(k + 1)β−1 , (n + 1)β

thus, essentially, we get the Riesz transformation of order β, therefore Theorem 2.17 also follows from Theorem 4.1. (iv) If the function φ(t) in (42) satisfies q 1 0 ≤ φ(t) ≤ K(q) log with q ≥ 0, t then easy calculations yield that (k + 1)q αk (ω) ≤ K(q) (ω + 1)q+1

ω ω+1

k ,

this shows that (23) for any q ≥ 0 follows from Theorem 4.1. Some further applications of Theorem 4.1 can be found in [36]. Finally we recall the following Corollary 4.6. Let p > 0, α > 0. If 0 < γ < p−1 and (5) holds, then

n 1 α−1 An−k |sνk (x) − f (x)|p α An k=0

1/p = ox (n−γ )

(44)

almost everywhere in (a, b) for any increasing sequence {νk }. In spite of the wide applicability of Theorem 4.1, unfortunately, in the most important special case g(ω) ≡ 1, it cannot be used to estimate the approximation rate of the partial sums sn (x) of the series (1), because then (36) does not hold for any µ. Consequently Theorem 4.1 does not include the result of Theorem 3.3 in the simplest special case when λn ≡ 1. This was the reason why in a further joint paper with A. Meir [37] we proved another general result. In formulating this result we shall use the notations as above and assume hence forth that the following conditions hold: γ(µm+1 ) ≤ N γ(µm ), g(µm+1 ) ≤ N g(µm ) and

n

γ(µm )2 ρ(m) ≤ N γ(µm )2 ,

m=0

where ρ(t) denotes a nonincreasing positive function defined on [0, ∞).

466


Theorem 4.7. Suppose that there exists a natural number q such that for all k and m αk (n) ≤ N

q

αk (µm+i ) with µm < n < µm+1

i=−q

and

µm+1 −1 ∞ γ(µi )2 i=0

g(µi )2

αj (µi ) ≤ N ρ(m)γ(µm )2

j=µm

hold. Then (37) implies that An (f, p, ν; x) = ox (g(n)/γ(n)) almost everywhere in (a, b) for every p, 0 < p ≤ 2 and for every sequence ν. In [37] we showed that Theorem 4.7 implies Theorem 3.4, thus Theorem 4.7 yields certain results for the rate of approximation achieved by the partial sums of (1), as well. In 1995 H. Schwinn [49] published a very interesting theorem. Namely he involved to the condition (37) a new type condition claiming a certain degree of approximation of some subsequence of the partial sums, e.g. that sni (x) − f (x) = ox (γ(ni )−1 ),

(45)

where {γ(n)} is the factor-sequence appearing in (37). Then he states that the conditions (37) and (45) together imply a certain rate of approximation of a general method.

5

On the strong approximation by Ces` aro means of negative order

In the paragraph 2 we already established some theorems where the partial sums were replaced by Cesàro means of negative order. E.g. Theorem 2.3 is a quite general result. Unfortunately its conditions do not hold if αnν = (ν + 1)β−1 (ν ≤ n) and 0 < β < 1. Consequently, for 0 < β < 1, we cannot apply Theorem 2.3 to get an estimate for the following strong Riesz means 1/p n (n + 1)−β (ν + 1)β−1 |f (x) − σνd−1 (x)|p , ν=0

but such means are frequently used in connection with strong approximation. Therefore, in [30] we tried to weaken the assumption given in Theorem 2.3. We also mention that e.g. Theorems 2.17, 2.20 and 2.21 contrary to Theorems 1.1 and 2.3 do not claim the extra restriction γ < 1. This is great advantage of these theorems, but they do not allow of approximating with Cesàro means of negative order as Theorem 2.3 does. The reason is the following. The common kernel of the proofs of Theorems 1.1 and 2.3 is a very interesting result of T.M. Flett [9] and a useful lemma of G. Sunouchi


467

[50], and the latter lemma, unfortunately, requires the assumption 0 < γ < 1. The proofs of Theorems 2.17, 2.20 and 2.21 run on a perfectly different line, and these proofs do not use the assumption γ < 1. This is the reason that our generalization of Theorem 2.3 that is recalled below includes Theorems 2.17, 2.20 and 2.21 only if γ < 1. To extend it for γ ≥ 1, unfortunately, remains as an interesting open problem, in my view. Using the notations introduced previously we can formulate our results proved in [30]. Theorem 5.1. Suppose that p > 0 and 0 < γ < 1. Moreover let us suppose that there exists a number ρ > 1 with the property ρp ≥ 2, ρ−1 and that with this ρ and n(() := min(2 , n), 2m < n ≤ 2m+1 , An :=

n

αnk ,

k=0

 m  n( +1) =0



ρ αnν (ν + 1)ρ(1−γ p)−1

ν=n( )−1

1/ρ  

≤ K g(n) An n−γ p ,

where g(t) denotes a nondecreasing positive function defined for 0 ≤ t < ∞. Then, for any d > 1 − ρ−1 , (5) implies ρp

n 1 αnν |σνd−1 (x) − f (x)|p An ν=0

1/p = Ox (g(n)1/p n−γ )

almost everywhere in (a, b). If, in addition, for every fixed (,  1/ρ  n( +1)  ρ αnν = o(g(n)An n−γ p ), as n → ∞,   ν=n( )−1

then the Ox above can be replaced by ox . This is a very general result, but not very easy to perceive. In order to show the applicability of Theorem 5.1 we establish some corollaries of it. Corollary 5.2. If 0 < γ < 1, d > max(1/2, (p − 1)/p) and 0 < pγ < β, then (5) implies

n −β (n + 1) (ν + 1)β−1 |σνd−1 (x) − f (x)|p ν=0

1/p = ox (n−γ ).

468


Corollary 5.3. If α > 0, 0 < γ < 1, 0 max(1/2, (p−1)/p, (p−α)/p), then (5) implies 1/p n 1 α−1 d−1 An−ν |σν (x) − f (x)|p = ox (n−γ ). α An ν=0 Corollary 5.4. If {µk } is an increasing sequence of natural numbers and σnβ (µ; x)

:=

n 1

Aβn k=0

Aβ−1 n−k sµk (x),

furthermore 0 < γ < 1, d > max(1/2, (p − 1)/p) and β = pγ > 0, then (5) implies 1/p n (n + 1)−β (ν + 1)β−1 |σνd−1 (µ; x) − f (x)|p = ox ((log n)1/p n−γ ). ν=0

In the proof of Corollary 5.4 we used along with Theorem 5.1 the following very crucial lemma as well (see e.g. in [25]). Lemma 5.5. Let δ > 0 and {δn } be an arbitrary sequence of nonnegative numbers. Suppose that for any orthonormal system {ϕn (x)} the condition ∞ δ ∞ δn c2m < ∞ n=0

m=n

implies that the sequence {sn (x)} of the partial sums of (1) possesses a property P, then any subsequence {sνn (x)} also possesses property P. It is easy to see that with the use of this lemma everywhere in the assertions given above the means σνd−1 (x) can be replaced by σνd−1 (µ; x). In this paragraph we shall recall one more general theorem. The inspiration to prove it came from the following facts. Comparing the theorem of Sunouchi (Theorem 1.1) and the Corollary 4.6, we see that among the assumptions of Corollary 4.6 the restriction γ < 1 does not appear. This is a great advantage of Corollary 4.6. But if we consider Theorem 2.3 in the special case αnν = Aα−1 n−ν , then (13) has the advantage regarding (45) that in (13) we can approximate the function f (x) by Cesàro means of negative order, although then among the conditions the restriction γ < 1 appeared again. So it is natural to ask whether in the general case, or only in the Cesàro case, if we want to approximate the function f (x) by Cesàro means of negative order, then the restriction γ < 1 can be omitted or cannot. Unfortunately we are unable to answer these questions, but the answer would be very interesting result. In [31] we barely proved that in the special case γ(t) = tγ , treated in Theorem 4.1, and under the restriction γ < 1, the function f (x) can be approximated by Cesàro means of negative order.


469

The main tool in the proof of the next theorem is the following proposition verified in [30]. Proposition 5.6. If p > 0, 0 < γ < 1 and d > max(1/2, (p − 1)/p), then (5) implies

2n 1 d−1 |σ (µ; x) − f (x)|p n k=n+1 k

1/p = ox (n−γ )

almost everywhere in (a, b) for any increasing sequence µ := {µk }. Using the notations introduced in Theorem 4.1 with γ(t) := tγ we can formulate the following general result. Theorem 5.7. Let p > 0, d > max(1/2, (p − 1)/p) and 0 < γ < 1. Suppose that there exist r > 1 and a constant K(r, µ) such that for any ω > 0 ∞

p µm ρm (ω, r)µ−γ ≤ K(r, µ)(g(ω)/ω γ )p . m

m=0

If (5) holds, then Aω (f, p, d, ν; x) :=

∞

1/p αk (ω)|σkd−1 (ν; x) − f (x)|p

= Ox (g(ω)/ω γ )

k=0

almost everywhere in (a, b) for any increasing sequence ν := {νk } of positive integers. If, in addition, for every fixed m, ρm (ω, r) = o((g(ω)/ω γ )p ), as ω → ∞, then the Ox above can be replaced by ox . We point out that the most important special case of Theorem 5.7, in our view, is when g(ω) ≡ 1. Then we get that Aω (f, p, d, ν; x) = ox (ω −γ ). It is easy to verify that Theorem 5.7 with d = 1 includes Corollary 4.6, too. In this point, finally, we mention that similar consequences as were recalled after Theorem 4.1 can be formulated using Theorem 5.7, as well.

6

The sharpest result

In this point we present the sharpest result at the moment of writing, as far as we know. We again call the attention to the facts that the proofs of Theorems 1.1, 2.3 and 5.7 are based on a very interesting result of T.M. Flett [9] and a useful lemma of G. Sunouchi

470


[50]. Unfortunately, Sunouchi’s lemma requires the assumption 0 < γ < 1, furthermore Flett’s result works only with factors having the form nγ , γ > 0. In [33] we generalized Flett’s result replacing the factors nγ by more general factors β(n). Having this generalization of Flett’s result (see below Proposition 6.A) and after extending the lemma of Sunouchi by similar way in [35] (here see below Proposition 6.B), the time and the preliminary conditions were ready to generalize Theorem 5.7 and Theorem 4.1 in a certain range of the functions γ(t) appearing in Theorem 4.1, such a way, that the partial sums in (38) were replaced by Cesàro means of negative order. Before formulating this general result we recall a definition and three properties of the function γ(t) to be used in our next theorem which was proved in [35]. A sequence {γn } of positive numbers is said to be quasi geometrically increasing (decreasing) if there exist natural numbers µ, ν and a real number K ≥ 1 such that 1 γn+µ ≥ 2γn and γn ≤ K γn+1 γn+µ ≤ γn , γn+1 ≤ K γn 2 hold for all natural numbers n ≥ ν. We shall say that the function γ(t) has the following properties: P1 : the sequence {γ(2n )} is quasi geometrically increasing; P2 : the sequence {γ(2n )2−n } is quasi geometrically decreasing; Pp,r : the sequence {γ(2n )2n(1−r)/rp } is quasi geometrically decreasing with some r > 1 and p > 0. Avoiding the unnecessary reoccurrences we shall use the definitions and notations given in paragraph 4. Now we can state the following very general result. Theorem 6.1. Let p > 0, d > max(1/2, (p − 1)/p) and let γ(t) be a positive nondecreasing function defined for 0 ≤ t < ∞ with properties P1 and P2 . If there exist r > 1 and a constant K(r, µ) such that for any ω > 0 ∞

µm ρm (ω, r)γ(µm )−p ≤ K(r, µ)(g(ω)/γ(ω))p ,

(46)

m=0

then

∞

c2n γ(n)2 < ∞

(47)

n=1

implies that Aω (f, p, d, ν; x) :=

∞

1/p αk (ω)|σkd−1 (ν; x) − f (x)|p

= Ox (g(ω)/γ(ω))

(48)

k=0

holds almost everywhere in (a, b) for any increasing sequence ν := {νk } of positive integers. If, in addition, for every fixed m, ρm (ω, r) = o((g(ω)/γ(ω))p , as ω → ∞,

(49)


471

then the Ox in (47) can be replaced by ox . It is easy to see that Theorem 6.1 in the special case γ(t) = tγ with 0 < γ < 1 reduces to Theorem 5.7; and thus it is a slight improvement of Theorem 5.7. But Theorem 6.1 is only partly a generalization of Theorem 4.1, namely we claim more about γ(t) in Theorem 6.1 than in Theorem 4.1. Next we recall two propositions playing key-role in the proof of Theorem 6.1. Before formulating them we give some further definitions and notations. Let k ≥ 1, α > −1 and β(t) be a positive nondecreasing function defined for 1 ≤ ∞ t < ∞. We say that a numerical series an is summable |C, α, β(t)|k if the series ∞

n=0

k −1

β(n) n

|τnα |k

is convergent, where n=1 mean of order α of the series an .

τnα

α := n(σnα − σn−1 ) and σnα denotes the nth Cesàro

Proposition 6.A. [33] Let r ≥ k > 1, α > −1, β ≥ α + k −1 − r−1 , and β(t) be a nondecreasing positive function defined for 1 ≤ t < ∞ so that with some C > 1 lim sup t→∞

Then, if the series

∞ n=0

thermore

∞

β(C t) < C α+1 . β(t)

an is summable |C, α, β(t)|k , it is summable |C, β, β(t)|r , fur1/r

β(n)r n−1 |τnβ |r

n=1

≤K

∞

1/k β(n)k n−1 |τnα |k

.

n=1

Proposition 6.B. [35] Let γ(t) be a positive nondecreasing function defined for 0 ≤ t < ∞ with property P2 . If (47) holds then b ∞ ∞ 2 −1 α−1 α 2 γ(n + 1) (n + 1) |σn (x) − σn (x)| dx ≤ K c2n γ(n)2 a

n=0

n=1

for any α > 1/2. Next we lead those results which can be derived from Theorem 6.1 in the special case when g(ω) ≡ 1 and both (46) and (49) are satisfied. Let αk (n) denote the coefficients of a regular Hausdorff transformation given in (40). Then we get as a generalization of Theorem 4.2 the following Theorem 6.2. Let p > 0, d > max(1/2, (p − 1)/p) and let γ(t) be a positive nondecreasing function defined for 0 ≤ t < ∞. Suppose that αk (n) are given by (40) where φ(t) ∈ Lr (0, 1), r > 1. If γ(t) has properties P1 , P2 and Pp,r with these p and r, and (47) holds, then 1/p n αk (n)|σkd−1 (ν; x) − f (x)|p = ox (γ(n)−1 ) (50) k=0

472


almost everywhere in (a, b) for any increasing sequence ν := {νk } of positive integers. Hence it follows Corollary 6.3. Let p > 0, d > max(1/2, (p − 1)/p), α > 0 and α∗ := min(1, α). Suppose that αk (n) is the matrix of a Cesàro (C, α) or a Hölder (H, α) transformation, and ∗ that the function γ(t) has the properties P1 and P2 , furthermore {γ(2n )2−nα /p } is quasi geometrically decreasing. Then (50) also holds almost everywhere in (a, b) under the condition (47). If we consider a regular [J, f ]-transformation whose coefficients are given by (42), we have the following result. Theorem 6.4. Let p, d, r, γ(t) and φ(t) have the same meaning and properties as in Theorem 6.2. Suppose that αk (ω) are given by (42). If (47) holds then ∞

1/p αk (ω)|σkd−1 (ν; x)

− f (x)|

p

= ox (γ(ω)−1 )

(51)

k=0

almost everywhere in (a, b) for any increasing sequence ν := {νk } of positive integers. This theorem plainly implies a corollary concerning the strong Abel summability if αk (ω) are the coefficients of an Abel transformation. Some further applications of Theorem 6.1 can be read in [35].

7

Relation between ordinary and strong approximation

In this last paragraph we intend to examine the relation between ordinary and strong approximation of orthogonal series. First we again recall some definitions. Let T := (αik ) (i, k = 0, 1, . . .) be a double infinite matrix of real numbers. We say that series (1.1) is T -summable to f (x) at a point x ∈ (a, b) if ti (x) :=

∞

αik sk (x)

k=0

exists for all i (except perhaps finitely many of them), and lim ti (x) = f (x).

i→∞

Let p > 0. Series (1) will be called strongly T (p)-summable at x if the relation lim

i→∞

∞ k=0

αik |sk (x) − f (x)|p = 0


473

holds. Let {γn } be a positive nonincreasing sequence tending to zero. If the relation Tn (p, f ; x) :=

∞

1/p αnk |sk (x) − f (x)|p

= ox (γn )

k=0

holds, then we say that the strong Tn (p, f ; x)-means of (1) approximate f (x) in order γn at x. At the end of the sixties, it was known that for the most frequently used summability methods, T -summability and strong T (2)-summability of series (1) coincide up to sets of measure zero. These results were proved by various authors and by individual methods. E.g. for the classical (C, 1)-summation process this was proved by A. Zygmund [58] (see also K. Tandori [55]), for (C, β > 0)-summation by G. Sunouchi [50], for Rieszsummation by J. Meder [40] and L. Leindler [14], and for the generalized de la Vallée Poussin summation also by us [17]. Then F. Móricz [42] raised and answered negatively the following interesting problem: ”Does, under the condition c2n < ∞, (52) T -summability of series (1) almost everywhere imply strong T (p)-summability for any permanent T -process?” Móricz’s result reads as follows: Theorem 7.A. There exist a uniformly bounded orthonormal system {φk (x)} on (0, 1), a sequence {ck } of coefficients and a permanent T -summation process such that (52) is ∞ satisfied, the orthogonal series ck φk (x) is T -summable almost everywhere in (0, 1), but the relation

k=0

lim

n→∞

∞

αnk |sk (x) − f (x)|p = ∞

k=0

holds almost everywhere for any p > 0. We wanted to prove the analogue of Theorem 7.A for strong approximation. Our conjecture that this can be done was born out by the fact (see e.g. [27], Theorem 1.9) that if f and its conjugate function belong to the class Lip 1 and (a, b) = (0, 2π), {ϕn (x)} is the trigonometric system, T is the classical (C, 1)-summation, then tn (x)−f (x) = O(n−1 ) everywhere, but Tn (1, f ; 0) ≥ C n−1 log n (C > 0), that is, the strong (C, 1)-means do not approximate as well as the ordinary (C, 1)-means in this special case; at least not everywhere. Our aim to prove a similar result for an arbitrary decreasing sequence {γn } instead of {1/n}, of course giving up the trigonometric system and the (C, 1)-summability as hopeless, became unsuccessful. We had to assume some restrictions on {γn } in proving the analogue of Theorem 7.A. Our results proved in [32] reads as follows:

474


Theorem 7.1. For any positive nonincreasing sequence {γn∗ } there exist a positive nonincreasing sequence {γn }, a uniformly bounded orthonormal system {Φn (x)} on (0, 1), a sequence {ck } of coefficients and a permanent T -summation process such that γn = 1, ∗ n→∞ γn

0 < γn ≤ γn∗ and lim ∞

c2n γn−2 < ∞

n=0

and the orthogonal series

∞

cn Φn (x)

(53)

n=0

satisfies the following relations: tn (x) − f (x) = ox (γn ) and limγn−1 Tn (p, f ; x) = ∞ almost everywhere in (0, 1) for any positive p, where tn and Tn denote the nth ordinary and strong T -mean of series (53), respectively. We do believe that Theorem 7.1 holds with γn = γn∗ , too, but we have not been able to prove it. In my view, to prove this, is an interesting open problem. The method of proof of Theorem 7.1 follows similar lines as that of Móricz’s theorem, which was mainly based on a direct construction of T -summability in question using the classical scheme of D. Menchoff [41]. The fundamental lemma of Menchoff reads as follows: Lemma 7.B. Let ν > 3 be a natural number and let K > 1. Then there exists in (−1, K) a system {ψkν (x)} (1 ≤ k ≤ ν 2 ) of orthonormal step functions with the following properties: (i) |ψkν (x)| ≤ K1 (1 ≤ k ≤ ν 2 , −1 ≤ x ≤ K); (ii) for every point x ∈ (1/2, 1) there exists an index ((x) depending on x such that 1 ≤ ((x) ≤ ν 2 and (x) ψkν (x) ≥ K2 ν log ν. k=1

Herewith the cease our survey.

Acknowledgment The author was partially supported by the Hungarian National Foundation for Scientific Research under Grant No. T 042462.


475

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CEJM 2(3) 2004 478–492

Weights in the cohomology of toric varieties Andrzej Weber∗† Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland

Received 18 September 2003; accepted 18 May 2004 Abstract: We describe the weight filtration in the cohomology of toric varieties. We present a role of the Frobenius automorphism in an elementary way. We prove that equivariant intersection homology of an arbitrary toric variety is pure. We obtain results concerning Koszul duality: nonequivariant intersection cohomology is equal to the cohomology of the Koszul complex IHT∗ (X) ⊗ H ∗ (T ). We also describe the weight filtration in IH ∗ (X). c Central European Science Journals. All rights reserved. Keywords: Toric varieties, equivariant intersection cohomology, weight filtration MSC (2000): 14M25, 14F43 (55N33), 32S35

1

Introduction

Let X be a smooth toric variety. If X is complete then its cohomology coincides with the Chow ring A∗ (X). Therefore the Hodge structure is not very interesting: H k,k (X) Ak (X) ⊗ C and H k,l (X) = 0 for k = l. If X is not complete, then the cohomology of X is equipped with the weight filtration constructed by Deligne, [13]: W0 H k (X) ⊂ . . . ⊂ W2k H k (X) = H k (X) . Since X is smooth Wk−1 H k (X) = 0. In the toric case the weight filtration has the property: W2l H k (X) = W2l+1 H k (X) for each l. The pure Hodge structure on W H k (X) = W2l H k (X)/W2l−1 H k (X) Gr2l ∗

Email: [email protected] Supported by KBN 2P03A 00218 grant. I thank Institute of Mathematics, Polish Academy of Science for hospitality.

†

A. Weber / Central European Journal of Mathematics 2(3) 2004 478–492

479

consists only of the Hodge type (l, l). If X is singular we replace ordinary cohomology by intersection cohomology. Since X can be given a locally conical structure in a metric sense, the intersection cohomology may be interpreted as L2 -cohomology of the nonsingular part, see [9, 10]. Nevertheless we avoid talking about the L2 –Hodge structure. We are interested in the weight filtration, which is defined via reduction of X to a finite characteristic field Fp , see [4]. The weight filtration is the filtration coming from the eigenvalues of the Frobenius automorphism acting on the étale intersection cohomology. Fortunately, for toric varieties the Frobenius automorphism of intersection cohomology can be induced from an endomorphism of the complex points of X. This map preserves orbits and it is covered by a map of intersection cohomology sheaves. Finally, the group IH k (X) is decomposed into the direct sum of the eigenspaces with eigenvalues pl , l = k2 , k2 +1, . . . , k. We stress that although the concept of weights is highly nontrivial and abstract, for toric varieties whole theory reduces to easy computations involving an action of a down-to-earth map. On the other hand we consider equivariant cohomology taken with respect to the big torus T = (C∗ )n acting on X, [2]. If X is singular we prefer to replace usual cohomology by the intersection cohomology. We prove that the equivariant intersection cohomology of X is pure even if X is not complete. This means that the only eigenvalue which can occur in IHT2k (X) is pk , whereas IHT2k+1 (X) = 0. Koszul duality allows one to recover the usual intersection cohomology from the equivariant one [18, 22, 27]. In general one has to know not only the cohomology groups, but also whole complex in the derived category of H ∗ (BT )-modules. One has a spectral sequence with E2k,l = IHTk (X) ⊗ H l (T ) converging to IH k+l (X). The differential in the E2 table is the usual Koszul differential: We identify H ∗ (T ) with the exterior algebra of the dual of the Lie algebra t∗ , and the cohomology of BT with the symmetric algebra. The differential d(2) = dKoszul has the form: d(2) (x ⊗ ξ) =

n j=1

xλj ⊗ iλj ξ ,

for x ∈ IHT∗ (X) , ξ ∈ Λt∗ .

Here {λj } is a basis of t, the elements λj of the dual basis are generators of H ∗ (BT ) = St∗ , and iλj stands for the contraction. Using the weight argument we prove that all the higher differentials of the spectral sequence vanish if X is toric. Therefore the nonequivariant cohomology of X is the cohomology of the Koszul complex. Moreover, the weight filtration coincides with the filtration given by the spectral sequence. The Koszul complex itself splits into a direct sum of subcomplexes, each computing the graded piece of the weight filtration Gr∗W IH ∗ (X). We note that the nonequivariant intersection cohomology of X is pure if and only if the equivariant intersection cohomology is free over H ∗ (BT ). The properties of the weight filtration are reflected by the behavior of Poincaré polynomials, which are the weighted Euler characteristics. Cohomology and intersection cohomology of toric varieties have surprisingly attracted many authors. Jurkiewicz [25] and Danilov [11] have computed cohomology of complete

480


smooth toric varieties. Instead of presenting a long list of papers, we suggest the reader to check the references in [8]. A complex computing intersection cohomology of a toric variety was described in [24]. Our paper is based on the approach of [17], [2 – 3] combined with [15]. We will assume that the reader is familiar with the basic theory of toric varieties [21], intersection cohomology [23] and equivariant intersection cohomology [6]. Our goal is to expose the role of the weight filtration. Toric varieties serve as a toy models. The reader who is not familiar with intersection cohomology may replace it by usual cohomology and assume that X is defined by a simplicial fan, i.e. X is a rational homology manifold. Now, we would like to explain terminology concerning formality which plays an important role in our consideration. (1) A manifold is called formal if the algebra of differential forms Ω∗ (X) is quasiisomorphic to its cohomology as a dg-algebra. In the definition of formality for an arbitrary topological space the algebra of forms is replaced by the Sullivan–de Rham complex. Except for Kähler manifolds, the classifying space of a connected Lie group BG is an example of a formal space. (2) Suppose B is a formal space and X → B is a map. Then Ω∗ (X) is a module over Ω∗ (B). A natural notion of formality over B would be the demand that Ω∗ (X) is quasiisomorphic to its cohomology as a dg-module over Ω∗ (B) H ∗ (B). Formality of ET ×T X over BT implies that H ∗ (X) = H ∗ (Ω∗ (ET ×T X) ⊗ Λ∗ , dKoszul ) = H ∗ (H ∗ (ET ×T X) ⊗ Λ∗ , dKoszul ) . This is exactly the content of our Theorem 5.4 for simplicial toric varieties. Recently M. Franz [19] has shown that ET ×T X is formal over BT (even with Z coefficients) for smooth toric varieties. (3) If B = BG and Ω∗ (EG ×G X) is quasiisomorphic to its cohomology as an algebra over H ∗ (BG) then X is called G-formal in [26]. Smooth toric varieties are formal in the above sense by [28]. (4) The notion of equivariantly formal space was introduced in [22]. Before it was called totally nonhomologous to zero. The name does not fit to the scheme of the previous definitions. It just means that Ω∗ (EG ×G X) is a free (up to a quasiisomorphism) dg-module over H ∗ (BG). It is equivalent to the statement that HG∗ (X) is a free module over H ∗ (BG). I would like to thank Matthias Franz for valuable remarks and comments.

2

Mixed Hodge structure

Let X be a smooth possibly noncomplete algebraic variety. According to Deligne [13] one defines an additional structure on the cohomology of X. One finds a completion X ⊂ X, such that X \ X = αi=1 Di is a smooth divisor with normal crossings. For 0 ≤ k ≤ α let D(k) denote the disjoint union of the k-fold intersections of the components of the divisor D and set D(0) = X. Deligne has constructed a spectral sequence with E1k,l = H 2k+l (D(−k) ) ⇒ H k+l (X)

(1)


481

(the coefficient are in C). This spectral sequence degenerates on E2 . The limit filtration is the weight filtration. The quotients are equipped with the pure Hodge structure, they are decomposed into the (p, q) summands: GrkW H l (X) =

p+q=k

p,q H(l) .

The groups GrkW H l (X) are quotients of subgroups of H l−2k (D(k) ). If X is a smooth toric variety then there exists a smooth toric variety compactifying X. It can be chosen so the divisors at infinity are smooth toric varieties as well as each component of D(k) . The resulting weights can only be even. Moreover: W H l (X) is of the type (k, k) and Proposition 2.1. For a smooth toric variety each Gr2k W Gr2k+1 H l (X) = 0.

Remark 2.2. There is an easy method of constructing a spectral sequence converging to the weight filtration. Just consider the Leray spectral sequence of the inclusion X ⊂ X. Then E2k,l = H k (D(l) ). The limit filtration has to be shifted in order to have Wl H l (X) = im(H l (X) → H l (X)). This spectral sequence degenerates on E3 . In fact, up to a renumbering it is isomorphic to the Deligne spectral sequence. The construction of Deligne was motivated by the previous work on Weil conjectures, [12]. One considers varieties defined over a finite field. Instead of the Hodge structure l (X) can have one has an action of the Frobenius automorphism. The eigenvalues on Het k absolute values equal to p 2 with k = l, l + 1, . . . , 2l if X is smooth. Each complex variety is in fact defined over a finitely generated ring; toric varieties are defined over Z. The étale cohomology of the variety reduced to the finite base field is isomorphic to the usual cohomology for almost all reductions. In the toric case every reduction Z → Fp is good. The weight filtration coincides with the filtration of the étale cohomology: l (X) = Wk Het

Vλ ,

(2)

k |λ|≤p 2

l (X) is the eigenspace of the eigenvalue λ. The eigenvalues appearing here where Vλ ⊂ Het belong to Q C in general, but in the toric case λ is a power of p.

3

Frobenius automorphism of toric varieties

As noticed by Totaro [32], the toric varieties are rare examples of complex varieties admitting an endomorphism which coincides with the Frobenius automorphism after reduction. Let p > 1 be a natural number. Raising to the p-th power is a map of the torus ψp : T → T . There exists an extension φp : X → X, such that the following diagram

482

commutes:


µ

T ×X ψp ×φp ↓ T ×X

−−→ µ

−−→

X ↓φp X .

(3)

Here µ is the action of T . The map φp is given by a subdivision of the lattice. It has been recently applied to study equivariant Todd class in homology of singular toric variety in [7]. The following observations (already made in [32]) illustrate the nature of the weight filtration very well. To prove them one applies elementary properties of cohomology.

3.1 X complete and smooth Let X be smooth and complete toric variety. Then every homology class is represented by the closure of an orbit. The restriction of φp to a k-dimensional orbit is a cover of the degree pk . Therefore the induced action of φp on H2k (X) is the multiplication by pk . The conjugate action on H 2k (X) is the multiplication by pk . We say then that the cohomology of X is pure (see Definition 3.3 below).

3.2 X smooth If X is smooth, but not necessarily complete, then one has Deligne spectral sequence 1. W The induced action of φp on Gr2k H l is the multiplication by pk . It can be verified by comparison with the étale cohomology or in a elementary way: if Y is the closure an orbit of the codimension one, then the following diagram commutes H ∗ (Y ) φ∗p ↓ H ∗ (Y )

i

! −−→

pi

! −−→

H ∗+2 (X) ↓φ∗p H ∗+2 (X) .

Here i! is Gysin the map induced by the inclusion via Poincaré Duality. (It is a general rule that the Gysin map shifts the weight by the codimension of the subvariety.) The differential in the Deligne spectral sequence 1 are induced by the inclusions. Therefore, to have an equivariant spectral sequence with respect to φp one encodes the action in the usual way: E1−k,l = H 2k+l (D(k) )(k) , where the symbol (k) denotes action of φp via the multiplication by pk . The spectral sequence degenerates on E2 . This is because the higher differentials d(i) : Eik,l → Eik+i,l−i+1 l

do not preserve the eigenspaces of φ∗p . The domain of d(i) has the eigenvalue p 2 and the l−i+1 target has p 2 (the relevant values of l and l − i + 1 are even). For the limit filtration of H ∗ (X) we have:


483

W Proposition 3.1. The action of φ∗p on Gr2k H l (X) is the multiplication by pk .

Again this is just a simple characterization of the weight filtration for toric varieties. On the other hand, not appealing to the general theory, we can define weights in the following way. To be consistent with the usual terminology we will say that: Definition 3.2. A vector space V is of weight k if • φp acts on V as multiplication by pk/2 for k even, • V = 0 for k odd. Definition 3.3. The cohomology group H k (X) is pure if it is of weight k. It turns out (see Corollary 4.11) that our definition of purity agrees with the one in [3], §2. Namely the condition on the eigenvalues in even cohomology redundancy. It is implied by the vanishing of odd cohomology. In fact the decomposition of H ∗ (X) into the eigenspaces of φ∗p splits the weight filtration into a gradation, [32]. Example 3.4. Let X = P1 × P1 \ {(0, ∞), (∞, 0)}. Let X be the compactification of X obtained by blowing up two removed points in P1 × P1 . The Deligne spectral sequence has the following E1 table 0

C2 (2)

C(2)

4

0

0

0

3

0

C2 (1)

C4 (1)

2

0

0

0

1

0

0

C

0

-2

-1

0

The second table is equal to E∞ : 0

C(2)

0

4

0

0

0

3

0

0

C2 (1)

2

0

0

0

1

0

0

C

0

-2

-1

0

d(1) : E1k,l → E1k+1,l .

484


The cohomology of X is the following: C, 0, C2 (1), C(2), 0

in the dimensions 0, 1, 2, 3, 4. The group H 3 (X) is not pure, since it is of the weight 4.

3.3 X arbitrary For singular X we study intersection cohomology ([23]) instead of usual cohomology. The map φp preserves orbits, which form a stratification of X. Therefore φp induces a map of intersection cohomology groups IH ∗ (X). Moreover this map exists on the level of sheaves. Thus it is natural with respect to inclusions. Avoiding the formalism of [4] we can define a weight filtration by the formula 2. Later it will be clear, that the action of φ∗p decomposes IH ∗ (X) into a sum of eigenspaces with the eigenvalues pk as in the smooth case, see 5.4.

4

Equivariant cohomology

Although the equivariant cohomology and equivariant intersection cohomology are complicated objects, they have surprisingly nice properties for toric varieties. Equivariant cohomology is defined by means of Borel construction: HT∗ (X) = H ∗ (ET ×T X) .

IHT∗ (X) = H ∗ (ET ×T X; ICT• ) .

For the basic features we refer to [6] and [2].

4.1 Smooth case Equivariant cohomology of a smooth (or just simplicial) toric variety can be easily recovered from the fan. If ∆ is a simplicial fan, then the associated toric variety X∆ is a rational homology manifold. The equivariant cohomology can be identified with the algebra of the continuous functions on the support of the fan, which are polynomial on each cone of ∆. The odd part of the equivariant cohomology vanishes also for noncomplete fans. From the commutativity of 3 it follows that φp acts on the equivariant cohomology. We will prove a key theorem: Theorem 4.1. The equivariant cohomology of a smooth toric variety is pure. Remark 4.2. Since the space ET is a limit of algebraic varieties (as defined in [6]) or a simplicial variety (see [14] 6.1) it is equipped with a mixed Hodge structure. This structure is not only pure, but also of Hodge type (i.e. (k, k) in 2k-th cohomology). Every class is represented by an algebraic cycle. Remark 4.3. The same statement holds for arbitrary smooth G-varieties consisting of finitely many orbits. In fact HG∗ (X) is generated by algebraic cycles. We will develop


485

this remark in a subsequent paper [20]. We want to keep the present paper as elementary as possible. Therefore we do not treat the case of general G-varieties here. This would demand at least relying on the theory of mixed Hodge structures (in the smooth case). We prefer to exploit the map φp . Proof of 4.1. Our proof is the induction on the number of cones. 1) If ∆ consists of one cone σ, then HT∗ (X∆ ) = H ∗ (B(Tσ )) , where Tσ is the subtorus with Lie algebra spanned by σ. The isomorphism is induced by the inclusion of the minimal orbit T /Tσ $−→ X∆ . The action of φp on H ∗ (B(Tσ )) is as desired. 2) Now suppose that X is decomposed into a sum of toric varieties X = X1 ∪ X2 . The Mayer-Vietoris sequence splits into short exact sequences: 0 → HT2k (X) $−→ HT2k (X1 ) ⊕ HT2k (X2 ) − → → HT2k (X1 ∩ X2 ) → 0 . The groups HT2k (Xi ) are pure, therefore the subgroup of their direct sum is pure too. ✷ Remark 4.4. m From the intuitive point of view Theorem 4.1 is clear: If one identifies the equivariant cohomology with the piecewise polynomial functions (Stanley-Reisner ring in the complete case), then the gradation is given by the homogeneity degree. Purity means that homogeneous functions are homogeneous.

4.2 Singular case The construction of the equivariant intersection cohomology in terms of the fan is more complicated, IHT∗ (−) can be described axiomatically ([2], Def. 3.1). As in the smooth case the odd part vanishes. The even part is equipped with an action of φp . The map IHT∗ (X) → IH ∗ (X) is φp -invariant. We generalize Theorem 4.1: Theorem 4.5. The equivariant intersection cohomology of a toric variety is pure. Remark 4.6. Proceeding as in [5] we can prove the same statement for an arbitrary G-variety, provided, that it consists of finitely many orbits and singularities are not too bad. This is the case for spherical varieties. See [20]. Proof. The proof of the theorem is analogous except the first step. We will introduce an induction on dim X. If the dimension is one, then X∆ is smooth (X∆ = P1 , C or C∗ ) and the equivariant intersection cohomology is pure. We recall the definition of equivariant formality: Definition 4.7. We say that X is equivariantly formal if IHT∗ (X) is a free module over H ∗ (BT ).

486


Equivariant formality is equivalent to the degeneration on E2 of the spectral sequence: E2k,l = H k (BT ) ⊗ IH l (X) ⇒ IHTk+l (X) .

(4)

In the toric case it is also equivalent to the vanishing of IH ∗ (X) in odd degrees. Other equivalent conditions are stated in [2], Lemma 4.1 and [3], Theorem 3.8. For the inductive step we will need the following lemma: Lemma 4.8. Assume that X is equivariantly formal. Then IHT∗ (X) is pure if and only if IH ∗ (X) is pure. Proof. ”⇒” If IHT∗ (X) is equivariantly formal then the map IHT∗ (X) → IH ∗ (X) is surjective. Purity of IH ∗ (X) follows. ”⇐” If IH ∗ (X) is equivariantly formal, then the terms of the spectral sequence 4 are pure. Therefore IHT∗ (X) is pure. ✷ Proof of 4.5 cont. 1) If ∆ consists of a single cone, then (we divide X∆ by a finite group if necessary, as in the proof Theorem 4.4 in [2]) there is a decomposition X∆ T /Tσ × C(XLσ ) , where Lσ is a fan in tσ /lin(α) with α ∈ int(σ). The toric variety XLσ comes with an ample bundle and C(XLσ ) is the affine cone over XLσ . It is the toric variety associated to the cone σ considered in tσ . The variety XLσ is complete, thus it is equivariantly formal ([22]). The equivariant intersection cohomology is pure by the inductive assumption on the dimension of X, so IH ∗ (XLσ ) is pure by 4.8. The intersection cohomology of the cone is the primitive part of IH ∗ (XLσ ):

IH

n−i

(C(XLσ )) =

hi+1

ker(IH n−i (XLσ ) −−→ IH n+i+2 (XLσ )) 0

for i ≥ 0 for i < 0 .

Here h is the class of the hyperplane section. Again we see that IH ∗ (C(XLσ )) is pure. The cone C(XLσ ) is equivariantly formal (since the odd part vanishes). Therefore IHT∗ (C(XLσ )) is pure by 4.8. 2) The induction with respect to the number of cones is the same as in the proof of 4.1. One should remember that at this point the decomposition theorem ([4]) is used to justify that IHTodd (X) = 0. ✷ Remark 4.9. Proof that the odd equivariant cohomology vanishes and that the even equivariant cohomology is pure may be done in one shot. Corollary 4.10. Intersection cohomology of X is pure if and only if X is equivariantly formal. Proof. The vanishing of IH odd (X) implies degeneration of the spectral sequence 4. To prove the converse suppose X is equivariantly formal. Equivariant intersection cohomology is always pure by 4.5. Purity of IH ∗ (X) follows from 4.8. ✷


487

We can restate 4.10: Corollary 4.11. Intersection cohomology of X is pure if and only if it vanishes in odd degrees. Warning: Our corollaries do not hold for an arbitrary algebraic variety with a torus action.

5

Koszul duality

Let X be a topological space acted by a torus T . Borel construction produces a space and a map: XT = ET ×T X −−→ BT . The homotopy type of X can be recovered from XT by a pull-back diagram: X

∼ ET × X

= ET ×BT XT ↓ ET

−−→ −−→

XT ↓ BT

,

The map ET × X → XT is a fibration with the fiber T . In the stack language it is the quotient map X → X/T , [30]. Remark 5.1. The T -homotopy type of X is not preserved by the procedure X → XT → ET ×BT XT = ET × X since the action of T on ET × X is free and the action on X does not have to be. On the level of cohomology one has a spectral sequence E2k,l = IHTk (X) ⊗ H l (T ) ⇒ IH k+l (X) .

(5)

(Note that XT is simply-connected.) To recover IH ∗ (X) one would have to know a complex defining equivariant intersection cohomology: precisely an element of the derived category of H ∗ (BT )-modules. This is a form of Koszul duality as studied in [22], [1], [27], [18]. Remark 5.2. After certain renumbering of the entries, the Er table of the spectral sequence 5 is isomorphic to Er−1 of the Eilenberg–Moore spectral sequence ([16, 31]). The analog of the Eilenberg–Moore spectral sequence for intersection cohomology is studied in [20]. It is shown there that the weight structure is inherited. In the toric case the spectral sequence is acted by φp . The cohomology of T is not pure: H (T ) = Λl t∗ is of weight 2l. Because (by 4.5) the equivariant intersection cohomology is pure, the weight of E2k,l is k + 2l. The differentials l

d(2) : E2k,l → E2k+2,l−1

488


do preserve weights. Proposition 5.3. The higher differentials d(i) : Eik,l → E2k+i,l−i+1 for i > 2 vanish. Proof. The weight of the target is k + i + 2(l − i + 1) = k + 2l − i + 2 < k + 2l for i > 2. ✷ Hence d(i) vanishes. The differential d(2) has the usual Koszul form: d(2) (x ⊗ ξ) =

n j=1

xλj ⊗ iλj ξ ,

for x ∈ IHT∗ (X) , ξ ∈ Λt∗ .

Here {λj } is the basis of t and the elements λj of the dual basis are identified with generators of H ∗ (BT ) = St∗ . We obtain a description of the nonequivariant intersection cohomology: Theorem 5.4. Intersection cohomology of a toric variety is isomorphic to the cohomology of the Koszul complex IHT∗ (X) ⊗ Λt∗ . The filtration induced from the gradation of Λt∗ is the weight filtration up to a shift. Remark 5.5. The analogous statement holds for arbitrary G varieties with H ∗ (G) instead of Λt∗ , provided that IHG∗ (X) is pure. The proof will appear in [20]. Remark 5.6. In the simplicial case the complex HT∗ (X) ⊗ Λt∗ has appeared in [8], 4.2.2 and [18]. Ishida complex [29] is related to it. If X is a simplicial toric variety, then our complex contains and is quasiisomorphic to the Ishida complex. Let us describe the weight filtration precisely. The Koszul complex splits into a direct sum of subcomplexes IHT∗ (X) ⊗ Λt∗ =

2n l=0

∗ C[l] ,

2(k−l)

i C[l] = IHT

⊗ Λ2l−k t∗ .

(6)

The weight filtration of intersection cohomology also splits into the sum of eigenspaces of the Frobenius action W ∗ Gr2l IH k (X) (IH k (X))pl = H k (C[l] ). ∗ Although the complexes C[l] might be nonzero in high degrees, their cohomology vanish: ∗ ) = 0 for k > l or k > 2n . H k (C[l]

In particular we have:


489

Corollary 5.7. The pure component of IH k (X) is equal to the cokernel of d(2) : IHTk−2 (X)⊗ t∗ → IHTk (X). Equivalent descriptions of Wk IH k (X) are IHT∗ (X)/mIHT∗ (X) = IHT∗ (X) ⊗St∗ C , where m = S >0 t∗ is the maximal ideal. Example 5.8. Let us come back to the Example 3.4. It is the toric variety associated to the fan consisting of two quarters in R2 having only the origin in common. According to §4.1 the equivariant cohomology can be identified with the pairs of polynomials in two variables having the same value at the origin. The spectral sequence 5 has the E2 table 2

C(2)

0 C4 (3)

1 C2 (1) 0 C8 (2) 0

0

C6 (4)

C8 (5)

0

0 C12 (3) 0 C16 (4) 0 C20 (5) . . .

C

0 C4 (1)

0

C6 (2)

0

C8 (3)

0

1

3

4

5

6

2

0 C10 (6) . . .

0 C10 (4) . . . 7

8

The E3 table is 2

0

0

0

0

0 0 0 0 0 ...

1

0

0

C(2)

0

0 0 0 0 0 ...

0 C 0

0 C2 (1) 0

0 0

0

0

1

4 5

6

7 8

2

3

0

...

Remark 5.9. The dual spectral sequence 4 does not have to degenerate on E3 . The counterexample is C2 \ {0} ∼ S 3 with the action of C∗ × C∗ on the coordinates. The tables E2 , E3 and E4 are equal. The differential d(4) is nontrivial: d(4) (1 ⊗ x) = λ1 λ2 ⊗ 1, where x ∈ H 3 (C2 \ {0}) is the generator and λ1 , λ2 is the basis of t∗ .

6

Poincar´ e polynomials

For a toric variety we define the virtual Poincaré polynomial by the formula: IPcld (X) =

k,l

W (−1)k dim Gr2l IH k (X)q l .

The subscript cld indicates that the supports of cohomology are closed, whereas the usual Poincaré polynomials are taken with respect to compact supports. One can find the same formula for cohomology with compact supports in [21] §4.5 and for intersection cohomology in [15] §5.5. Our approach does not refer to the abstract theory. The polynomial IPcld (X) can be treated as a Euler characteristic with weights: IPcld (X) = χW (IH ∗ (X)) :=

l

W χ(Gr2l IH ∗ (X))q l .

490


Due to purity of equivariant intersection cohomology the equivariant Poincaré polynomial is simpler to define: IPcld (XT ) = χW (IHT∗ (X)) =

l

dim IHT2l (X)q l .

In general it would be the alternating sum as before. Poincaré polynomials were studied in [3], §4. The recursive definition of IPcld (X) applies in our case, as it will be clear from the following propositions. We note that for complete toric varieties our Poincaré polynomial differs from the one of [3] by a substitution q = t2 , whereas for noncomplete X the polynomial of [3] can be defined as the weighted Euler characteristic of intersection cohomology with compact supports. Proposition 6.1. The Poincaré polynomials are related by the formula IPcld (XT )(1 − q)n = IPcld (X) . Proof. The weighted Euler characteristic is multiplicative, therefore IPcld (X) = χW (IH ∗ (X)) = χW (IHT∗ (X) ⊗ H ∗ (T )) = χW (IHT∗ (X))χW (H ∗ (T )) . The Poincaré polynomial of H ∗ (T ) is (1 − q)n , thus IPcld (X) = IPcld (XT )(1 − q)n . ✷ Our Poincaré polynomials IPcld (X) are not additive with respect to union of strata, but: Proposition 6.2. The polynomials IPcld (X) and IPcld (XT ) are additive in the following sense: if X = X1 ∪ X2 , then IPcld (X) = IPcld (X1 ) + IPcld (X2 ) − IPcld (X1 ∩ X2 ) and the same for IPcld (XT ). Proof. The additivity of IPcld (XT ) follows from the fact that Mayer-Vietoris sequence ✷ splits into the short exact sequences. The additivity of IPcld (X) follows from 6.1. The recursive formula for IPcld (X) is the following: IPcld (X∆ ) =

σ∈∆

∂IPcld (XLσ )(1 − q)n−dim σ ,

where ∂IPcld (X) = τ> dim X (1 − q)IPcld (X) 2

if dim X ≥ 0 and ∂IPcld (∅) = 1. The symbol τ> dim X denotes truncating the coefficients 2 of the monomials of degrees which are not > dim2 X . The above formula holds for equivariantly formal X by [3] and Poincaré duality. Hence it holds for an affine X. By additivity it holds for any X.


491

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CEJM 2(3) 2004 493–493

Erratum Janusz Matkowski∗ Institute of Mathematics, University of Zielona G´ ora, Podgórna 50, PL 65-248 Zielona G´ ora, Poland

In Theorem 1.1 of my paper On subaddidive and ψ-additive mappings, CEJM 4 (2003), 435-440, I assume that ” 5. ψ(t) < t for all t > 0”. It has turned out that the notion of the ψ-additive mapping introduced by Professor G. Isac in [4] is based on the conditions 1, 2 and, instead of ”5”, on the essentially weaker condition ”ψ(t) < t for all t > 1”. Therefore the statement (Remark 3.1) that every ψ-additive mapping is additive is not true for the mappings which are ψ-additive in the sense of Isac. I am indebted to Professors G. Isac and P. Gãvruta for calling my attention to this important fact.

∗

Email: [email protected]

Central European Journal of Mathematics

Central European Journal of Mathematics

Central European Journal of Mathematics