Continuous Transformations of Manifolds

290

MA THEMA TICS: S. LEFSCHETZ

PROC. N. A. S.

CONTINUOUS TRANSFORMATIONS OF MANIFOLDS'

By S. LiwscHiTz DEPARTMZNT OF MATH1MATICS, PRINCZTON UNIVI:RSITY Communicated May 12, 1925

1. I propose to communicate here further results along the line of a previous Note.' The product of two complexes Cp, Cq, in the sense of Steinitz (image of their point-pairs) will be written Cp X C., and is not to be confused with their intersection Cp.Cq Let W", W' be the manifolds of the first Note, W' a copy of W. To assign a transformation T of W, into itself is equivalent to assigning a correspondence between point-pairs A, B of W. and W', where B is the image of T. A on W'. The complete representation of T is then given by the point set A. X B of W, X W,,. We impose upon T the unique restriction that the set shall constitute an n-cycle r, of the product manifold-practically a mild restriction indeed. Let r' correspond to T'. We are after the numbers (rP. r) for whose interpretation see loc. cit. 2. Denote generically by Ch, Ch complexes, respectively, in W", lrl. Then by means of indicatrices and in the light of another Note2 (Ch * Cn-h) (Ck- Cn-k) (Ch X Ck * Cn-h X Cn-k) = (-1)k(fl.h) (Wn X B.A X Wn) =+ 1 (Ch X Ck * Cn-h-i X Cs-k+i) = 0. 3. Let R,, be the A-th connectivity index of W, (1 + RI, is the 1,-th Betti number), and let yl, .y.. zR, be the cycles of a canonical fundamental set for the IA-th dimensionality, which are independent.3 Let also 68 be the image of y', on W'. We shall denote homologies "with division allowed" (i.e., with zero divisors neglected) by in place of the usual --. It may then be shown that the cycles ,, X _ constitute a and the n cycles of W,, X W'. Hence fundamental set for the relation c.'M,, mod W, X W'l.(1 X, 7y' XaThere is a similar relation for r' with the E's replaced by other integers say 77. 4. By combining the formulas of No. 2, with an important theorem of Veblen's on the Kronecker indices of fundamental sets,4 we may derive, readily enough, a formula for (rF, . r'I) in terms of the e's and tq's. It is, however, of distinctly more interest to obtain the expression of the index in terms of the coefficients in the transformations induced by T and T' on the cycles of W,. Thus for T:

VoL. 11, 1925

291:'

MA THEMA TICS: S. LEFSCHETZ2

a,,Ej-y

mod W., 70 . J would correspond to the effect on the A-cycles. There will be similar formulas for T' with O's in place of a's. If yp - T . -Y,, we prove by means of indicatrices this all important relation: (r,"T, x ain-;&) = (-1) ^;, -1n) (2) the indices being computed as to W.f X W,, and W,,, respectively. If we substitute for rn its expression from No. 3, for yA, and 5,,J, the cycles of the canonical sets and apply Veblen's theorem we obtain the desired expression of the e's in terms of the a's, hence also of the q's in terms of the a's; whence ultimately: (a) n W 2, mod 4

(r.r) =

(3) OI,1

}X=0

(b)

4m + 2

n

R

2m

(r rn= t+

E(-) i R2m O+ 1

t ,

j=1

(a.'

An-

(C,2h 2)h2k P2m+1 _E t°2m+1 |

+***) (4)

-

a2h-1, a2m +1

-2h,2k P2m+1

+ .2, t***J

h, k-i

the terms omitted being such as to make the formula symmetrical. 5. Interesting special cases.-(a) Fixed points of a single T. It will correspond to T' = 1, j= 0 of i j, = 1. Hence if r. corresponds to the identity: R, (rn.ri) = E (-i) E ax'

O.'

;-O

i=1

which, as it happens, holds whether the fundamental sets are canonical or not. This includes for example Alexander's recent results for n =2.5 (b) T' is of the class of T and n is odd. Then (i.r,r) O. (c) T is of the class of a deformation. Then

(ro. irn)=,-)

R.

Hence this striking theorem,; The minimum number of fixed points under a T of the class of the identity is equal in absolute value to the Euler character., istic. In point of fact. this remaip4s true if T Qperatinzg on,any cycle, merey increases it by a zero divisor. Again when n is odd, the number is zero. For n = 2 and if p is the genus of the W2? the number is 2p -2 as found by

Birkhoff.

292

MA THEMA TICS: H. S. VANDIVER

Phoc. N. A. S.

(d) Huruwtz's coincidence formula. Thanks to a theorem proved elsewhere7 it is a direct corollary of (4) in the case n = 2.8, In conclusion let us state that most results obtained along this line by Brouwer and his school follow from (3) and (4). 1 These PROCSgDINGS, 9, p. 99 (1923). In the present volume of these PROC$XDINGS. See my Borel Series Monograph: L'Analysis Situs et la geom6trie algibrique, p. 15 (1924). 4 Trans. Amer. Math. Soc., 25 (1923)540. 2

6Trans. Amer. Math. Soc., 25 (1923)173. S Trans. Amer. Math. Soc., 18 (1917)287. 7 Monograph, p. 19. S The product manifold was indeed used for the first time in the same manner as here in connection with algebraic correspondences by Severi in his well-known paper of

the Torino Memorie of 1903.

LAWS OF RECIPROCITY AND THE FIRST CASE OF FERMAT'S LAST THEOREM By H. S. VANDIV1ER DEPARTMENT OF MATHEMATICS, UNIVURSITY oF TsXAS Communicated April 15, 1925

Suppose that

xP +

yP + zP=o

(1)

is satisfied in rational integers x, y and z, none zero and each prime to the odd prime p. Then (x + ay) (x + a2y) ..(x + aPly) = u where a = e2 /, and u is a rational integer. It follows from the unique decomposition of an ideal into its prime ideal factors that the ideal (x + a'y) is the pth power of an ideal in the field Q (a), a 0 O (mod p). If w is an integer in Q (a), q an ideal prime in Q (a) which is prime to (co) and (p) then we set N(q)-1 w~ q P ~ f 9}(mod q)

where N(q) is the norm of q and wl/q) is some power of a. It follows that w is congruent to a pth power in (, modulo q, if and only if,

{q}=1