Topology

rllOPOLOGY

Lipman Bers

Notes by: and

Jacqueline Lewis Esther Rodlitz

27

New York University 1956-19.57

-1-

TABLE OF CONTENTS Chapt. I

POINT SET TOPOLOGY NO'cation and Preliminary Definitions ........... . Vector Spaces Scalar Products and Hilbert Spaces • .c: _ • • ,. • • • • • • Normed Vector' Spaces . Metric Spaces c • • • • • • • • • • • • • • • • • Continulty in a Metric Space •••••••••••••••••• 0 ••••••••••••••••••••••••••••••••

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Cartesian Product of Two Metric Spaces ••••••• Topological Spaces (Definiti9n) ••••••••• Ha~sdorrf Spaces •••••• o • • • • • • • • • • ~.~ • • • • < • • • • o Closed Sets o • • • = Interior, Exterior and Boundary Points $

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I!lduced Topology ....... eO" . . . . . . . . . . . . . . . . . . . . . . . Continuity in a TopologicB.l S.pace ••••••••••••• Homeomorphism ••••••••••••••• ~.~ ••••••••••••••• Ax1.:;ms of Countjab:tli ty ••••• " ........ ~ •••••••••• Convergence .~ ••••••••••• Completion of Hetr:'..c and Normed Linear Vector Spaess •••••••••• ~.Q.&~oo.~.o • • • • • • • • • Baire Catego~y Theorem ••• ~ ••••• ~~ ••••••••••••• Compactness , •••••••••••••• l« •••· . . . . . . . . . . . . . . . . Fundamental Cube in (Separable) Hilbert Space • Connectedness c •• ...... . Topological Product ••••••••••••••••••••••••••• Metrization Theorems ........................... . 0

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SIHPLEX AND COJVIPLEX Closed Convex Hull of a Set • • • • • • • • • • • • • • • • • • • Points in General Position ..................... The Geometric Rectilinear Simplex ••••••••••••• Ba.rycentric Coordinates ••••••••••••••••••••••• Definitions of Faces of a Simplex and Properly Situated Simplices •••••••••••••••••

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24 25 26 27

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CARNEGlr IRSTITUi~

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OF CONTENT.§. (Oontinued)

The Canonical n-Simplex ••••••••••••• , •••••••••• Geometrio Oomplex e c • • • • • • • • • • • • • Dimension of a Complex ••••••••••••••••••••••••• IKI, The Space of the Complex K ~ •••••••••••••• Polyhedra and Triangulations ••••••••••••••••• Subcomplexes o • • • • s • e _ • • • • • • • • • • r-Dimensionc.l Skeleton •••••••••• "•••••••••••••• 5t&r of a Vertex ••••••••••••••••• ~ •••••••••••••• Abstract Complex ., •••• $, •••••••• " •••• " ........... . Gcometl'ic Re8.1iz6. tion of an n-Dirnens ional AbstJ..'Clct Ccmplt.:;:;r in R.2n+l ,. •• " ••••••• < • • • • • • • • Sil:-1plic. ial JYIe.ppings •••• ~ • 0 " 0"" .......... " • Isomorphism of Two Abstract Complexes ." ........ . •

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Ordered and Oriented Simplices •• ~ ••••• ~ •••••••• Support of a Simplex •••••••••••••••••••• "••• e , • Integral Oriented r-Ohains •••••••••••.••••••••• Elementary Chains •• Boundary Opera.tor CJ ••••••••••••••••••••••• , ... .. Support of a C.l:':.a:i.n ............................. . Poincare Rela·::;iol1, 6 2 x = 0 cc~.v." • • "~e·~e • • • • • ~ Boundaries, Homologous Chains ~ a!:d Cyoles ••••• " r-Dimensional Integral Homology Group of K ••••• Scalar Product of Oriented Simplices 0

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33 34 34

35 35 35 35 35 36 37

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42 Coboundary Operat:)r 0* ........................ ~ 43 Coboundaries, COhoillologous Chains and Oocycles • 43 and Chains • ., •••.

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r-Dimensional Integral Cohomology Group of K e • • Incider.. ce and Coincidence 1'latrices • Chains Over Arbitrary Abeleare Groups •••••••••• Cochains ••••••••••••••••••••••••••••••• d as a Cochain Mapping ••••••••••••••••••••••• Hr.(K,G) and Hr(l'C,G,r) ••••••••••••• Homomorphisms f, of Chains ~ Induced by Simplicial IVlappings •••••••••••••••••••••••••• 0

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-i11TABLE OF CONTENTS (Continued) Chapt. ~f == fd • • • • • • • • • • • • • • • • • • • • • • • • Induced Cochain Homomorphism (f )

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••••••• ~ •••••••••••••••• o • • ~ • • • • • • • • Baryoentrio Subdivision of an Abstract Complex e Alg. Dim. GmK = Alg. Dim. K ••••••• 0." ........ . 53 Baryoentrio Subdivision of an Abstract Complex • 53 Maoh of a Ge:nnetrio Complex ........ c . . . . . . . . . . . . . 53 Cones over Simplioes and Cha.ins ••••• 54 K:;:ooneoker In';:·~!.~c t'D r ••• " •••••• ,. ••••••••••••••••• 54 Ba~ycenters

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Subdivision ~f Ch~ins ••••• e.~ • • • • c • • • • ~ • • • • • • • • Superdivision of Chains •••••••••••••••••••••••• .... -m ==0-m""\. rr'*~ m* = \ J r;-!l1* CJ ...... 00 DIU \:31 ••••••••••••••• Spe:t'ner Mapping, (S5.mplioial ApproxilflS'.tion to the Identity) •• " ••••••••••••••••••• ... m Spernel" 's Lemma, l () x = x • ., ••••••••••••••••••

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DIMENSION THEORY .. Coverings: Ordar of, Finite, Refinement of ••••• ~9 Definition 1 of Topological Dimension •••••••••• 59 Monotonic1ty of Dimension •••••••••••• ~ .......... 60 Lebesque Number of an Open Covering •••••••••••• 61 Diameter of a Covering ••••••••••••••••••••••••• 61 Definition 2 of Topological Dimension •• " ••••••• 61 Nel"ve of a Colleotion of Sets •••••••••••••••••• 6,3 Nerve and Order of a Covering •••••••••••••••••• 63 Stars ........ 64 Top. Dim. lKj Alg. Dim. K •••••• ~ ••••••••••••• 65 e-Mappings "." e. • • • • • • • • • • • • • • • • • • • • • • • • • • •.• • • . . . . 68 ~ Pal"t1tion of Unity Subordinate to a Covering ••• 70 Dimension Theorem for Compact Subsets of 73 Unit Cube I in RZn+l ............ ~ ................. 74 TheSpaoe IX • o •••• " . . . . . . . -• • • • • • • • • • • . • • • -. • •• • •• .14 II • • • • • • I I . ' • • • • • • • 0 " . . . . . . . . . . . . . . . . . .

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CO,NTENTS (Continued)

Chapt.

Page

Imbedding Theorem for Finite Dimensiona.l Compact Metric Spaces •••••••••••• a • • • • • • • • • • • Lebesgue Definition of Dinension for a Separable Metric Space •••••••••••••••••.••••• Urysohn-Menger Definition of Dimension (Inductive) ••••••••••• Compactification •• c • • • • • • • • • • • • O • • 6~O.~Q • • • C • • 9 Dime~sion of a Separable Metric Space in Terms of a Gompactification •••• ~ ••••• c • • • ~ n-Sphere ~ ••••••••• ~ ••••••••••• &o • • • • • • • • • • e Top. Dim~ R_.... = n ••••••••••••• e • • • • • • D • • ~ • • • • • • • 0 ••••••••••••••••••••••

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~rinciple

78 78 79 79 79 79

OF DOJYIA IN

of Contracting Mappings ••••••• ~ •••••• Principle of Non-Expanding Mappings ~ ••••••••••• Lebesque Number of a Collection of Closed Sets in a Compact Space •••••••••••••• ~ Covering Theorem for an n-Simplex ••• c . e • • ~ • • • • • Brouwer Fixed Pain t JJ.1heore:r.n •••••••••••••••••••• Retraction Mappings O • • • • • • • • • • QO • • • • • • • • , • • • • • • Theorem on the Invariance of Domain •••••••••••• Appltcation of tile Theorem on In'ITariance 0f Domain .... ~ ••.' ••••••• " " ••••••• oe c • ~ . . . . . . ~ • • Approximation of a ConveA Compaut Subset of a Banach Space by Finite Dimensional Spaces .0 Schauder Fixed Point Theorem (Weak Form) ••••••• Mazur 1 s Lemma ..................................... Schauder Fixed Point Theorem (strong Form) ••••• The Spaces Co and 01 ••••••••••••••••••••••••••• Holder Continuity •••••••• G • • • • • • • • • ~ • • O • • • • • • • • Equicontinuous Functions and the Theorem of Arzela (Aseoli) ••• ~ ••••••••••••••• , ••••••• Applications of Fixed Point Theorems ••••••••••• V

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H01"10LOGY THEORY (Part I)

Groups •••• o • • • • • • • • • • • • ~ • • • • • • • • • • • • • • • • • • • • • • 103 Factor (Quotient, Difference) Group ••••••••••• 104

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TABLE OF CONTENTS (Continued)

0.

Natural Homomorphism of A onto AlB ............. 104 Linear Dependence and Independanoe ••••••••• ,.$ 105 Rank of a Group .............................. ".105 Maximal Linearly Independent Set •••••••••••• ,. 105 Rank A = Rank AlB + Rank B ........... " •••• "...... 107 Generators oo •• • • • • • • • • • • , .. ·• • • o • • • • • • . . . . . o 109 Direct Sum o • • • • • • Q• • e • • • 109 ".~

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Cyclic Groups

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Fundamental Theorem of Abelain Groups ••••••••• Betti Numbers and ~~rsion Coefficients •••••••• Nctaticn Q.~ • • • • • • o • • • • • • • ~.o • • • • • • • • • • • • a •• s • • Euler-·P~Jince.rs Formula •• 1 •.•••••••• ~ ••••• Product of a l-Curve and an n-Curve e~

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HOMOTOPY GROUPS (Part II) Commutator Subgroup ••••••••••••••••••••••••••• Paths ••••••••••••••••••••••••••••••••••••••••• Elementary, Special and Simplicial Curves (Paths) •••••••••••••••••••••••••••••• Eimplicial Equivalent of a Special Curve •••••• Mapping, 4·., af Special Paths into Chains ....... Homotopio Curves are Mapped (by into Homologous Chains •• " ••.••••••••••• "" •••••• ".. For K Connected, H~(KJGo) ~ lIr (IK/) ••••••••••

*)

XII

175 175 178 178 178 180 180 181 181 181 182

185 18& 187 187 187 188 191

CONTINUOUS JYIAPPINGS OF THE n ...SPHERE INTO ITSELF Homology Groups of the n-Sphere ••••••••••••••• 192

;"'viii-

TABL?. OF' CONTENT§. (Continued) Chapt. Fundamental Group of the n-Sphere ••••••••••••• Degree of a Mapping ••••••••••••••••••••••••••• Construction of a Mapping of Arbitrary Degree • Veetor Fields " Singularities • • • • w • • • • • • • • • • Degree of a Vector Field •••••••• , ••••••••••••• Vector Fields of Exterior and Interior Normals to Sn •••••••••••••••••••• ~e" • • • • c • • • Brouwer Theorem for Sn (n Even) Brouwer Theorem or.. Tangentia.l Fields ••• ,. ............ lfopfis TheoI'em: E,(f) = S(g) imp:!..ies l' ':;' g .. cu.

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XIII HOMOLOGY AND COHOMOLOGY THEORY OF COMPACT SPACES Partia.lly Ordered Sets •• o.c • • • • • • • • • ~ • • • • • • • • • 202 Directed Sets .............. ~ ••• v • • • • • • • • • • • • • • • 202 Direot System of Groups ••••••••••••••••••••••• 203 Inverse System of Groups •••••••••••••••••••••• 203 Projection Mappings ••••••••••••••••••••••••••• 204 HI' and Hr as Direct and Inverse Systems •••••• ,. 205 Direct Limi~ of a Direct System of Groups ••••• 209 v r-DimensioneJ., (Oech) Cohomology Group.:! of X, a Compact Space, H1'(X} = lJ~H~(N(a)) •••• ~O~ 209 Strings •••••••• a , • • • • • • • • • ., • • • • • • • • • • • • • • • • • • • 2:09 Inve~se Limit of an Inverse System •••••••••••• 210 Hr(X) = .Jjm Hr(N(a.» .......................... 210 Cafina1 Sets •••••••••••••••••••••••••••••••••• 213 \/ Isomorphism of the eech and the Simplicial ' Groups £01' X = IKt •••••••••••••••••••••••••• 214 Alexander Co~mology Group ••• o • • • • • • • ~ • • • • • • • • 217 v Equivalence of the Cach and Alexander Groups for X Compact •••••••••••••••••••••••• 218

XIV

INDEX

Part I - Finite Dimen-

Urysohnfs Lemma ••••••••••••••••••••••••••••••• 219

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TABLE OF CONTENTS (Continued) Chap~.

Page

Tietze Extension Theorem ••••••••••••••••••••••• Degree of a Mapping (on Sn) ••••••• ~ •••••••••••• Properties of Degree ••••••••••••••••••••••••••• Rouchets Theorem ••••••••••••••• w • • • • • • • • • • • o • • • Admissible Mapping ~ ~ Degree of a Mapping .,o • • • • • •••••• e •• o •• 2,trong Form of Homotopy Invariance and R01:;.che fS Theore:;.n ;: •• eo . . . . . . . . . .

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Jordan Curve Theorem Invariance of Do:':nain I11dex ~*~ • • • ~4I'.".,.C! • • • • • • • • • • • • • Q.,.e • • • • • • Applications to Function 'I'heory ........ '

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233 235 235 237 240

LOO]'.L DEGREE J"ND INDEX (Part II - Banach Spaces,

-;[chaud~r-L~ray-~~Ciry)

Lemmas on Mappings in Rn ••••••••••••••••••••••• Admissible Mappil1gs ~ •••••••••• o • • • • • Completely ContiLl.2011S Mappings ........ Extentions of Contlnuous I:1applngs Allowable Ap}J~'oximations . 0 . 3 • • • • :> • • 0 . . . . . . . . . . . . Jordan-Brou't-"er .. Leray Theorem ,. .................. .,. ~ Invs-l'iance of Domain ••••• " .............. ~ 0

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ta and Addenda

The page ts given followed by the line. An asterisk means counting from the bottom of the page. ~'lhen the original text Is not given, insertions and corrections are underlined. f some x G X ••• ••• is ~n abelian group with respect to addition for which a product ax ~ V is defined whsnever •.. . .• in a normed l:inear ••. .. 31 should read 3 i whenever d(xo'x) < 6 l.u.b. d(fy,gx) • •• 0

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- 12-> a c Y. It is read IIX is included in ytl or Ux is a subsel of ytt. (-.-d) X:jY = L alacX or acYl is called the union of X and Y. XflY = [alacx and a8Y) is called the interl J ~ction of X and Y. e) Let Y eX. The set of element s in X that are not in Y is called the complement of Y with respect to X. It is written X - Y. f) Let Ct be a collection or family of sets. v,Te define the union and intersection of sets A in t Y, we shall mean a rule that assigns (uniquely) to each x ~X an element y IS Ye We shall denote the element y 15y fx or f (x). Note that we have said that X is mapped in:~£ Y. This means that not necessarily every y~ Y has the property that fx :::: y for some x ex. For the case where ~y y e Y is the image of some x Ii: X, we shall say that f maps X ~ y. For a subset At: X, we define: f (A) :::: {y Iy

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Y and y :::: fa for some

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1 r 1 f- (B) ==tx1x€.X and fx€:BS

Also [-l(y) :::: r- 1 ({Y}) ::::

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Using this language, f maps X ~ Y whenever f(X}

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Let x" Xl ~ X with x :f x' • If fx t:- fxt, then we say that f is a lI one -to-one" or 1-1 mapping of X into Y. In other words, for ye:.Y, f-ly is either a set ronsisting of one element or the empty set.

3. We may now state precisely what we mean by a finite or denumerably infinite set. a) S is a finite set if there is a 1-1 mapping of S onto the set-of natural numbers ~1,2, ••• ,n:rfor some n. b) S is said to be denumerably infinite if there is a 1-1 mapping of S onto the set of the natural numbers. By a countable set, we mean one that is either finite Or denwnberably infinite. Compos~ mappings: Let X, Y, Z be three arbitrary sets. Assume that we have two mappings: g: Y---ll>o Z f: X ~ Y, which we may write as

X~Y.J~Z Now fx6 Y and so gf'x£. Z. ~omposit~ mapping:

Thus we may define the

h=gf:X~Z

Algebraic Concepts: The following definitions and elementary facts will be needed in subsequent discussions. \'I!e assume that the reader is familiar with the concept of a group, in particular, an Abelian group. A ~ctor space V, (or linear space) over the real numbers in an Abelian group with respect to addition for which ax €: V whenever ~€.. V and a. is a real number" Also, for a.,{3 real numbers, and x,y€ V:

i) ii)

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(o.+{3)x = a.x + {3x

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The elements (vectors) Xl' x 2 , ••• ,xn , in V are said to be linea;r',ly de:l2~n~ent if there exists real numbers Yl' Y2'··.'Yn with at least one Yi

o.

y.x. ::: l

l

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0 such that

otherldse the vectors are said to be linear'ly

independent~

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A vector space has (alge'l?,.J?aic) dime~ion n if there exist n linearly independent vectors and if every n + 1 vec-

tors are linearly dependent. Every fini1J8 dimensional vector space with dimension n is iR0IDoX'phlc to the vector space whose elements are ordered n-tupl.es .. A linear map of an n-dimensional space into another ndimensional space is onto i.f and only if it is

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Scalar. Pl:2.duct s __8n~ .Hi.lbe~t Spa..£~~: L scalar P.2:9_duct in a vector space over the real numbers is a rule which assie;ns to every pair of elements x,y a r'eal nlli11ber denoted by (x, y) sat isfying the following properties: 1) (0,0)::: 0 (~:be first tl·!O appearances of 110" are as the null

elemen~

(vector), the third is as the real number 0)

f

2)

(x ~ x) > 0 H' x

3)

(yx,y) =,y(x,y) for y any real number.

4)

(x,y)::: {y,x}

5)

(x + y, ~) ::: (x;z) + (y,z)

0

As an immediate consequence of' these properties, we have the Sch1-Jarz inequality: (x,y)2

~ (x,x) (y,y)

A complete linear space in which a scalar prodl..lct is de.fined is called a Hilber~ Spa~. (A space is . 0 , x ~ y 3) d(x,y) = d(y,x) 4) d(x,y) < d(x,z) + d{z,y) 2)

Any linear space in which we have introduced a distance function d(x#y) satisfying 1) - 4) is called a ~etric space. Although every normed linear space is automatically a metric space, we cannot alwa,)Ts introduce a norm into a metric space which VIill Give us be.ck the original metric. For example, consider a space whose elements are infinite sequences of real numbers. We define

=L co

d(x#y)

1

That will satisfy all the metric space axioms, but an attempt to derive a norm from this will lead to a violation of the third norm axiom. Let S be a metric space. The diameter of S, written diam S is defined as dj.a.m S ::: 1 ",1.l1J b.

d tx>Y')

x,y,€,S

--- -

S is said to be bou.nded if and only if diam S < cD • S will be cD.lled totally bounded, if for every € > 0 # . S can be covered by a. fipite number of sets Si such that n diam Si < E for all L· That is, Set) 3i 'lrJhere dia.m. Si
0, there is a

o

> 0 such that d(fx o ' fx) < C whenever d(xo'x) < S (although X and Y are diffe}~ent metric spac6s, we have used the same

lid" to represent tbeir respective dist ance functions. This I!inconsistencyll in notation will be repeated, and nG doubt similar on~ as well, whenever there is no danger of confusion). f:X ~y is contihuous on X if it is continuous at every point in X. The

netri.~ Space ~ Let X,Y be metric spaces.

Y is bounded.

Consider the

0

space whose elements are the mappings f: X ---i>' Y. (If X has n elements and Y has ill elements, then there are evidantl;v

n?

maps of X into Y.

This suggests calling this space of functions duce a metric into yX by defining

yX).

We intro-

1.(,)

d{f,g) = lub

(fx,gx)

x€.X

Axioms 1) -3) are trivially satisfied. inequality, we note that

To prove the triangle

d(fx,gx) < d(fx, hx) + d(hx,gx) ~

is true for

~

d(f,h)

+

d(h,g)

X€X and consequently for

lub d(fx, g:x.) xE.X

= d(f,g) car~~n

PE.2...duct of Two Hetr ic Spaces:

--

Let; X,Y be metric spaces. The cartesian Product of --".---..X and X XY = Z, is a metric space whose elements are ordered pairs (x,y) such that x G:X and y€ Y. The distance between two points zl = (x11Yl) and z2 = (x2 ' Y2) is -..--,,-

r.,

Examp1!: Rnx Rm

= Rn+m

{up to isometry}

Topological Spaces We have already defined an open set in a metric space. In order to generalize this notion we start with a collection of sets which satisfy the propertie.s that we have found to be true for open sets in metric sps.ces. We define a topological space X to be a non-empty set of elements in which we distinguish certain subsets, called ~~en sets, having the follm..ring properties: i) O,X are open .ii} IfCl is a collection of open sets, then ~A is open.

11

iii)

If A,B'are open sets, then Af),B is open (the same s.t~roent for any finite number of open sets follows ..J.,mm.ediately by induction). Rema~k: A metric space is automatically a topological space~ Arbitrary unions of balls (open spheres) in a metric space are the open sets of the topology$ In general, we shall be interested only in a particular kind of topological space. A space is called Hausdorff if, in addition to satisfying the conditions required for it to be a gereral topological spacej it satisfies the so-called H~us~2Ef! sepaEa~ion ~xio~:

iv) Giv,en tHO distinct points x,y€ X" there exist open sets G, G' such that x€.G, y£G t and G(lG' o. Remark: A metric space is Hausdorff. Indeed,

=

x -:f y G

'::d;

d (x, y)

={zJd(Y,Z)


o.

G

= ~ Jd (x, z)


O.

But for j > some N, diam

1

Fj


about

....

-_. ---.-.........

... " .. -

• whose ra d ~us r l < E..;,.". Ie w~ t'a S '1 C G-1 0

'vJe can rn8.ke r l

8 1 C 3.

small enough so tlla t

Clearly 31 contains an ODen

sphere S~ with center at Xl-

Since

\. ~........

G2 == X, d(x l ,G 2 ) ;;: 0, so that thel'e is an x 2 E. G2 such that x 2 €

s~ • Now

- -,._

.......

../

... ....

-

.'

19 G2 is also open.

This gives us a closed sphere S.2CG2'

whose center is at x 2 and whose radius r2:
ly AC U G ,which means A is compact" °i 1

Tb.eorem.:

Let Y,' be compact.

Then f:X

~

Y is 1-1 onto, and

continuous -=) f is a homeowlorphism. Pr~f: vle must show that f- l is continuous, or that f is an open map.

Suppose G is an o}')en set of X.

Then X - G is

a closed subset of a compact space _ ) X - G is compact =)f:(X - G) is cornpact is open.

-..::> f(X

- G) is closed =-=)Y-f{X-G)

A Hausdorff space X is called sequentially co~~act if every sequence of elements in X contains a subsequence Hhich converges to some element in X. A Hausdorff space X is called conditionally sequentially compact if every sequence contains a subsequence which

C01"1-

'verges) although not necessarily to an element in X itself. Remark: In Rn , compe.ctness( ~,sequential compactness ie Heine Barel TheoremBolzano weierstrass Theorem. x is said to be an accu,,"llulation P2!nt of a set A, if every o)en set cont8.ining x contains a point of A other than x itself.

Incidentally~

x need not

oe

in A at all.

22

Theorem: If X is cO!l1p-:::,cl_aruL S~1ri-sl"ies the first countability axiom, then X is sequentially compact. Proof: Let ~ x n } be a sequence of points in X. i)

If xn

=x

for infinitely many n, (x,x ••• )

is a convergent subsequence. ii) If no element is repeated infinitely many times, then we may consider the subsequence obtained by removing those elements which are identical with elements that have appeared before. Let us denote this subsequence of distinct elements by ..fx ~. Nowfx ~ must have an accumulatl... n j ( nJ ion point. If not, X may be covered by a far-lily of op en sets such that each contains only a finite Dllil1ber of

{Go,-zj

the xn -

Go.

n

But X is compact, and so

xcU

Ga.. which implies

1

that some G

a.i

~

contains infinitely many x n co

Let p be an

L

8.~cumulation

:Joint of..fx.1. Every open set L loj contains P, also contains infinitely many of the

Xn

-

At p, there is a nested basis (first countability axiom),

call it [GiJ.

Letfx i

i be a subsequence of [Xn}chosen in

the obvious way, letting with i

I

j, etc~

~

=

some Xi ~ Gl ,

x2

= some Xj E G2

Clearly

After proving a few lemmas, we shall state and prove a partial conv.srse of the above theoI'em: In a metr'ic space, sequential compe.ctness ;> compactness_ !!emrn~: A sequentially compact metric space is complete. pro£!: Suppose that [xnS is a Cauchy sequence in a sequentially compact metric space X. Th_ some subsequence converges to an element x in X. Clearly, this implies that the original sequence converges to x. Indeed

23 d(x,xn ) The term d (XUi

;,lXn

= d(x,x~ni } + d(x ,x ) n ) < E for n, n i > some N since [Xnl is a

Cauchy sequence, and d(x,x

.

xn'i

~

x as n i

n.

\
some M since

~

~ 0, there are a

finite nUlY.ber of points aI' a 2 , ••• ,a n in X such that for any xE"X, d(xJa j ) <E for at least one a j • Let us assume that for some E > OJ we cannot find a finite number of points with the desired property. Then, choosing any point in X, call it Xl' there is a point x 2 in X such that d(x l ,x 2 ) > € , an x3 such that d(X3 ,xl ) > d(X3 ,X 2 ) >E and so ono fx n

e

and

1rJe have constrLlcted a sequence

3with the property that

d(x.,x.) >G'for i J.

J

f

It is

j.

quite evident that this sequence carmot contain a convergent subsequence, "\rJh.lch contradicts the assumption that X is sequentially compact. Theorem: In a metric space, sequential compactness

Sicom-

pactness.

~oo!:

G [Go.}

Let = be an open covering of X. show that G contains a finite subcovering.

We want to

To each point p E X, we [flay associate a re a1 positive number *r(p} such that the sphere of radius 'Y (p) about the point p is contained in some G. This can be done since a Go. is open and every II

CX

is in some Go.-

It will be shown

that there is a 13 > 0 such that y (p) > 13 for all p € X. Let us assume that no such 13 > 0 exists. This assumption leads to a sequence of points ~xi1 i11. X such that y(X i ) < viously Y(*i)

f.

tends to zero as i tends to infinity.

ObIt fol-

24 lows from the sequential compactness of the space that a subsequence of f~ i1 converges to some point Xc:o in X. But around Xro there is a sphere that lies entirely in some Ga. This sphere contains infinitely many of the xi' vihich contradicts the assumption that Y(x i ) ~ O. Therefore, a i3 > 0 exists such that y(p) > ~ > 0 for all p. X is a sequentially compact metric space :::::9' X is totally bounded ? X may be covered by a finite number of spheres of radius (3/2, Each such sphere lies entirely in some Ga.. The finite union of these Ga will cover X. S~ Th~£~~: Al\compact metric space is separable, SUDpose X is a compact metric space. Then X is totally bounded "X.J.nl,SUCh that if x is any point in X, d(x,xli ) < 1 for at lea~t one xli in the set. Si~ilarly

!!2~f:

there exists a finite set of points x2l,x22,···,x2n

2

such

that if x is an~T polnt of X~ then d(x,x 2 :t ) < ~ for some x 2i in the set, and so on.

In this manner we constru.ct countably many finite set:], Hhose union is also countable. This sequence is dense in X. For, 3:i..ven any x€ X and any E > 0, there is a positive integer N such that 0 < < ~, and con-

t

sequently a point x Ni of the sequence for Hhich d(x,Ni )


i

M

there is a positive inte",.'er II such

> 0,

.L (~ - S~)

2

and so on.


0 ~

n

This representat :10l1 is unique for if y

But

;>

I-Li-Ai

(i-Li-Ai) =

= O.

° and since the a i

= >-1=0

!-Liai then

are in general position,

The Ai are called the barycentric coordinates of

the point a We see that the si;rnplex is a bounded closed set; in RN and is therefore compact. Definitj.on: A p-dimensio:qal face of a simplex is a subset: p p {Y Iy Z AiXi' 2: Ai = 1 Ai =. 0, i ~ P < n; Ai = 0 for i >p}

=o

0

Hence a face is itself a simplex. An o-dimensional face is a vertex, a l-dimensional face is called an edge, Definition: Two simplices are said to be properly situat~~ if their intersection is either empty or a common face. We see i;rnmediately that two faces of a simplex al'e properly situated. Definition: An inner poi~~ of a simplex is one for which Ai > 0 for all i. Every point of a simplex is an inner point of a uniquely determined face.

33 Theorem: Every inner point of a simplex is a midpoint of a segment lying entirel JT vdthin the simplex. r Proof: Let x (!;A ::; l,a o" •• ,ar } Then x = Aia i and since C i=l x is not a vertex at least AiAj f 0, i ~ j. Choose e such that

'2:

Yl

Let

=x

+ f2{a i -a j )

e (a i -a j )

y2 = x -

Yl+Y2 2

Then

Y1 Y2 ~A and x:::

Theore8: a face o

No vertex can be the midpoint of a segment lying in

Proo.£.:

Assume a K =

•

~(YO+Yl)' YO'YI r

Yp

1

since Y1

1

i

~(AO

=~ 1.=0

p

Y2 there exists an i,j

i

1

j

j

6- A

= 0,1 i

+ Al) > 0, 2(a o + a 1 ) > O.

I-

j

i

j

such that AO > 0, AifO

But from the uniqueness

of the barycentric coordinates we must have

a K = Oan+···+laK+···oar which is a contradiction. Theorem: Proof:

The vertices of a simplex are uniquely determined. Il11l11ediate consequence of the two previous theorems.

The canonical n-simplex is the set of all x that admit representation in the form

~hition:

34 Then x = (AO •• ,An ) and we see that the barycentric coordinates AII ••• ,An are equal to the cartesian coordinates while AO

=1

n - ~.

~=r

Ai'

It is easy to see that any suaplex is

homeomorphic to a canonical simplex and hence any two simplioes of the same dimension are homeomorphic. In the special case of an N-simplex in RN the irmer points of a simplex are the interior points. The points which are not inner points form the set theoretical boundary or frontier. Theorem: The diameter of a simplex S is equal to the lenGth of its longest "edge,

.

Proof: Let x and y be Rny two points in the s,mplex. Assume x is not a vertex, and that diam S = d(x,y), Then x is the midpoint of a segment lying entirely within the simplex. Denote the ends of this segment by PI ,P2" Then sinco S is convex the segments yP 1 and yP 2 will be in S and since we are in

RN either d(xy} < d(yPl ) or d(xy) < d(yP 2 ). diam S.

Hence d(xy)
1ap

INI

onto

IKI

by assigning to the point l'

: : >-F:i

A. c.

~ ~

elK I.

It is

left to the reader to show that f is a homeo::norphism. Hence every realization of A is homeomorphic to the natural realization. Theorem.: An abstract complex A with vertices a O"" ,a K of dimension n has a geometric realization K in R2n +l , and the vertices may be prescribed arbitrarily close to any given set of k+l points in R 2n +1 • Proof: Given k+l points in R"c_n +1 we can move them into general position by an arbitrarily small disp1acement# Let cO,.~.,cK denote these points in general position. point c j •

2 =

If S ::: tso,ou,srJ is any simplex of A, denote by

f Go, ..

ponds to Sj.

Assign to a j the

o'Or}

the geometric simplex in which

OJ

corres-

It follows immediately that the resulting set,

K, of simplices of R 2n+1 satisfies the first condition in the definition of complex. To show that k satisfies the second condition let S ::: [so, ••• ,sr] S' ::: {s;,'••• ,s;

, T

1'

= rto, •• o,tp~be

simplices of A and

T' ::: {to, ••• , t;J be the corresponding geo-

metric simplices of K.

Let V ::: {v 0'.' .,

"t}

denote the set

of all points vJhich are vertices of either S,t or T'. Since dim A ::: n, r ~ n, p ~ n and therefore t ~ 2n+l. Hence V is a

38 simplex in R 2n+l and since Sf and T' are faces of V they are properly situated. ,l)efini t iOJ:?: .. _Let· '-a.O~· ......... ,a..r'·be··the·-ve!"t ia~g'6f Ha--c-omplex K and bO, ••• ,b s the vertices of a complex L. of K into L f: K

~

The simplical mapping

L

is given by f(a i ) == bj where v,Te require the mapping to be such that vertices which span a simplex in K map into vertices which span a simplex in L. Let x ~ IK! • Then x = 2,:.-A i a i where Ai ::: 0 except for the vertices of a simplex.

Then the 'ipiecewise fl linear mapping

determined by the vertex mapping is given by f(x) == ~Aif(ai). We call f a sinlplica1. lilapping of J.2.xaTI!p.~:

IK I

into

IL I.

Prove f is cant inuous.

Definit:'Lon: Two abstract con~lexes are called isomor9hic if there exists a 1-1 ma::>ping of the vertices Hhlch preserves simplices. Two geometric complexes are said to be isomorphic if their corresponding abstract complexes are isomorphic. Theorem: If 2 abstract complexes ai'e isomorphic, then their geometl'ic realizations are homegmorphic Proof: The proof is left as an exercise. If Land K are any two complexes,L is called a subdivision of K if

Defin~tio~:

1)

ILl

==

IK/

2) to any subcompl ex Kt of K ther e corresponds a subcomplex L' of L, such that /Lt I = !Kf I.

39 Chains and Cochains: Def~n~ti..~: An ordered simplex is a simplex whose vertices are given in a specific order. Notation: A = [aO, ••. ,ar ]. To every l' simplex there are (r+l) 1 ordered r-simplices. De£.19:.?-~i0t.:!:

Two ordered simplices are said to be similarly if one is an even permutation of the other.

~.!:!!~d

Definition: Two ordered simplices with the same vertices which are not Similarly oriented are said to be oppositely orient£~. Notation! If Sand 'r are oppositely oriented simplices we will write S = -T. Using similar orientation as an equivalence relation lrJe define an ?riented si~plex as an equivalence class of similar'ly oriented simplices. An oriented simolex will be d enot ed by giving one of its representatives in round brackets, i.e., (a O,a l ,a 2 ) is a representation of the class consisting of (a O,a l ,a 2 )p (a l ,a 2,a O) and (a 2 , aO,a l ). DefiE~ti~:

DefiI2l:~:

The support of an ordered (or oriented) simplex [aO, ••• ,a] (or (a.O, ••• ,a r )) is the geometric simplex -l

r .-)

(,. a. 0' • ar S Definition: An integral oriented r-chain, or simply an r-chain, over the complex Z is a functLm, x, which assigns an integer to every oriented r-simplex so ti.lat x(-Si) = -x(Si) (Le., chains are odd functions). If x(S.) = a. we denote the chain x as a linear combin~ ~ ation., ,?u.S., with the provision that if S~= ~ ~ ... -SJ"' then 0 •

,

-

a i = -a j

O

We extend the definition 0,1'. an for oriented k-simplices with kfr.

l'

chain by letting x = 0

Definition: Consider the set of all oriented simplices of a complex. Then a chain Xs where

3

=T

3

= -'r

otherwise is called an elementary chain. In standard notation an elementary chain is denoted by the oriented s ir,plex to which it assigns the value one. There is a 1-1 correspondence between oriented simplices and elementary chains. Theorem: Any chain can be written as a linear combination cf elementary chains" Proof: Let 31 " ",SK be oriented r-simplices such that Si

f

~ Sj for i ~ j, iGe., no two simplices have the same

support~

Then if x is any chain k

x

= --;,: :

i~

x(S.) x. l

S1

where tne sum of tlrJO cnains is defined as the sum of their functional valUes. The sum of 2 r-chains is an r-chain and we have that the r-chains over a complex K form an abelian group, denoted by Cr{K). In this group the chain t-Jhich assigns the value 0 to every simplex is the zero elenlent. define multiplic8.tion of a chain x = :>ais. by an -

l

integer ~ as ~x =,2:~aisi

Defini~i.2!!:

The boundary operator () is a mappinG of r-chains into (r-l)-chains C> : Cr (K) -'> Cr _1 (K)

such that () (x+y) = a.x + () (a xi = aOx

for an elementary chain C)( a x) = 0

Oy (a is an integer) for r = 0

41 and for r > 0 '"\

o(aO,a1,···,a r )

=~ i> (-1) i (a O,a1 ,···a1\ i ,···ar ) J.=o

. /'I:'

where a K denot'es the-·~rm--ti')-be·le-f't· out of- th-e-·--¢hain. We see that 0 is a linear opere.tor, i.e. a homomorpb.j.sm and hence comple te1y defined once the boundary of an elementar·y chain is giyen. Por example ()(aO"a l ) = a 1 -a O i.e. it is a chain which takes on the value 1 at a 1 and the value -1 at a Oo

C)

(a O,a 1 ,a 2 ) = (a 1 ,a 2 ) - (a O,a 2 ) + (a O,a 1 ).

Definition:

A

simplex

S

is said to enter in a chain x if

x(S) :J O. Definition: The support of a chain x, denoted by txl is the set theoretic union of all the simplices which enter in the chain. Ixl is compact and 1) Ia,x I = Ix I a + 0 2) !x+YI C Ixl u Iyl

3) loxl C

Ixl

2;d X;: a2x=o, where x is any Theorem: Poincar~.2:el~~ chain =0 Proof: 1) if x is an O-chain then J x = 0, dUx = " = O-chain and therefore 2} if x. is a I-chain then dx adx = o 3) Since ~. is linear it is only necessary to prove that

uo

02

of

In thlssum each term appears' twice, once when the i th term . is dropped first, and again when the jth ter!Q is dropped first. In the former case the coefficient is (_l)i (_l)j-l since after the i th term is dropped the j th term becomes the j_l.st term, and in the latter case the coefficient is (_l)i(_l)j. i . -1 . j Since (-1) (~l)J + (-l)J(-l) 0, the terms cancel each others

=

Definition% A chain x is said to be a boundary, or homologous to....Q, x !"V 0, if x : .: dyo Definition: A C;h~~,x x Ny, if (x-y) tVO .. Definition:

13 ~!i.d

to be homologous to a chain y,

A chs.in x is called a ~yq~ if C)x = o.

°

The sum of tltJO c;{'..;les is a cycle: Q)x = 0, dy = implies that () (x+y) = d x + (J y = O. Therefore the r-cycles form a subgroup Z (K) of the group of chains over K. I' Every boundary is a cycle. The SUIIl of two boundaries is a boundary: x = d y , x, (;)y' implies that x + Xi ;;y +~T= d(Y+Y')4 Hence the bounds.ries form a subgroup Br(K) of the group of chains over Kc Thus:

=

Definition: comp~~

=

The r ... dimensional integral horilology group of a

K is the factor Group

The elements of H {K} are homology classes of cycles, I' where t1/lJocycles are called homologous if their d:i.fference is a boundary. Definition:

Let Sand T be two oriented simplices and define

their scalar product

43 S = T S = -T

otherwise

Definition: The scalar product of tHO r-chains (x,y) is defined by the following properties 1)

(x+x t , y) = (x"y) + (x'"y)

2}

(x,y)

:::

(y,x)

3)

(ax,y)

:::

a(x,y}

4)

(x,x) >

o

if x

f o.

Choose a set of oriented r-simplices LSi ~ such that no two of them have the same support. If x,y are two chains, x

=? ~ J.

aiS., y == J.

>

f)-;S-; then (x,y) =). aJl.; and hence we ...- J. ~ J.

--,-... ... J.

J.

haVe an orthonormal system for oriented simplices.

The .£.C?£2.~ndary oper'ato!:, CI"'\

and ar- 1 columns.

r - \ r-l = ~ c.. r ~ ~ iJ· J,K

i - LIS i· (/ S . j J J

Since r. 'It:;;

r-l Sr-2 J·K

K

we have E r E r-l

=0

The coboundary of an elementary r-chain is given by

We

by

45 where (eI j ) is called the r-co~nc~~ence matrix. ( G>~~S:: l'

:r

sr+l);:: ~ e r

K

ij

(Sr+l j

,

Now,

Sr+l)

K

;:: e.r K s 1.' nee (Sr+l " 1. J

1 ) = > '-.1 r 1

_ ~()i( 1\ ) -..c:.:::... -1 a O"·" .a i ,.· oar i

f

-

i

h.

'> (-1) (faO, ••• fa., ••• fa r ) 0 (a 0' ~ • • "ar ) ::: ........ 1 1

case ii)

Two of the fa i coincide while all the others are distinct. Without 10ss of generality we can assume that faO ::: fal f ( a 0" •• ~ a r ) ::: 0

c)f ::: 0 "'l --i A C/(aO, ••• ,ar ) ::: ~(-~) (aO,···a., ••• a ) 1 1 r

f(;)(a o'· ",sr) = (fal,···fa r ) - (fa o,fe'2,···fa r ) since faO

= fal

definition.

terms i,n which fa O and fal a',?pear are zero by

Hence f

cJ (a O' . ' . ,ar ) ::: O.

Case i3, i) If any 3 or XlJ.ore of the fa i are equal then it is easy to see that everything is zero. ~'Je define a hOlTIol"llor-phism f-l:- of r cochains over L into r-cochains over K f~~r(L) _->Cr(K)

by requiring that if xQCr{F::), t;E:,Cr(L) then

f~tx ::: ~fx where

fx~ C r

Theorem: Proof:

(L)

aOa l = bCT(a1

- ao)

Since a 1 -a O is a O-chain b

Cd x = b(al

- a O)

= (aOb)

+ (ba l )

56 By induction on m we define the mapping

by 1)

C;'nlx

:::; x if x is a O... chain

tl~ansposed

\ille shall call the (!m·:~

er' «(j InK)G, I')

cr m

~x

homomorphism of cochains

~

er

::

S (,J'mx

(k, G)

• • . a super d ~Y..l. ..~.?:.£E.. Theo~:

(1) The subdivision operator commutes t-vith the boundary operator .. c;:) (T fiX = () me) x. (2) The superdivlsion operator commutes with the cobounda.ry operator, d-l~ Om':4 = (j'm* (')~s. ~ro0f.:

We shall give a proof of 1) and leave it to the reader to

fill in the details for 2). Let m = 1 If x is a O-chain d (jx = c)x = 0

u Dx = vo

: ;: 0

If x is a I-chain x = ()x = (aob) + (ba l )

c)

Ux

The proof will be done by induction.

(ao'~)

= b - a o + a 1 - b = a 1 - a o = () x ::; (/c)x

Assume the theorem true for r-l chains r > 1. r-chain then o-'x ;::: b (lVc,lX Using tbe forrrllla for the cone we have that

Let x be an

UU.x = 6" Ox - b f) O....... t0 x where now c)x is an r-l chrl.in and hence by our induction hypothesis

c)vx = v-ox and since C)2;x. :: 0, ClO

=0

b

(JUcJx

and bO :::; 0

du;x. =u G. ,G. ,9'.' ,G. . l ~l ~2 ~+l

xQX is in

o

called an ~

I

but for !J;J..l subsets fG. ,G i ,0 L ~1 2

some point of the Gi

of G,

,I +2 •• ,G .. JI. G.

r)

~i:.+2 ' 1

~j

= o.

f1em~:r'k:

Al though only ~.!.~ cuverings vlere defi::.1ed, we might just as well have considered infinite coverings, since all spaces to be considered (for the time being) wi11 be compact.

ii

Defi£itio£.: SU1?Pose that = ~ Ga-~ and )J :;::tHp?; aye both cov~I t J ~ - Q er ings of X. .n is s8.id to be a Eefir!eme'p~ of 11 if every H{3 is contained in BCmE. Go!

For the major part of tb.is cUscuss ion of Dimension 'l'heory, we shall assume that the space X is COD~act and metrizable (and consequently se~arably metrizable o Chapter I, p. ). Two defin.itions of top r.Jlogica1 dlmenslon VJ]:d.ch Hill be shown to be equivalent for crn!~act metric spaces, will be presented, At the end of the discussion, a few i,-rords vJi11 be said about separable (non-compact) metric spaces, and the dimension of such a space will be given in terms of the di~ension of its "compactificationl!. Definition: The (topological) di;.Wens:b0n of X, written dim Y..., is not greater than n if every open covering has a refinement of order not greater than n. Dim X = n if dim X < n, but dim X is not::::. n-l; that is, there is an open covering of X wh:i.ch has no refinement whose order is less than n.

60 If no such positive integer n exists" then t-Je write dim X co •

=

-

It is easily seen that dimension is a topological invariant" Indeed, the definition involves only £PE!~ ooverings, and open sets are preserved under homeomorphisms. ~~:

Topological dimension is an absolute property of a set. 'rhat is, for ACX, dim A n regardless of our choice of coverings by sets open in X, or sets open in the induced topology on A.

=

~f.:

Suppose that ACGI UG 2 U

X then[A()G1 ,Af\G 2 , ••• ,A()Gk,5 ing of A by sets GA

"". UGk ,

where each Gi is open

= [GA1,GA2, ••• ,GAk1 is

a cover-

Let dim A < n mean

which are open in A.

i

that there is a refinement [HI, ••• ,H.t} of [Gl, ••• ,Gk3whOSe order is not greater thBn n. a refinement of [ GAl""'. n.

Clearly {Ar)Ul, ••• ,A()H e ] is

,G~} whose

order is not greater than

Conver'sely it is easy to see that if din! A

~

n in the in-

duced topology, then dim A ~ n using sets open in X. As an immediate consequence of the definition: i) dim 0 = -1 (Given any open cover.ing, ta.ke the covering consisting pf no sets as a refinement. The order of this covering is -I). ii) dim fx}= 0 if x is a point (from a given covering, choose a refinement consistin~ of any single set of the given covering which contains x). Theorem:

Let X,Y be co.mpact.

Then XCY

~

dim X .:: dim Y.

Proof: Suppose that XCG1 U G2 U ••• UGr where each Gi is open in Y. X is a compact subset of a compact space =) X is

Y-Xl

closed:::::;> Y-X is open ::::::;:>{G1 "" .,Gr , is an open ooveriDg of Y. It dIm Y;=::2l,then{G1P ..... ,Gl',Y-X}has a refinement of order ~ n. Let [HI, ••• ,Hs} denote S~Ch a refinement. Consider the suboollection il , ••• lk ) of this refinement,

[H

,H

61 for which we have thrown away all Hi which are contained in Y-X.

Clearly

i Hil , •• • ,Hik Sis

a ref'inement of

f Gl , • • • ,Gr ~

whose order is < n. In order to present the second deflnition of topological. dimension, it is necessary to first prove a lemma, ~:

Suppose that X (compact metric space) has an open covering. Then there exists a 0 > 0 such that if i)ACX, ii) dlam A < 0, then AeG a for some Ga.in the coveringo (5 = lub 0 is the Lebesque numb'er of the cavering.. p.rq 0 a will be called a. Lebesque l1iitrlber) 0 Proof: vJe must show ths.t there exists an r > 0 such that if xEX then Sr{x) is contained in some Ga. We may then take 0=2r.

:s

Assume that no such I' > 0 exists. Then for r I -- 1/2, there is an Xl€- X such that Sl/2(xl ) is not contained in any Ga - For r2

= 1/22 ,

there is an x 2 €-X such that S,,\_2(X 2 ) is not cant::

tained in any Go.- Similarly for any positive integer n there is an xn such that S n(x ) is not in a.ny Gr , . This defines a 2....... n "" sequence [xi} in y.~ In a metric space, compactness implies sequential compactness. 'rherefore a subsequence {Xi .-~ converges to a point Xb X.

Now

fGa.~ covers

J

Xi so x is in some Ga.

Fur-

thermore this Go. is open; hence there exists an r>O such that S (x) CO , contrad.icting our assumption. I' a 4 Eefinition: Suppose that )1 = a is an open covering of X. We shall say that the £2~ ~ove~g ha~ diameter d, written

fG 1

diam b~_~" if lub(diam Go.} a

= d.

Definitio~1:

Let X be a compact metric space. Then dim X < n, if for e';;ry ~ > 0 there is a i'lnite open covering of XI such that diam =:. ~ , and the order of is ~ n. As before, dim X n if i) dim X ;: n, ii) ther, is some (;Z > 0 for which there is no finit.e open covering!f, where

It

diam

!J.:::. G

ft

=

and order of

/J :l.s

~

nr.I.

fo

62 Theorem: The two definitions of (topoloGical) dimension are equivalent. Pr~f:

Suppose that dim X < Consider the open covering

h

n~

using the first definition. = tS--E/2{X)J where xEX. Since

X is compact, there exists a finite subcovering

It'

=

f s ~/2{xl)'·"'S e/2(Xr )J. it

Using the triangle in-

equality, diam t ::: diam j} ~ E. Since dim X ::: n, there is a refinement)) = of //, v1hose order is < n. Finally each Hi C some S ~ /2 (X j ) --? diam Hi ::: diam S E./2 (x j ) .::: C:,

tHI" .. ,HsJ

satisfying the conditions of the second definition. ii) Now sup~ose that dim X ::: n, according to the second definition. Let jJ be any open covering of X. It has a Lebesql.le number 0, we havEl found a fini te open covering /; :; tst (a~) such that diam ~ E- and the order of

J

~iS

n.

But this means that dim X

~

§

n.

ii) lATe shall show that all open coverings of IK I by sets of arbitrarily small diameter, have order not less than n. Suppose that Gl , ••• ,Gl is an open covering of II~I such that diam Gi ~ € where (/t,' is less than the Lebesque number of the ' covering fst(ai>j of ,IKI. The covering!t has a nerve L, and a Lebesque number 5. A positive integer m may be chosen so that

i

"l

We shall construct two simplicial maps, tl: that

rY

~

G'~ ~L

t2: L -..,. K in such a w~y

,

= t2tl is a Sperner mapping.

First let us define the vertex mapping tl: a-mK---!>L in the following way: Suppose that c is a vertex of vIDK. Then c(2st(c}CG i for some i, since m was chosen so that diam (st (c» < 5. live define This defines a vertex mapping of (/MK into L. Does tl map a simplex of O'lilK into a simplex of' L? Suppose that c i , ••• ,c i 0

1p

. 1 ex span a S J.mp follows that

J.' n

vr.-mK •

Then f) st (c i )

o

:f

p

0, from which it

j

p

() where st(c J,) C Gi • j

°But then gi

#gl , •• "Si

o

1

span a simplex

p

in L, proving that tl is a simplicial mapping of

u-mx

into L.

67 Now let us define the vertex mapping

in such a way that

= ai

tZg i

where a i is a vertex of K and Gi cst(t 2 gi ) = st(ai)(Recall thgt diam Gi < E which is a Lebesque number of [st(a i )} .) To show that this is a simplicial mapping, we must look at the image of a set of vertices of a simplex in L. gi , ••• ,gispan a simplex in L~ ~

a i I.'. ,a i

o

p

p

no

p

G. J. j

f

0

0

-7 . n0 St{t 2g i j ) Y. 0

~

span a simplex. in Ko

r.--1= t2tl is the product of two simplicial mappings, and

therefore is itself a simplicial mappinge It is easily seen that 'i': 0"I11:K .-,. K is a Sperner map:

{C)€? Gi

V n1K -=) cG-st(c)CG i where tlc = gi"

CSt (t 2 g i )

= st (t 2 t l c) = st( 'L c).

Furthermore

Therefore

c ~ st ('t' c) ..."

,

showing that i really is a Sperner map, In Chapter II we discussed the manping 'i":Cr( umK)

--!>

Cr(K)

(where Cr (O""mK ), Cr (K) are int.egral chains over (/~ and K ' respectively.) for r=O,l, ••• ,n. Since alga dim K = n, K has at least one n-simplex and consequently at least one elementary n-chain, which we shall denote by x (aO, ••• ,an ). By Sp erner s Lemm.a,

=

'LO"'mx = x This says that there is a chain

68 C"

,....."

such that /..y = x. there is a chain

But x = l Y = t 2 (t l y)

tly such that t2z

= x,

Hhich means

= zG-Cn(L)

= x 1 o.

Since

is a homomor'phisrrl, z f: O. Thus there is an n-chain over. L, and consequently an n-simplex in L, proving that the alg. dim L > n. But L is a geometric realization of the nerve of the co;ering 1.. i ~ , so that the order of i ~ is not less than n, which coru.ple te-s the theorem. To satisfy the m.Ore impatient reader p VIe might pause here and pr evie'~l some of -ell.e coming events of the chapter. First we shall define the concept of an f -mapping, and then prove a theorem (Theorem II) which char'acterizes the dimension of a compact metric space in terms of such a mapping.. If the notion

G

?G

of an ~-mapping, or even Theorem II itself, does not appear to be intrinsically interesting, it is at least necessary to introduce both in order to prove the main results of this chapter, which will be; Let X be a cOIDDact subset of R. Then . n o 0 X i) dim X n if X 1 0 (X is the interior of 0) o i1) dim X < n if X 0

III: ..Theorem .

=

=

Theorem IV: A compact metric sr.)ace of dimension not greater than n is homeomorphic to a subset of the unit cube in R2n +1 • Definition: i)

¢:X ~ Y is cS.lled an E -mapping, if

¢

is continuous on X

i1) d iam

¢-1 Y
0, there is an E -mapping f, which maps X into the space of a complex Hhose dimension is not gre ater than ne Theo~em

II:

,~

..

'

-pp.o.oi' ~ -fruppos-e that dim X < n.

ite

~pen

covering

,f;

Then-:ror-&>-O, X has a f'in=fG1,: •• ,Gr ] such that

i) diam

$:: e

ii) order of iii) Gin X

:f

f

X

is not greater than n

for any i

Let L be a geometric realization of the nerve of We are going to construct a particular f:X ~

.fJ,

with

e -mapping

ILl

which is known as the cF.~nonical mappin~ of X into We first define the r real-valued functions

IL I

The,Xi have the following properties: i) ,/1,.,1 . ./ is continuous on X ii)

Since each x

ex

=

° for

xQX-G~.....

is in at least one Gi , the function

° and continuous for Consider the function

all x EX.

70 The ~ have the following properties: i) ~i is continuous on X ii) ~i = 0

for xl2 X-G i

iii) 4i > 0 for xQ Gi r

iv)

L1 4.J..

= 1

(The r' funct ions 4icons t i tute a part it ion of unity subordinate to the covering

1;.)

We shall nov.T produce a candidate for the desired G -mapping of X into IL I" To this end we define the function r

fx ==

L1 4i (x)gi

f is certainly continuous on Xo We know that the order of /; is < n and so the alg~ dim L :::! n. What we want to show is that f'x € the sum"

IL I.

For a par·ticular x~ X, omit all 4i (x) This gives us: k

fx =

2. k

f'x G:

IL I

so if

J

each

4i

J

. (x) == 1.

It is now clear that

J

if g. ,gi , ••• ,g. span a simplex in L. II 2 l.k

X G. 1

lj

q.lj (x)

-1

0.

> 0.

from

~i. (x)gi.

j==l

where each ~i. (x) > 0, ~ J J=l

=0

But we 8.lready know that x

e

This will be

0 1

G.

lj

since

Therefore f is a continuous mapping of X

where the alg. dim L < n. Is f an ~ m8.pping? Suppose that for x,x 1 X, fx=fx l =y ~ IL I. y is an inner point of a uniquely determined simplex in L, into

ILl

e

say [gl, ••• ,gm~. Then x,xt~Gi for i that d(x,x') :::! diam Gi :::

e .

= l, ••• ,m

which implies

71 'l'1ms we have shown that if' dim X < n: then for every

e

> 0, there is an G -mapping

f:X -~

ILl

where L is a geometric realization of the nerve of a covering of X stl.ch that. diam ~ & and order of is < n. Naturally dim L < n too. Conversely, we now assume that for every G > 0 there is .,;-. an ~ -m8.pp lng

/J

f;

1;

where dim IKI :; n. We vJant to shovJ that dim X < let us prove the following

n.

First

Let ~ be an Q -lilapping~ Then there exists a 6 > 0 ;~that diam q,-l(A) < G whenever diam A < 6. Lemma:

Proof: If we assume that no such 6 > 0 exists, then there is a sequence of sets siC'IKI such that lim (diam Si) = 0 i->co while

for all i. 'rhis means that there is a double sequence of points in X, x~ ,x~ and a corresponding double sequence in IKI,

i 4Q, j

~xi}

3

t

yi]SUCh

=tYl,

that

lim d(Yl' Y~) = 0 i->co while

E.1. where lim co

i~

~

i

=0

Since X is compact, lim i~ co

xl

=

x'

lim i~ co

=

II

X

72 lilrom the continuity of ~,

exist.

lim i..,..

00

But we also know that .,

til)

lim d'Yi lY i

= 0

1->00

=

Therefore ~xt 4x", which means there is a point y ~.xl 'x" in IK I for which

=

=

diam ~-ly > ~

contradicting our assumption that ~ is an ~-mapping This proves the le~na. Given any E=> 0, we shall construct a finite open covering )Iof X, such that diam 'J/ e a:J. d the order of is not greater than n (second definition of dimension). Let ~ be an e-mapping of X into IKI where dim IKI;£ n.

tI

=:

From the lemma, there is a 5 > 0 such that whenever l (A) < ~ • diam A < 5 for A elKl, then d1am Let 11 •• D,Grl be an open ,cQ"t.;rering of IK I such that diam /;::: 5 and the order of is ~ no Set

4-

/j = fG

/J

The following properties are satisfied i} H. is open (~ is continuous) ~

ii} xC

r

0

Hi

1

( I KIe

r.

IJ Gi) 1

iii) diam Hi :: E. (Lemma) iv) order of

fli ::;: [HI' ..·.,~3

order of )J were grea tel" than n tl·~en some

.

n+2

n 1

Gi

~ O. j

But the order

is ::. n

~2

(If the

t · 'J1 Hi j ~Q.. impJ.ying that of )j is =: n by cons£ruction)

This completes the proof of Irheorern II

73 Theorem III:

-

---

Let X be a compact subset of R. n

0

i) dim X

=:

if X

n

0

1i) dim X < n

0

if X =I 0

0

Proof: i) Suppose that X :f O. -

o

"I

Then

Then there 1s a f> >

o

and a

0

point as X such that So (a)c..XCX.

In other words, about the

o

point a in X, there is a sphere of radius 0 lying entirely in

~.

Consider a set of points £a,bl, ••• ,bn~ with the property d(a,b i ) < 6/2~ By an arbitrarily small displacement, these

points may be bro 11ght into general position while still remaining in 8 6 (a). The displaced points 'E,blP .. ,bn span an n-simplex 6'. Therefore

G'c s ~ (a) C Xo C XC Rn and alg. dim u- = dim 10 i < dim X But a compact subset of Rn is bounded. Therefore X is contained in some complex K. Since K is a comp~ex in Rn , we have dim X < dim IKI =: alg. dim K < n n

=:

Therefore dim X =: n~ o ii} To ShDW that dim X < n if X =: 0, we shall construct an ~ -mapping and use Tbeorem II. v'Je already know that since X is compact, and hence bounded, it must lie in same n-simplex in Rn' call it K. For f > 0, consider the roth barycentric subdivision of K, G/mK, where mesh

C-lnI(

< ~

Letting L denote the n-l skeleton of crIDK (L consists of all simplices in o-mK whose dimension is not greater then n-l) we define f: X ~

as follows

ILl

71-!-

a} fx = x if x ~ IL/ b) If x $ 111, then x is an inner point of a unique n-simplex in V-mK. There is a point y, in this sir..iplex o -\ such that yq X (if not X -:J 0). Project the / " , point x through y to the bowldary of the simplex which is contained in IL I. Denoting ! >( / \ this projection on the boundary by z, we take // \.~ fx z. (See diagram for n 2). It is not I \ / \, \ difficult to see that f is continuous. And 1- _ _ .. ~~., since mesh (i'mK < G/2, f is an E: -mapping of X into the spa ce of an n-l COlilplex L. From theorem II, dim X < n. The definitions and lemmas that follow are needed to prove that every compact metric space of dimension < n is homeomorphic to a subset of the unit cube in R2n +l • Actually the theorem holds for non-compact separable metric spaces, but this more general result will not be proved here. The reader miCht take note of the fact that lemmas 2 and 4 will set the stage for the application of the Baire Category Theorem (See Chapter I)n Definition: The unit cu0e I in R2n +l is the set of points "

\

/

,

'3

=

=

_._._.------..~i.

{yJ = {(Yl ,··· 'Y2n+l)] s::ch that

0 < Yi < 1.

Definition: Let IX denote th.e metric space of continuous mappings of X into I. For f, ge IX we define d (f , g) = 111b d ( f

x(CX

x J gx )

(See Ghapter I) Lemma l~ IX is complete. Proof! Suppose that [fn is a Ce.uchy sequence in IX. any point x E X d(.fn(x), fm(x) ~ d(fn,i'm)

!

F'or

so that for a fixed x, ['f n (x).s is a Cauchy sequence in R2n +l , which is complete. Therefore lim l1-!!>

co

75 In addition, convergence is uniform, (see first inequality) so that the function

°

is continuous. Finally, ~ f(x)~ 1, which results from the fact that 0 < f (x) < 1 for all n. Therefore f E IX and n lim d(fn,f) 0, proving that, every Cauchy sequence in IX converges to an element in IX.

=

°

Lemma 2: For a fixed ~ > let Ce denote the set of all f -r!lappings of X into I. Then C 0 such tL.tI.t d(fx f ,fX"} < E whenever d(x l ,x"} < 5. No·w let f E Cf: e Choose any g e IX for which d{f"g) < 0/2. Suppose g(x1 ) = g(x 2 ) = y G I. Then d(fxl'Y)~ d(f,g) < 5/2 d(fx2'Y)~ d(f,g) < 6/2

Usir.:.g the triang1e lnequality: d(fx l ,fx 2 ) < 6, from which it follo't·!s that d (Xl ,x 2 ) < e • Therefore g is an f: -mappj.ng. This means that for every f Q Co , there is a 5'(6' = 6/2) such tha t g E C~ whenever "-" d(f,g) < o~

-

Lemma..1: A continuous rna~)ping of e. compact metric space into a metric space is unifo!'mly conthluouS. Let X, Y be metric spaces, X being compact.

tro~;::

Suppose

f :X~ Y is continuous. Take a point Xo G X. Consider S5(x o }' where 0 > 0 is chosen so that d(fx,fx o ) < E whenever d(x,x o ) 0, there is an -mapping g such that d(f,g) < ~ t. To simplify the not-

e

ation let u,s use ~ to denote a pO,sitive number not greater than either the original (2 or fr I .

76 Then our task is to find an -mapping g such t;f1.at d(f ,g) < r;' • Since dim X -!: n there is an open covering l:f = [ Gl , ••• ,Gr whose order is < n and whose d5.a'Tleter can be made arbitrarily small. v-Ie shall requh"e that diam /:; < .; where s is chosen in the following way:

3

i)

s<E

ii) d(fxl ,fx 2 ) < e/2 whenever d(xl ,x 2 } < .; (uniform continuity). For each i, let xi be a fixed point in Gi • Then fX i = Yi GI. The set of points tYl"'.'Yr3 mayor may not be in general position. By an arbitrarily small displacement, we may bring them into general position in such a way that the "displaced!! points, which we denote by tYl' ... • ,yrJ, remain in I. vJe require

To be sure that the Yi remain in I, we move the points Yl'." ,yr , into general position by displacements not greater than ~/4. If some of the points lie outside, we may perform a contraction mapping on these points if.Jhich will "pull them into I" in such a way that no point is moved by more than a distance e /4 by this contraction. At worst a coordinate of 12 E one of these points may be either 1 + ~ or - 1r. Then we must find the number

t,

such that

V(l

(]

+

e -"21) = -4

(1

1 "2 + E) Ij: ="'2

If Yl' •• "Yr are in general position, but lie outside of I, then "'Je per .form the contraction mapp ing given by '(/ (Yi -~)

= ~i

-

~

The poi-n,ts Yl' •• ' 'Yr are still in general position and lie inside I. Let L denote the realization in ~2n+l o~ the nerve of The vertices of L are Yl ,··· ,Yr , (or Yl' •• ' iYr.a if we had to perform the contraction mapping), and Yi ' ••• 'Yi will span a s I s simplex in L if Gi 1: o. j=l s

/J •

n

77 ~Je

shall construct the canonical map g: X ~

ILl

in the same wa.y ae··-was. do-ne-in--rrheorem II, taking gx ;;::

L

~i (x)Yi

ELis an G :~:!.8~: For, assume that gX1 = gx 2 " Then x l ,x 2 are both in some Gi , since 41 (X j ) ~ 0 X j cG i for j ;;:: 1,2. To show that d(f ,g) < E: , take any x EX.

II

d(gx,f'x) ;;::

e:x-fJ:i!;;::

if.::

L:

~i(X)~\-f(x)

~(t'i (x) II Yi~'fxll.:: .2- 4i (x) [11 Y-Yill toJhere x

II

Qi(x}"~

+

II

+

~

Yi-fXIl} =

£. Gi

2:¢i (x) since

Then

f/I

Yi-Yi

ll


0, there is an open covering [; of X,' such that diam < G and order of is ~ n. ---~

A

!J

The dimension of the empty set is .:1.. dim X ::. n at a point xe X if there are arbitrarily small neighborhoods of x "Jhose boundaries have dimension at most n-l. A space X has at most dimension n if dim X ~ n at all xt'.. X. dim X=n if dim X ~ n, but dim X ~ n-l is false. That is, there is a point x~ X at which the dimension is not < n-l. 11Ie shall use a third definition which involves the notion of compacti~ication. prysor~~ger deZin~tion:

79 Definition: A is called a cOIDPactification of X if . .i) A is compact ii) AOY where Y is homeomorphic to X. F'rom the Ur~Tsoh1Lk!etrization Theo~ (Chapter I, p. 27) every separable metric space is homeomorphic to a subset of the fundamentql cube in the separable Hilbert space. It has been shown that the fundamental cube is compact. Thus every ~ep­ ~able metric sp~e_~a~ a compactification. Definition: The dime~sion of a separable metric space is the smallest possible dimension of a compactification. It can be shown that these three definitions of dimension are equivalent in separable metric spaces. Using the third definition;) we shall prove the monotonicity of dimension (already shown for compact metric $paces) and that dim Rn=n. Let A, B be separable metric spaces, A B.

Theorem:

Then

dim A < di111 B. Proof: Let C be a compactification of B. Then since B is a closed subset of a compact space, it too is compact. Thus

-

ACBCBCC so that B is a compactification of A. dim

A
OJ S~(a) C Ao

Clearly 3 r (a) contairls an n-simplex S "t,rho se barycents:r is a. We already know that a 1-1 continuous mapping of an n-simplex into R is a homcomorphlsm. n Let 11 cenote the boundary of S. Diagram 1 Now .i).. and (2.} are both compact, disjoi~t subsets of R , and hence f(a) and f(Sl) are both n compact (and therefore closed) and di s joint" Thus ther" is an n-simp:,.·:';x So; with f(a) as its barycen!:;er and .J..

If all the points of Sl are images cf points in S we are through. ]1,::-: Sl1:..1'1e the::6 is a point

C

Diagr'am 2 G 8 1 which is not the image of

any POl:::.lt in S. StJ.b.di vide S = [ao$ ••• ,an1 once. Let St(a i ) ~enote the star of the vertex ai in a'F;,. vIe see easily that

n n

st(a;)

=

Dj,8gram

~

Let Fi = f(St(ai»1) [llJ_-w..~re· fl1 is the boundary of Sl" Clearly [Fi} is a colle-etlon of n+1 closed sets, and since the . map- is l-~-.and a 1s not mapped into ill

n

Ilo Fi = 0

•

Let 5 be a Lebesgue num"Ger. Qf the collection fFt}- We may construct a covering !j{ = {Ho, .... ,HLJ of fil where diam Hi ply theorem 1 to

I

1\

~i

= 4i

-

"'2.

i-

That is

and the hypothesis of theorem 1 is satisfied. ~!xed Poi~t

Theorem for Banach Spaces vJe sh~ll extend the Brouwer fixed point theorem to Banach spaces, i.e., complete normed vector spaces. The theorem one gets is the Birkhoff-Kellog-Schauder theorem. This appears in a weak. form due to Schauder$ and a strong from due to Tv'Iazur. ir,Je shall derive both forms.

Lenllna l: Let B be a Banach space, S a convex, compact subset. Then for all e > 0 there exists a continuous map)ing, h, of S into a finite dimensional subset, T J of S, such that Ilh(x) - xII < e. Proof: Since S is compact we can find a finite number of points xl, ••• ,xr in S such that every point in S will have distance < e from at least one of the XiLet~(t) be any continuous function such that

vd (t)

r

, = to

t

~i

'> ="f(:i~). \,.::

Form \j!i(x) Let ~i(x)

=-

*i(x)

.

4:W j (x) J

.

0

->1

0 < t < 1

•

-

Then

>"

T

'¥ i (x)

>

0 on S.

Then ~i{x) is continuous on S,

94 ~i(x)

=0

if d(x,x i ) ~ e and ~ ~i(x)

=1

~

Now let 'r be the convex hull of [xl',.,. .•• ~~l. since S is convex, TC S. Define r

,hex) =

fg'=1

Then

4· (x)x i • J,

Then h(x)eT, and

~

>-

~i (x) Ilx ...x.1J<e ~

II xi-xli

< e

hex) is continuous since it is the sum. of continuous functions, and hence our construction is complete. Theorem: .§chauderts Therorm (weak form). Let B'be a Ban&ch Space~ S a. convex compact subset of B, and F a continuous mapping of S into itself. Then F has a fixed point. Proof: By the previous lemma, there exists finite dimensional subspaces Tn of B and continuous mappings h n such that n==l, •.• ,k and

1/ h n (x)-xl/ -< 1 n Consider Fn (x) ping of S into Tn.

= hn [F'(x)].

Then Fn is. a continuous mapConsider the reC&RstrJction of Fn to TnnS

By Brouwer',s fixed point theorem each Fn has a fixed point, say xn; i.e., Fn(xn ) ::: xn" Now

II F(x)-Fn (x) II < 1.n and as n

~

a:>

This holds in particular for xn' and hence

Since S is compact the sequence (xn ) has a convergent subsequence (xn ). Set lim x = x m • Now i ni

/!F(Xn

i

) -xn

i

/1-->0,

theI'efore

and hence xCI) = F( x oo ) is a fixed point. Le~~~: (Mazur)~ If T is compact, the closed convex hull, H, of T is compact o We will prove that H is totally bounded, which implies that H is compact~ Let H~r be the union of all convex hulls of finite subsets of T, 1 .• e., P~oof:

'r.

H"" =

[:

x/x =

r =>""

i=I

Aix i where r is any finite integer, 2)'i = 1,

Trivially H~l-CH and hence H-;"C H = H. H* is convex for if x, yeH'~ then, by the definition of H*, the closed convex hull of . x, y lies in H~, and hence the segment between x and y lies * ~je will show that -* -* There in H. H is convex. Let x, yeH. exists xv' y" e H* such that x = lim xv' Y = lim Yy. ~

y .,..co

v ..

CD

.)}

Since H

is convex we have tx y + (1 - t }yy

y

~oo

= z.,eH-lI-.

and taking the limit as

we have tx + (1 - t)y = Z e

--* H • -it--

-'~

=

convex and H~3H. Thus H H. Since T is compact, T is bounded. For simplicity assume that if xG T then Ixl < 1. There exists a finite number or points al, ••• ,ar in T such that every point in T has distance < € from at least one of the ai' 1. e., min II x-a i II < ~ for x~ To Now xe H~4 and hence x = 2: AiX i • i Consider y = LAia ji Then

" x - y

II
Ai II x., - a J, .I. i

-

II

< E..

Divide the interval rOtl] into n equal intervals in such a

1 r: way that -n < .s:..r·.

Let z =

Pi is an integer

~

M.

Pi > T n

a. where Ji

Pi

11n

Then

Ila·lIa(bc) = (ab)c iii) A contains an identity element 1 such that f.oral.l-.a· in A, a'l = loa = a tv) 11'0 every ele:nent a in A, there corresponds an element a -1 st1.ch that Remark: The binary operation, as represented above, is multiplication. However, we shall usually be concerned with additive groups. The change in interpretation is obvious. We shall however, in talking about groups, use the notation of multiplication, as this will facilitate any necessary reference made to texts on group theory. Definition: A group A is called Pbelian (commutative) if for all pairs of eleMents, a,b in A ab

= ba

Definition: Let B be a subgroup of A. Choose any element a in A. By the product as 1f.le mes.n the set of elements in 1-\ which are of the form ab, where b is in B. That is

"-

aB = tab

I

beB}

It can be shown (see any text on group theory) that a group A caD be considered as a ?isjoint union of sets aB, where B is any subgroup. Th5s is called a decomposition of A into cosets of B. Since we shall only be interested in Abelian groups, it 1s true that for any subgroup B,

104

aB

= Ba

for all a in A. This leads us to the notion of the factor (SE~­ tien t) group (or differe,nce group if A is additive). Suppose that

A

= alB

+ a 2B + ••• aLB

where aiB () ajB = 0 if i 'I j. Then the 1. sets, alB, a2B, .•. ,a1.B themselves form a group, known as the factor gro2}E of A with respect to B, and denoted by A/B. Denoting the elements of AlB by Bl , B2 , "', B~ we may define multiplication of two elements in AlB as follows: Choose an element b i e Bi for i=1,2, ••• A. Define

where the element bjb k is in the coset Bm' It is quite clear that the elements of AlB are equivalence classes of elements in A in the follol-Jing sense: Two elements of A are called equiva- . .!ent !!'!9d'Ll:~.! B if they lie in the s a.me coset of B in A. liJe shall forego a.dditional explanation, assuming that the reader has been throu~';h 9.11 this before. If not he should consult the literature. Remark: The factor group of A with respect to B exists when A is not Abel:tan, provided that B is an l.nvad.ant subaroup. That is, if for all a in A aB Ba

=

irie define the "natural U homomorphism of A onto AlB in the follow1.ng way: Let f:A .-... AlB be such that for a in A,

if' a is in the coset aiB. (The reader should check that this is indeed a homomorphism.) Recall that f:A ->- C, where A,C are ~roups, is a hO~9.!:phism if' f is a single valued mapping of A into C that preserves addition. That is, for all pairs a l ,a2 in A

105 If f is 1-1 and onto, it is called an iso~!phism. ---:A:1:thoU'gh ~'Je shaTl iibt"pr-ove'lt:"( see Ledennann;. The Theory of Fjnite G,r?.:lps; for example) it turns out that all homomorphisms are "na.tural" homomorphlsms, in the sense of the following: Theorem:

If B is an invariant subgroup of A (If A is Abelian,

all subgroups are invar-iant.) then A can be mapped homomorphically onto A/B. Conversely, if f:A .-;::. AI is a homomorphism then the elements of A which are mapped into the unit element of A I , form an invariant subgroup B of A such that A/B ~ At (j!lIt means

is homomorphic to"). homomoI'phislll. II

B is called the kernel of the

Definition:

A set of elements x l ,x 2 '" . xi.. , of a group A are said to be linearly ind~pendent if Alx l + A2 x 2 + ••• + A.ix.t :::: 0, all Ai be 1ng integers, implies that Al :::: A2 :::: ... :::: AI. :::: o. 'rhe set is called .9:~end~.~ if for at least one Ai =I 0 the integral combination Alx l + •.• + A,.(x( :::: 0.

The reader should convince himself that the following statements are true:

i) If A is a group of finite order (finite number of elements), then E.:..1.l sets of elements ara linearly dependent (hint: take A.1. :::: order of A). . i1) In any group (:i.nfi.nite order), every collection of elements, each of finite order, is linear'ly dependent. iii) Any subset of a collection of independent elements is itself independent. iv) A set of elements containing a subset of dependent elemen.ts is itself dependent. pefiD:i t.~Q.£: The x.~nl!: of dependent elements in A.

!

is the maximum number of linearly inIf, for all pos i ti ve integers 11, tf.:2I'C

exist n linearly independent elements in A, then the group is said to have infinite

~k.

'rhe set of elements x l ,x2 , •.• ,xk in A, is said to be a maximal 1inear:ly indep9.Lldent set, if for any x e A, the

Definitio..!!:

set [x,x l ,x 2 , .•. ,xk} is linearly dependent.

106

Theorem: Suppose {X1 ,X2 , ••• ,xk) .is a maximal linearly independent set in A. Then I'ank A = k. Proof: i).A contains at least k linearly independent vectors, namely x 1 ,x 2 ' ... ,xk " Therefore rank A > k

•

ii) We shall show that any k+l elements Yl'Y2' ••• 'Yk+l in A are linearly dependent. Since the xi form a maximal linearly independent set, for all x in A, x,x l ,x2 ,. ",xk are linearly dependent. And so

where the A's are integers and not all of them are zero. In fact AO .; o. For if 11.0 ::: 0 then either the xi are dependent or x ::: 0, both of which 'V:e have assumed to be false. Thus, if x e A there is a non-zero integer AO' such that k

AOX

: : L1

Aix i

•

Therefore, there exist k+l non zero integers 1-I-j such that Zj

= ~jYj

k

::: f~l ajix i

,

j

= 1,2, •.. k+l

This represents a system of k+l e'quations in k unknowns. Since it has a solution, we know that some linear combination (rational coeff:i.cients) of the z. must vanish. That is, J

Y1 z 1 + •••• + Yk+1zk+1::: 0 where the "'(1 are rational and not all zero. Letting Y be the least common denominator of the "'(i' we see that yy1zl + •••• + Y"'(k+lzk+l where the YYi are integers.

=0

Since Zj ::: ~jYj' we have

107

proving that Yi are linearly dependent. Theorem: Let A be an additive Abelian group with finite rank n and let B be a subgroup_ Form the factor group A/B (actually A-B would be more appropriate since elements of the factor grotp are sets Bl c A which have the property that if x,y are in B1 , then x-y is in B. That is, for any two elements x,y in Bi , x = y mod B). Then rank A = rank

A B +

rank B

Proof: Since A has finite rank, A/B and B must each be of finite rank. Indeed B C A; hence B cannot have more than n li~ nearly independent elements. As for A/B, the fact that A/B is a homomorphic image of A guarantees that rank ~ :: :rank A i) rank A > r~nk B + rank AlB: Let Yl'Y2'" 'Yr. be Ie independer.1t elements of B~ And let Zl,Z2'" .Zp be p linearly independent elements (cosets of B) in A/B. From each Zi' select a repr·esentative element Zi' '\tie shall show that the set

is a linearly independent set in A, proving that rank A > r·ank B + rank A/B. Assume that the set is dependent. Then

where

Ai'~i

are integers, not all zero.

we may conclude that

From

108

which means that in A/B Sinc.e "the Z1 are inde'penden:t '" tJr~

=

1J.2

=

= ~p = O.

Therefore

with some Ai "I 0 which means that tYI J ' " ,Ytt:;1 is a linearly dependent set. i i) rank B = k ==> the re is a maximal linearly independ.ent set Yl'Y2""Yk in B. rank A/B = r;~ there is a maximal linearly independent set Z~.1. ,Z2""'Z r in A/B • Letting zi be an element of Zi' we shall show that {Yl""'Yk' zl""'z;} is a _maxima~ linearly independent set in A. We must show that for any x in A, the set

is linearly dependent. We have two cases: a) !;~: Tnen x = L A.y., and the set is dependent. ]. l. b) x i B: '.rhen x is in some other coset X :/ 0 (remember B is the 0 element of A/B). But since the Zi are a maximalli .. nearly independent set, tx,Zl"",Zr-} is a dependent set. 'l'hus

---

A(I-I.OX -

L

k

I-I.i Z i) =

~ Ai Yi

proving that {Yl""'Yk' zl""'Zr} is a maximal linearly independent set, and that rank A

=n = k

+ r

=

rank B + rank AlB

•

Definition: Let A be an Abelian group. If there is a collection of elements a 1 ,a2 , .•• ,~t in A such that for all x £ A,

109 j,

X ::::

L: n ,a, 1 1

1

where the n i are integers, then A is said to be finitely gene~ted. 'l'he set {aI' a 2 , ••. , aL are called ~~~~ of A. v-ie s tate without proof the following:

1

Theorem: If A is a finitely generated Abelian group and B is any subgroup, then i) B is finitely generated

ii) AlB is finitely generated. Definj.tion: Let A l ,A 2 , ••• ,A1 be ~, denoted by

1

additive groups.

The d:l.rect

is the collection of i.-tuples (a l ,a 2 , •.• ,a-t) wher-e a i e Ai. The rea-ier should verify the following: Theorem: A:: Al (t) A2 CD ••• 8.) AJ!, is an additive group with addition defined as component-wise. Theorem: Let Al,A2, •.. '~t be additive Abelian gr·oups. A :: Al l:t} A2 (£) •.• (~) A/... is Abelian. Definition: cyclic .B.£0uE.'

Then

A group which has only one generator is called a

If' the order of the generator, a, is infinite,

i. e. na =I 0 for all n, then the grou.i.J is called a fr~ cyclic group. Clearly such a gro'J.p is isorn0rphic to the additive group of integers, whi.ch 'we deflote by GO' If a group has one generator· a, whose order is n (order=n mes.ns that ne. :: 0 and ma -I 0 if m < n) then the group is isomorphic to G , the residues modulo n. n

lrTe shall state, without proof, the Fundamental Theorem on -;'1belian Groups. For further discussion see Leder-mann; ~ ~he2roy of Finite Grou.p1., (Chapter VI) ; University ~.t[athema tical 'Texts; or Bers; ~ntrodu£~io~~o Topology, (Chapter XVIII), Lecture notes of 1954-19~;5 term, N.Y. U. ) Theorem:

Every fini tely gener'ated Abelian gr·oup A, can be writ-

ten as the direct sum of r free cyclic groups and q finite cyclic groups,

110 A = Al

(f)

(f')

A

r

®

G1:'

1

®

,'+1 ",,-,,

.

'tl > 1

Gy

q

where 7:'1 is the order of G.1:: ' and i

I

t'l

I .. . I

1";....

1'2

Lq

f

(a Ib means a divides b, or am = b for s orne integer m). The number r is called the £etti number of A, and the ' i are called the torsion coefficients. These numbers are uniquely determined by A. Thus the elements of A may be written as (r+q}-tuples (a l ,u2 , ••. ,ar'~1'~2'···'~q) where the a i e Ai and ~i e G~i' Two elements x,x', in A are equal if i)

ai

:::

ai

fo r i ::: 1,2, ... , r

ii) 'til{I\-~i)

for i

= 1,2, •.• ,q

Theorem: Let A be a finitely generated Abelian group. rank A = r, where r is the Betti number. Proof:

Then

The r elements Xl = (1,0,0, ... ,0)

x2 = (0,1,0, ... ,0) r-th place xr

:::

(0,0, ... 1,0, ... 0)

form a linearly independent set. If we consider the collection X ~ where y. = {al, •. ea 'O'''''~j'.''O) we see that t(y j' x l ' e •• '"r j J r tjYj::: {t' j a l ,···1'j a r ,O,O, ••• O}::: '7.:'ja l x 1 + C j a 2 x 2 + ••• + 'tjarXr" From this consideration it is clear that the set

{y ,Xl' ••• Xr 1 where yeA, is linearly dependent.

Therefore rank A ::: r.

III

Definition:

G:::: G

'"(1

@ G

't. 2

(~

. •• (f) Gl

is called the q

torsion group of A. Clearly, G is the subgroup of .A which contains only elements of finite order. Since A is finitely generated, the order of G itself is finite. Before going on, let us recall some of the ideas introduced in Chapter II. It will be the purpose of the next few chapters to study the topological invariants of polyhedra. That is, whenever a space a.dmits'a triangulation (is-homeomorphic to a finite complex,) its topological properties can be investigated by studyin~ its horeology groups. The principal theorem to be proved in Cha.pter' VIr, states that homeomorphic spaces have isomorphic homology gro:.lps. That is, the Betti numbers and torslon coefficients aI-e topological invariants. The converse of this theorem, known as the principal conjecture of combinatorial topology, has never been proved. Hence the theorem that we shall prove will only allow us to S9.y that two spaces are not homeomorpn5.c if they do not have the same homology groups. 1I'Tben a finite t:'iangulation is not possible we may use the co-homologz groups. But more of co-homOlogy later. For the present we restrict ourselves to finite dimensio~al geometric rectilinear compleyes. Let us present a brief reS'LUlle of notation, definitions, etc J which have al:!:-eady been introduced in Chapter II and earlier in the present chapter. K will denote an n-dLlle!iSional geometric complex. G is a group. (i,~Je assume G is an additive Abelian group unless otherwise stated.) C (K,G) denotes tbe group of r-chains over K, with coeffir cients in G. To simplify notation we shall let r = O,+I,~2, .•. , defining Cr(K,G) to be empty for r < 0, r > n. 'tn1hen there is no danger of confusion we shall write Cr inste~d of Cr(K,G). An element x e Cr may be wr:!.ttea as

112 r

where the Si are elementar:cr-chains~over K, and the gi e G. The bo1indary operator and Cr_l(K,G)

d defines a homomorphism .of C (K G)

-

r

'

;i-x= 0 for all x e C (K,G)

r Zr(K,~ denotes the subgroup of cycles in Cr(K,G), where Z is a

cycle means dZ = 0

Cle arly Zr is the kernel of the hemomerphism ,~. C /Z

r

r

q:: ,)(

That is

C )

r

Br(K,G) denetes the subErcup of Cr{K,G} w'hich ccnsists .of bcundaries. That is, if b e Br , then there is an x e Cr + l such that ~x = b.

Cr

Iz r

~ d (C . r ) = Br- 1

rrhe factcr p.:roup Zr ~ hemclcgy group .of K.

IB r = Hr

is called the r-dimensicnal

Let us .once more remind the reader of what is tc come. The principal task of combinatorial topology is tc determine the tcpclogical invariants of pclyhedre.

The prirlcipal invariants

are the H.omclcgy (Betti) grcups; i.e. the B8tti numbers and the t.orsicn c.oefficients.

We begin by intrcducine the so-called

Euler-Pcincar~. ?ho.r~!!£Eistic, "j...(K) , .of a ccmplex K.

Let us ccns1der Cr(K,G.o ) = Gr , where G.o is the (addi.tive) group of integers. EVidently Cr is finitely generated. Fcr, if

sr

we let denote an or-iented r-simplex in Ie (and hence an ~l~­ mentary r-chain; see Chapter II, page 40), then every r-chain can be written as

As a re'sult, Zr' Br , and Hr are alse fini tely generated. As a result .of the Fundamental Theorem on Abelian greups, Hr may be described by its Betti numbers and its tersien c.oefficients.

113 Thus the complex K, assuming dim K == n, may be characterized by the n+l Betti numbers p , and the n+l sets of torsion coeffi. r r r c J. en t s ?; l' ••• ?: , I ' = 0,1, ••• , n. qr Let us denote by aI" the number of r simplices in K. The Euler-Poincare Formula (which shall be proved later) states that

It is quite c1ee.r that once we have shown that the Bettt numbers are topological invariants, we may conclude the j.(K} is also a topological invariant. wbat this means is that ~(K) (like the Pr,~r, ..• tr ) depends only on the space, IKI, of the qr'

complex ){, and. not on the complex i ts.elf • In this way, we see that the complex plays only an auxiliary role. Its principal function is to determine the topological invariants of the polyhedron. Before proving tbe Euler-Poincare Fo~!ula, let us see what it means in the following very simple case. Only the shaded areas anu the edges de:1.ote the space to be cons idered. The ~wo figures dr~wn below illustrate two triangulations of the sa.me polyhedron.

:t Let us calculate !riangulation...!:

r: (_l)r ar

1k for triangulations I and II:

0,0 (nu..'11ber of vertices) == 10 ~

(number of edges)

18

==

0,2 (number of triangles) ==

= 10

~

18 + 8

=0

8 CARNEGIf. lNiTlTOTa

O~ J:.fCrlNOlOGY. J..JSRARY

114

j(K)

Proof:

HI'

= Z!B I' I' rank HI' Br _l

=5 -

7 + 2

=0

•

1

= Pr

= Cr/Z r

~?:;;r =

~> a r

=

PI' + I3 r ?:;;r + I3 r - 1

since u r is the n(.lmber of r simplices, and consequently the rank of C •

I'

13_ 1 =

For r = 0, a O ; ?:;;O' since every O-chain is a cycle. Thus O. For I' < 0, r > n, we may define a = p = O. Then

I'

Ur

= ~r

+ I3 r - l

= PI'

r

+ I3 r + I3 r - l

Multiply5ng both sides by (_l)r and sumrrdng from -co to get

+00

we

The Betti number Po of Ho C9.n be given a very simple geometric interpretation. To motivate the more general discussion, let us look at a few pictures. This in no way is a deviation from our g e11e1"a1 policy of IIdoing topology without picture s II , since this will not affect the general discussion. 1~!e shall find that Po is the nu::nber of "connected compo-

nents ll (definition later) of the complex IC As one might easily guess, looking at the t!rJO complexes drawn below, K has two cO;'.Lnected components 8.nd L has one connected component. ll1Te recall that the elenents of II are equivalence classes of o-cycles. o Clearly, every simplex (a i ) or (b i ) is a cycle. Looking at the picture, we see that (13.1 )

r-.l'

(a 2 ) ' ...... (a 3 ).

That is

·

\'") A ,,:> .ll

115

I

,j/

" ,I }

!

/

!

,

b.

,

"--~----!-.

\-., .. Jv

L

D'i

--.

( L... _ " ) ',- --' ,

;

(a l ) - (a 2 ) =: deal a 2 ), etc. (twe cycles are.hemologous if their diffen~nce is a beundary). But it is net true that (a4) 81nce (a l ,a4) is net a simplex 'Of K. Therefere the rank 'Of He(K) = PeCK) =: 2 since (a1 ),(a4) are two linearly indenendent generators. By the same r'easening He(L) = po(IJ) = 1. That is, (b i ) ",-,(b j ), 1,5 = 1,2,3,4,5. (a l )

·V

Definition:

A cemplex K is said te be

al~ebraically

cennected

if, fer any twe 'Of its vertices a,b, there is a sequence of vertices, 8 1 ,a 2 , ... ,a 8 such that (a a l ), (a l a 2 ), ••• (a s b) are I-simplices in~, i.e., yeu can get frem a te b by travelling along the edges 'Of K. Theerem:

K is al[:ebraically cenn.ected

f

;> K is (tepelegically)

cennected. Preef:

i) Suppese K is''net (tepelegically) cennected.

That is,

n

K ::: L U M, L M = 0, L i 0, :tV! i 0, where L, N, are subcemplexes. Let a be a 7ertex 'Of L, b a ver,tex of ]\'1. AssUJ:ning that K is algebraically cennected, there is a sequence 'Of vertices as described abeve.

If a i is the last vertex in L, i.e. a i + l is in

M, then ~.a1+1) is net in either L 'Or M. But (a 1_ a'+l) is a sim1. _ 1. plex in K (defin:I.tien 'Of algebraic cennectedness). Thus K must be cennected. K.

1i) New supP'Ose K is c'Onnected. Choose any vertex b in If a vertex 'Of a simplex in K can be joined te b by the se-

quence described above, then all vertices of that simplex can be jeined te b.

The set 'Of all suOh simplices in K which can

be jeined to b ferm a subcemplex L.

The simplices 'Of K that are

116

not in L form a subcomplex M. Since K is cormected, IVi = O. Therefore K is algebraically connec-ted.-Definition: If L is a (topologically) connectea subcomplex of K such that K = L (j IY1, 1-1 (\ L = 0, M a subcomplex, then L is called a connected component of K. It is not difficult to see that if Kl , .. oKs are the connected components of K, then K = Kl U • ••

Ki

n

Kj

=0

U Ks

,

i

t

j

Since the continuous image of a connected set is connecte~ the number of connected components of a polyhedron is a topological invariant. Let us now recall the definition of arc-wise.connectivity'. The topological space X is said to be arc-wise connected if, for any two points a,b in X, there exists a continuous function ~:[O,l] -::.. X such that 40 = a,

ii) Suppose d'X O. Let a,b be any two vertices in K. Since K is connected there is a sequence of vertices al"'.Jaa in K such that (aa l ), (a l a 2 ), .•• (a s b) are simplices in K. Let y g(aa l ) + g(a1 a 2 } + ••• + g(ssb). Then ~y = g(a) - g(b). Thus (a) rv (b). And so every O-chain x over G is homologous to g(a). Since. x ~ g(a), dlX ~ g and so x ~ ~'x(a). Therefore .:3'x = 0 = } X O. Before we continue with the general discussion, let us compute the Homology groups of some polyhedra.

=

I"V

118

......,

,

1\ - ,-- - - - - - - -..-, -,'-

~) ' ~

1

.

~

_ -

1,

f

....~-.

',1

~

~-

C. ~.-- --+-

) --',)



-1) r

(

"--

U

, - ( -1) r P

::::)

I'

---

-

r

Substi tuting ao :::: 9, ul :::: 27, a2 :::: 18, Po :::: 1, Pl :::: 2, P2 :::: 1 into the above formula, we get ul + u2

::::

9

27 + 18

Po - Pl + P2

::::

1

2 + 1 :::: 0

a0

0

:;:

which checks. ii) The HHbius Stri;e! •.'. .-~~, !,,' ...,...,....... _ - - ....... ' ..~ .... -

i "~i.:~·

.i''-

I. II \\\ ;,: !I \. '\. 1 \\')1 \ Ii

it::··

1

~.. ~

I

\'

\.•. I.

,

\\

,'\

\.~

\',

: \ \'"

\

\

; ___ M _ _ _ _ _••

t-,

,j

.. _...... - -" _.... ;-_. -.~----~-P -'-"'-r

\\\

\,

'.\, !

_~

\

!

\.i

'" } __

.~_ • • _ _ • •

\~-.~---.--.--

_. _ _

~

figure 7 '

•

-• • •

. \

122

The NBbius strip is the "l-sided" surface obtained by joining two opposite sides of a rectangle so that diagonally opposed points are identified (figure 7). It is quite clear that figure 7 is an ade4uate triangulation of' the M8bius strtp. The reader should check that the configuration is a complex and that the identifications are maintained under this triangulation. Since the complex is connected, Po = 1 and Ho and Go. Clearly there are no 2-cycles. For if there were a 2-cycle z, every triangle would have to enter in z with the same multipli..city, to insure the cancellation of the tlinterior" edges (e.g. (ae), (cf) etc.) when ~z was formed. But then it is quite clear tha t we c01.l.ld not get rid of the ter-ms (ac), (cd) etc. Thus P2 = 0, and H2 is empty. The easi est way to get Pl is to use the Euler-l'oincare formu.la:

Therefore PI == 1. z

=

\rJe

shall show that

(a b) + (bd) + (d c) + (c a )

is a. I-cycle that does not bound and that all l . . cycles are homo ... logous to z. By the usual ar·gument, z can be a boUt."ld ary of a 2-chain x only if ~ the elementary 2-chains enter into x with the same multiplicity. But again, ~x will con ... ta.in (ac), (cd) etc. That is ~x I o. To see that all other I-cycles aI'e homologous to z, we eliminate, in order, the edges (be), (ae), (ef), (tc), (fa), and (ad) leaving us with figure 8. 'Ihe only other cycle appearing in the picture is

~ -.:-.---.-~ : I

i

:I j

i I

'

L----.-.. ?

...- . -.('i i.

It

I

I !

figure 8

l

123 Zl

but clearly

:::

Zl

(ba) + (ac) + (cd) + (db) + (ba) :::

-z, so that Hl has no torsion coefficients.

iii) The Pro.iectiv~_flane The real projective plane consists of sets of real triples (x l ,x2 ,x3 ), not all zero, where (x l ,x2 ,x3 ) is said to be equivalent to (Yl'Y2'Y3) if there is a real number a I 0 such that X.

1

:::

ay.

1

,.

i = 1,2,3

1~je

may think of the pro jecti ve plane as the set of all straig,ht lines (in R3 ) throue:h the orivtn, Hhere, to each line we may associate an equivalence class of triples {(x l ,x 2 ,x3 the xl' x 2 ,x3 being direction numhers of the line. By requiring that

1,

x~

+

x~

+

x~ ::: 1, we choose a particular element of each equi-

valence class and thus represent the projective plane by the surface of the unit sphere in R3 , where diametrically opposed points are to be identifi,ed. Finally, by projecting the hemispher'e onto the unit circle we get a representation of the projective plane by points inside and on the unit circle, diametrically opposed points being identified. A suit9ble triangu w la t1tion is shown below:

124

figure 9 Since the complex is connected, p o ~ 1, H0 ~ G. Again it 0 is clear that ther'e are no 2-cycles. Therefore P2 ~ 0 and H2 =0. To find PI we mightiJ.se the same method as was employed in the case of the torus. But the Euler-Poincare formula will also a 16, we get give the same result. From a o ~ 9 ' 1

Thus PI == O. HO"t\lever He do have a torsion coefficient 1:"1 = 2. hi'e shall see that i) z = (ad) + (dc) + (cb) + (ba) is a cycle that does not bound ii) 2z = ~x where x is the sum of all the triangles in the complex. iii) All other I-cycles are homologous to z.

12;;

Using the s arne argument as for the torus 'VJe may foy-get about (eo), (o,f), (o,g), (g,f), (fe), (eh), (hg); (ea), Cah), (he), (cg), (ga), (af), (fc), (ee) in this order, leaving us with figure 10. From the b picture we see that ii) and ",...,~---...-. . ,\ /'" '. , .... c>· ~// i '-> < )

Proof: Let 3 1 ,3 2 , ... ,Sr be the coherently oriented simplices of M. ffhe chain 3 = 3 1 + 3 2 + ••• I- 31' is a cycle, ainee every (n-l) elementary chain in as appears twice with opposite orientation. 11his follows fro;;l the definition of a coherently oriented pseudo-manifold. Clearly all other n-cycles are multiples of 3. For, suppose z is an n-cycle. Then if 8 1 appears in z wi th a mul tj plici t;,r k, then the other n-simplices which intersect 3 1 must also appear with the same multi~licity, so that when .jz is formed, the (n-l) simplex, which is the intersection of 3 1 and a neighbor, will well appear ti-Jiee with opposite or'tentatJon so that it will cancel out in ~z. Therefore g :::: k3 and p n :::: 1. Moreover in any pseudo-manifold, k[3 l + ... + 3n ], if it exists, is the only n-cycle possible, for no other n-chain will have a vanishing boundary. rrhus either Pn = 0 or Pn :::: 1 holds. If Pn :::: 1 ruld S :::: 3 1 + 3 2 + ... + 31' is a cycle, the pseudo-manifold is coherently ,. oriented. Otherwise pn :::: 0 and the orientation is not coherent.

128 Chapter VI Homology Theory - Part II Definitl.!2E:: Let f: Cr(K,G) ~ Cr(L,G) be a homomorphism for r:: 0,1,2, •.• . f is said to be allowable~ ~f:: fd. Example ~.._of A.llowable Homomorphisms: i) Sj.ml21:ic i81. Mar1pings :

'Ilo every s implic ial mapping f,

we have an associated homomorphism of chains (see pages 47-48) wi th 011' :: fd. ii) m-th B':l!'Ycentric Subdi vis ion:

(par"e 56)

Theo_~:

If f:Cr(K,G} ~ Cr(L,G) is an allowable homomorphism, then f induces a homomor'phism f.,: HI' (K,G) -> Hr (L,G) • ..~

Proof: --._X

i) x -is .a cycle ---_.- -in Gr (K,G) ===';?-fx -is -a cycle -in CI' (L,G): is a cycle ~.-:-·-).h :: 0 =~ ofx :: f2lx :: 0 ( ~fx is a cycle.

ii) x ~ ~ b~~?ar'y in Cr(K,G).w ~fx is §:. boundary in Cr ( L, G): x is a boundary chain 0, ~ Ai

k

Y =

L

°

A.a. + ~

~

fe

is the image of some point

138 k

where

L

o

~i

Looking at the expression for fy, we

see that: A. o = ~o' Al

= tlJ.'

!-LO + ••• + !-Lk

··;,-A.k··=···(~+-Vk)' Ak + 1

= Al

+... +

= vk+l'···A r

= vr ;

Ak -1 + !-Lk = 1.

so that f is an onto mapping which completes the theorem. The reader should not be surprised to see that the cyl:lndar construction induces a homomorphism

which is defined in the following way for elementary chains, (i.e. oriented simplices). First let us recall that in the cylinder cons truction we assumed s.n ordering of K, which in turn induces an ordering on CK and all its faces. Thus the geometric simplex {aO,al " . "a r } receives B.n induced ordering, and so may be written as an ordered simplex (aO,al" •• ,a r ) (the same subscripts are used repeatedly to avoid a cQmbersome sequence of sub-subscripts). !~re shall then write S for the oriented simplex which we associa.te with the eq1.J.ivalence class of ordered simplices to which the naturally ordered simplex [a O,al , ••• ,a r l belongs. (Remember that two oriented simplices are said to be equivalent if one may be obtained from the other by an even permutation of the vertices.) We write -3 if the order is not the natural one. The chain mapping is defined as follows:

111001110 + (aOBla2a2,··ar) + •.• ± (aOal···ara r )

As before, we may associate with (a Oa 1 ••• a r ) in C(K), two oriented r-chains over ~KJ which are the analogues of the lltopU and "bottom" of C,K, na,nely (a1Oa 11 • •• a 1r ) and (a 0Oa 10 ,· .ar0 ), If · 1 0 3 = (aOa1 ••• a r ), we sball denote by S and S , the top and

139 bottom respectively. A straightforward, but long and tediQus calculation usi:ng t.ha. . . definition of ~S = ~(aOal" .a r ) gives us the following .'. Lemma:

deS'" CJs

= SO

Sl.

shall perform the calculation for the very simple case (aOa l ). The idea is the same for the general r-chain.

!AJe

of S

=

C(aOal )

= (a~aga~) o

-

(a~aia~),

d(aOal )

= (a1 )

- (a O)

0

1 0 1 0 1 0 1 0 1 1 JG( aOal ) = (aOal ) - (aOa l ) ... (aOao ) - (alaI) + (aOa l ) - (aOa l ) = sO _ Sl C;~(

1 0 + (ala.o) - (ala l) o0

1 0 a Oa 1 ) = (a 11 a 10 ) - (aOa O)

1I\Te may extend the definition of ~K to the case where K is an arbitrary complex. Let K consist of the simplices Sl,S2""'S and all their proper faces. (v,ie assume that no Si is a proper face of any other S .). tl: If is the complex which is J the union of all the (PSI. With this, we .'(lay generalise the lemma sta.ted directly above. The proof is left to the reader. !heorem:

If x e C(K,G), then

= xo

- x

1

Definition: Let f,g, be two simplicial mappings of the complex K into the complex L. Then f and g are called combinato~11-x ~toEi£ if there exists a simpliCial mapping F: ~K ~ L which coincides with f on the bottom and g on the top. This means that FK ,FK ' the restriction of F to KO andK l respectively O... _:1.. .. .... "_' 'arethe "same lt as f and. i-in -the--following sense. We shall say

that f,g are combinatorially ~omot£Eic if there exists a simplici. al mapping B': CK ~ L for which FK : K O --;:.. L, f: K ~ L are

°

the seme homomorphism; and FK : Kl --;:.. L, g: K ......... L are the same homomorphism. . 1 vJe shall now show that if two si.rplicial mappings are com-

binatorially homotopic they induce the same homomorphism of the homology groll.ps. iNe have already shown tha.t two algebraically homotopic allowable homo~orphisms induce the same homomorphism of the homology groups (pare 129). Thus we need only prove the following: Theorem: f, g 9.1"'e cowbin9.torially homotopic braically homotopic. ~f

:

>- f ,g

are alge-

t-re must find a homomorphism

such that for all x fx - gx

£

Cr(K)

= FxO

- Fxl

= ~Qx

+ OdX

where F is the homomoI'phism of G;K into L that agrees wlth f on the bottom and g on the top. Since F is a.n allowable homomorphism and xO - xl = d ex + e...dX,

Thus we mt=l.y define Q for x e Cr(K) as Qx

= F~

Since ~x e Cr+l(~K), the right hand side is well defined and f and g are algebraically homotopic. So far we have dealt almost exclusively with homology. As the re.ader might guess, it is not difficult to derive similar results for cohom.ology. An allowable homomorphism f: Gr{K) -7~Cr{L) induces a homomorphism f : CI' (L) ~ Cr (K),

'*

such that

(fo)..~~;)x

== ';(fx)

for 13.11 x

s

C (K), € Cr(L). f* is r ,>' ~I' in the sense that 2/"f'''' == {''';/'' €

H

also an "allowable homomorphism tl (par,;e }.j..9).

\1

With this, we may state a theorem which is the ana-

logue of the first theorem proved in this chapter. The proof is exactly the same if one prefi..xes the words "chainlt, IIcycle 1t , "bound.aryll, and "ho.mologyU by "COli. Theore!!!: I~ f: Cr(K,G) --;;.. Cr(L,G) is an a1101..Jable homomorphism (and thus f"": Cr(L,G) --;;.. Cr(K,G) is alloVJable) then f induces a homomorT..)hism f~~: Hr(L,G) ~ Hr(K,G). It is also quite easy to see that

We leave the proof to the reader. -'~

Q~i:nitio!!.:

~~

We shall say that f' and g are algebraically homotopic :i.f there exists a homomorphism g{}: Cr+l(L) -> Cr(K) such that for all

s

We shall

€

Cr(L)

Sb01..J

that whenever f,g: C (K) -> C (L) are alger

r

braically homotopIc, then the induced mappings, f*,g {(-: Cr{L) -.;::.. CrUn are algebraically homotopic (in the sense of the immediately preceeding definition) and that the s arne homomorpilism

of th e cohomology groups.

f:,g:

are

Fi!lst we must

show that ~~

i) If f,g are algebraically homotopic, then f ,g

i~

are

algebraically homotopic, and then we must prove that

* *

* *

ii) If f,g are algebraically homotopic then f.;pg* are the same homomorphisms of the cohomology groups. In general this will allow us to dispense with a separate discussion of cohomology. For whenever a homotopy operator Q .,

is found, a g-'l' is automatically produced.

First suppose that

the two allowable homomorphisms f,g are algebraically homotopic. This means there is a homomorphism Q: Cr(K} -;.. Cr +1 (L) such that for all x e Cr(K)

142 fx - gx

= OdX

r *Sx -

g

+ dOX

'!il-

SX

Now r?l-,g *: Cr(L) ~ Cr(K), so that f-ll-s,g*s, e Cr(K) and

r

f?l-sx,g~-sx, e

r).

(remember (g*.;): Cr(K) ~ Since f*s and f,g are algebraically homotopic, it follows that ~~

f .;x

~

~~

g .;x

= sfx

- sgx

= ~(fx

- gx)

= s(JQx

+

= sf

QdX)

We may now define a homomorphism O';}: Cr(L) ~ Cr-l{K) for all x e Cr_l(K), ~ e Cr(L). It is clear the.t this expression is well defined. On the l.eft side, o-ll-.; e Cr-l(K) and thus (G*~)x e: On tbe right,

by requiring that

(Q~~)x

= s(Qx)

r.

r.

Ox e Cr(L), and ~.o ~(Oxt e Applying this de:finition to the express:Lon for r"X".;x - g'),.;x, we get

This takes care of i). Finally we must prove that: r,g ~~ s.re algebraically ,~ ..:~ homotopic -; ).f?}Jg~~ produce the same homomorphisms of tpe cohomology groups. Bllt the proof is exactly the same as for the homology groups. To conclude this discussion of cohomology, we shall say a few words about the cohomology groups of a O-complex and of a complex consisting of a single n+l-simplex and tts proper faces. First suppose that K consists of a single O-simplex (a). If e CO(K,G), then s(g(a) = y where g e G and y e c. is a homomorphismj so ~(g(a) + gl (a)) = y + .'(t. Therefore, we see

*

r.

s

that C:: G ~

r

1~.3

is a homomorphism . .~ . "'\'~ 1 l~ ow c' r:, £ C ( K , G) .

But Cl(K,G) is empty since dim K = O. Therefore (;) E. = 0 for all c.. e CO(K,G), proving that all O-cochains are cocycles. Furthermore no O-cochain is a coboun..!,}

dary since there are no (-l)-chains. Thus the O-dimensional cohomology group is the set of all homomorphisms of G into which is wri tten Hom (G,

r,

r ).

Now let us assume that K is a cone, b(a O'" .,an _l ). lrJe shall show that its cOhomology groups are the same as those of a complex cons"t sting of a single O-simplex, which we shall denote by L. In order to establish an isomorphism between the Hr(K) and the Hr(L), we must construct two allowable homomorphisms

such that f~~·g~~,.,..... 1, g~i-f~;·::: 1.

E9.r1ier in this chapter we com-

puted the homology groups of a cone using a homotopy operator. The operator Q, constructed in that proof, automatically defines a Q-l~ for this problem so that f:.g!: = 1 and g~:f!: HO(K) ~ Hom (G, and Hr(K) = 0 for r:/ O.

r)

= 1.

Therefore

144 Chapter VIr Homology Theory - Part III Untl1n:-ow' weha-ve--dealt e.x.elusively· 'with oriented homology and cohomology theory. That is, we have defined oriented simplices, oriented chains, etc. For some pur'poses it is convenient to use the .£Lq~~£ homology and cohomology theor·y. b'ortunately both theories give the same results about most things. As a matter of fact, we shall establish a natural isomorphism between the oriented and the ordered homology (and cohomology) groups. Defini ti.on: Let K be a complex. For every orEiered collection of vertices [aOa l •.• a r ], such that the a i are vertices of a simplex in K, we define an ~lementa!.y int~gE!! ordere~ chain to be a function that a.ssigns the value 1 to this ordered collection and 0 to every other collection. ~-Je shall denote this elementary chain by [aOa· l •· .a r ]. Remark:

We allow a vertex to be repeated any number of times.

That is, [aOalaOa08.2J is an ordered chain of dimension

4-

Definition: An. r-JJ.mens:t.ona1 ordered chain over K with ooeffioients in a g.!2u:p 52, is a linear combination -_._-----

x where each gi

£

=r

~------"""""#

."--

- -

-

-

---

g.S . ~

~

G, and each Si is an. elementary integral ordered

chain. Definltt..£.!}:

An elementar'y order-ed chain [aOal ••• a r ] is said to

be ~~~~r'~~ if for some i I j, a i = a j , e.g. [8.0al8.0aOa2]· Clearly, the ordered r-chains over K with coefficients in '" G form a group which we shall denote by Cr(K,G). Just as we did for oriented cbains, we may define a horno0 morphism ~: Cr .... Cr _l • For el~mentary chains, 0;)

145 ~[aOal •.• a ] : r

r

2 k=O

(_l)k [a a

0 1'"

where ~k denotes the vertex to be o~itted. o morphism, the eytension to Cr is obvious.

~~:

62

Aa

k'"

a] r

.

Since J is a homo-

= o.

Proof:

Same as for oriented chains. Def:tning cycles and bounda.ries in exactly the same manner as for oriented chains, we may form the r-c.irr..ensional ordered homology group by taking the factor group

Definition: The cone over [aOal' .. a r ] with vertex b, written b[aOal, .• a r ] is defined as [baOal, •• a r ] provided that [baOal ••• a r } exists, i.e. provided b,aO,al, •.. ,a r are vertices of a simplex in K. Just as for oriented cbains, we may prove the follo"Jing Theorem:

If x is an ordered r-chain, then dbx = x - bdx

provided that bx exists. The only important theorem about oriented chains that we can not state for· ordered chains is the Spar-ner Lemma. The ordered cochains may be defined in the natural way: Let Cr(T n.

r+;

'1'hus G::

1..._"

Gr

is a graded g!roup.

make G into a graded ring we must ciefine a multiplication.

To Just

as in the case of exterior differential forms, we define e t /\ e j :: -e j 1\ e 1 , concl 11ding as before that e i f\ e i :: 0, and in general, that e~ 1\ eo; /\ •• • f\ e i :: 0 if some i. :: i(;o, ·'·1 ... 2 r J .z.. j 1-1.. '1'11e remainder of the discussion is the same as for differentlal for-rns.

r

Let A = (f) An be a graded group and let be any J\belian group. Cons ider all hOIDomorp:lisms c: ~ A ~ 1-. We shall wr'ite Hom (A,

r)

to denote tlds collection.

diately from the fact thet

r'

is

Hom (A,r) is also an Abeli.an.

[l[J.

It follows imme-

Abelian group, that

In fact Hom (fl"r"') is a graded

group. For, if we let An denote tb.e n-dimensl.ol1al homogeneous elements of A then it is quite clea.r that Hom (A, 1-) = (i) .- Fom (A:n ,

Hom (L,

r)

r) .

We haV8 8.1r6 ady enc ount ered an example of in the discussion of the cohomology groups of the

cone. In .fact, let A = C = = C r ( K , G, Irhen

r ).

G-)

C (K,G) and Hom (P..

Derini tion: A is called a Fl;!§.ded operator tlet" if

r

f!1:£''-:l~th

a

r

,r)

diff~tial

158 i) d: A -...;:.. A is a homomorphism ii) d(A,) C (A,,+ ). I..

11

"-

s

This means that for all non-negative

integers A:, the homogeneous elements of dimension

,l. are mapped

into homogeneous elements of dimension 1+s, wher'e s is a fixed inte3er which is called the shift or degree of d. 2 -----iii) d :::: 0 • ~xampl.£s__ ....2..t.2rad~ G~ps

i) Let A:::: C::::

with Differential 0Eerators:

(±> C r and d ::::d.

Surely

a is a homomor-

phism and dC r e e r-l ' so that s :::: -1. Furthermore.;;2:::: 0 , pr'oving that;) satisfies all three properties of the differenti al ope rator. ii) Let A :::: C-l~:::: (£) Cr and d :::: ;j fx = fdu = dfu fx is exact. Thus f induces a dimension preserving homomorphism ~~

t,