on simply connected, 4-dimensional polyhedra

On simply connected, 4-dimensional polyhedra By J. H. C. WHITEHEAD, Magdalen College, Oxford

1. Introduction. Our main purpose is to show that the homotopy type of a simply connected, 4-dimensional polyhedron is completely determined by its inter-re]ated co-homology rings, mod. m (m ----- 0, 1 ,2 . . . . ), together with one additional element of structure. The latter is defined in terms of a product, which was introduced by L. Pontrjaginl), and which has recently been studied in greater generality by N. E. Steen- rodS). What we want here is Pontrjagin's method of associating a 2n- dimensional co-homology class, p x, mod. 4r, with every n-dimensional co-homology class, x, mod. 2r. We shall call p x the Pontrjagin square of x. If f is a co-cycle, mod. 2r, in the co-homology class x, then p x is represented by the co-chain which, in Steenrod's notation, is written as

fU l + lU, ~f .

The co-homology rings of a polyhedron, P , with integers reduced mod. m (m -- 0, 1 ,2 . . . . ) as coefficients, may be combined into a single ring by a method due to M. Bockstein a). We give this ring additional algebraic structure by introducing a certain operator A and also the Pontrjagin squares. We describe the result as the co-homology ring of P and prove tha t :

(1) any such ring, which satisfies the general algebraic conditions appropriate to a finite, simply connected polyhedron of at most four dimensions, can be realized geometrically. That is to say, it is possible to construct a polyhedron of this nature, whose co-homology ring is "properly" isomorphic to the given ring. Also

(2) two such polyhedra are of the same homotopy type if, and only if, their co-homology rings are properly isomorphic.

An example is given at the end of w 12 of two simply connected, 4-dimensional polyhedra, which are not of the same homotopy type though

1) See reference [1]. 3) See [2]. s) See [4].

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their co-homology rings are isomorphic if the Pontrjagin squares are ignored. They can be distinguished from each other by their third homotopy groups or by their Pontrjagin squares.

A note on the steps in the proof of the second of the two results stated above may be helpful. Let K and L be two simply connected, 4-dimensional polyhedra and let R(K) and R(L) be their co-homology rings, as we arc using the term. Let /*: R(L)-+ R(K) be what we call a "proper homomorphism" of R (L) into R(K) . Then we prove that [* can be "realized geometrically" by a map ] : K ~ L. That is to say, there is a map, [ : K -+ L, which induces the homomorphism [*. The result (2) above then follows from the following theorem. Let K and L be two simply connected complexes, of any dimensionality, and let I : K -+ L be a map, which induces an isomorphism of each co-homology group, H ~(L), with integral coefficients, onto the corresponding group, H ~ (K). Then K and L are of the same homotopy type and / is a homotopy equivalence 4). This theorem shows the importance of the "realizability" of a given homomorphism /* : R (L) ~ R(K). The conditions which we impose on the co-homology ring are designed to ensure that every proper homomorphism, f* : R(L) -+ R(K), can be realized geometrically.

We consider this matter of realizability in two stages. We first confine ourselves to the additive group of the co-homology ring, or rather to the "spectrum" of co-homology groups, of which the additive group of the ring is the "finitely generated" direct sum. By the spectrum we mean the aggregate of all the absolute and modular co-homology groups, related by the homomorphisms, LI,/x, defined in w 2 below s). We first consider the realizability of "proper" homomorphisms of the co-homology spectra by co-chain maps e), not by geometrical maps. I t is obvious from the definition of ~ and/~ that any homomorphism of the co-homology groups of L into those of K commutes with A and/~ if it is the homomorphism induced by a co-chain map. Lemma 4, at the end of w 2, states the converse of this. This is valid for arbitrary (finite) complexes K, L, but the resulting co-chain map cannot, in general, be realized by a geometrical map K -~ L . The step from the eochain map to the geo-

4) i . e . t he re is a m a p g : L - - > K such t h a t g ] ~ l , ] g ~ l , where ~ denotes t he relation of h o m o t o p y a n d 1 deno tes t h e ident ical map , b o t h in K a n d in L .

5) Our ~u inc ludes a n d can be defined in t e r m s of Bocks t e in ' s ~ a n d ~o b u t o u r / I is a n addit ional e l emen t of a lgebraic s t ruc tu re .

6) i. e. h o m o m o r p h i s m s of t he g roups o f co .cha ins which c o m m u t e wi th t he co-boundary operator (cf. [51 a n d pp . 145 e t seq. in [6]).

4 Commentaril ]~Iathematici Helvetici 4 9

metrical map depends on the multiplicative structure of the rings, in- eluding the Pontrjagin squares, and the special nature of the complexes.

2. The co-homology groups. This section is concerned exclusively with additive properties. I t is purely algebraic and depends only on a sequence, C ---- ( C n} (n ----- 0, 1 , . . . ) of additive Abelian groups, related by a "co-boundary" homomorphism, ~ : C n --+ C n+l , for each n, such tha t ~ ~ 0. We assume that each C n is a free Abelian group of finite rank, no two having an element in common, and tha t C n -- 0 for all sufficiently large values of n. We use the language of co-homology, but this is purely conventional. To translate this section into the language of homology we have only to delete the prefix "co" and re-write C ~ as C~_~, where N is such tha t C n ~ 0 if n ~ N .

Let H n (m) be the n-dimensional co-homology group, defined in terms of C and 6, with integers reduced rood. m as coefficients (m : 0, 1 ,2 . . . . ; H~(1) = 0). We shall write H~(0) -~ H ~, the co-homology group with integers as coefficients. For convenience at a later stage we agree that all the groups H n (m) have the same zero element. Given p > 0, q ~ 0, we define operators,

A~ : H n (q) - + H ~+1 (q :> O) , i~ , ,q : H ~ (q) ---> H n (p ) ,

as follows. Let x e H n ( q ) and xte x. That is to say x' is a co-cycle, rood.q, in the co-homology class x. Then (~x ~ = q y ~ , where yr is an (absolute) co-cycle. We define A q x as the (absolute) co-homology class of y~. Thus we could write Aq ~-- (1/q)~. Let c---- (p, q). Then

Therefore ( p ] c ) x r is a co-cycle, rood.p, and we define /~,qx as its co- homology class mod.p. I t is easily verified tha t ~laX and /~,qx depend only on x e H n (q) and not on the particular choice of x r e x. They are obviously homomorphisms. If p ] q, in which case c ~ p, and in particular if q -~ 0, then lu~,,qx is the co-homology class of the same co- cycle x z, but calculated mod. p instead of rood. q. Thus #~, o is the natural homomorphism of H n onto H ~ ~- H n - - m H n . We shall sometimes write Aq, #~,q simply as A , # .

As an immediate consequence of (2.1) we have

A ~/~p,q ~ ~ A ~ . (2.2)

50

Also I say tha t , if p , q > 0 , r ~ 0 , then

q (p, r) #p,q/~q,r : ( p , q) (q, r)/~p,r �9 ( 2 . 3 )

For let x e H a ( r ) , x l e x , and let a ~ - ( q , r ) , b : ( r , p ) , c : ( p , q ) . Then /~q.~x~Ha(q) is represented by y ~ = (q /a)x r and l% ,~#q .~x e H a(p ) b y (p /c)y r -~ ( p q / c a ) x ~. On the other hand / ~ , ~ x e H n ( p ) is represented b y (p /b)x ' . L e t d -~ (p , q, r) -~ (c, a). Then (ca/d) ] q, since c l q , a l q , d - ~ ( c , a ) . Also d l b . Therefore ca l q b . T h a t is to say qb/ca is an integer. B u t

p q x~ q b ~ _ . X !

c a c a '

whence bo th sides of (2.3), operat ing on x, are represented by the same co-cycle, mod. p .

The union of all the groups, H a (m), re la ted b y the homomorphisms, A,/~, will be called the co-homology spec t rum of the set of groups C. We shall denote i t b y H . Notice t ha t H is not the direct sum of the groups H a (m). An element of H is an element of an a rb i t ra ry one of the groups Ha(m) .

Let v : H a -~ H a be the endomorphism defined b y r x = m x, for a given value of m. Le t # ----- ~m. o, A - - Am. Then the sequence of homomorphisms

�9 �9 �9 -~ H n -~ H n -~ H a ( m ) -~ H a+l -~ . �9 �9 (2.4) A v I~ ~ v

is exact, meaning t ha t the kernel of each is the image group of the one preceding ~) it. Fo r i t follows a t once f rom the definition of v,/~, A t h a t �9 ' A - ~ 0 , / ~ r : 0 , A t ~ 0 . Conversely, let v x ~ m x - ~ O , where x e H a , and let x r e x . Then m x ~--~0. T h a t is to say m x r~ - ~u r, for some u~e C~-1. Then u t is a co-cycle, mod. ra, and x -~ A u , where u e H a - 1 ( m ) is the co-homology class of u I. Therefore r - l ( O ) : A H n - l ( m ) .

Let # x ~ 0 , where x e H a , and let x ~ e x . Then x ' is an absolute co-cycle, and /~ x is its co-homology class mod. m. Since/~ x = 0 there are co-chains u ' e C a, v~ e C a-x such t h a t x t - - m u r ~ 5v r. Then m ~ u r -~ O. Since C a+~ is a free Abelian group it follows tha t ~u ~ -~ 0. Therefore u ~ is an absolute co-cycle and m u ~ ~-~ x r. Therefore x ~- m u ,

where u e H a is the co-homology class of u ' . Therefore #-~(0) ----- r / / n .

~) H ~ is t h e (free Abel ian) g roup of co-cycles in C ~ whence v : H~ H ~ is an isomorphism of H o in to itself.

5 l

Let Ax-----0, where x e H n ( m ) , and let x ' s x . Let ~ x ' - - - - m u ~.

Then u ~ s A x and since A x = O we have ur-----~v ~, for some v ~ e C n. Therefore 5x ~ - ~ m ~ v r, or ~ ( x ' - - m v ~ ) : 0 . Th a t is to say x ~ - m v ~

is an absolute co-cycle. Le t y be its co-homology class. Then x ---- t~Y, since x~-~ x t - - m v ~, m o d . m . Therefore A-l(0)-----/~H n, which completes the proof t h a t (2.4) is exactS).

I t follows f rom the exactness of the sequence (2.4) t h a t ~d~Hn(m) --

~ H n+l, the sub-group of H n+l consisting of all the elements, x s H n+l,

such t h a t mx---- O. The group H'~(m) ( re>O) is a tr ivial extension of 9) /~m.oH n--- I t~ by m Hn+l. Hence there is an isomorphism A*: m Hn+l -->" H n ( m ) (into, no t necessarily onto) such t h a t A~A* -~ 1. We shall describe A* in detail , bu t first observe tha t , in consequence of this, H n ( m ) is the direct sum

H n ( m ) ----- H ~ + Z]*(,nH n+l) , (2.5)

whence H n ( m ) = O,~(g n -+- ,~H n+l) , (2.6)

where 0~ [ H n ----- lure. o, Om [ ~n Hn+l ----- Z]*m . I t will be convenient to allow m-----0 in (2.6), wi th 001 H~--- 1, 00H n+x--- 0 (N .B . oH n+~:-Hn+l) ,

bu t we do not define Ao*. We shall of ten wri te A* and 0n as A* and 0. We m a y subject the opera tor A*, which is no t de te rmined uniquely,

to the condit ion

Y = I y (Y (2.7)

To prove this we recall t h a t C n+l and hence H n+l has a finite number of generators . Therefore H n+l is a direct sum of a free Abelian group and finite cyclic groups U1 . . . . . Us. Obviously

~H n + l - - ~U1 + " . + ~U~

for any m > 0 . Therefore i t is enough to prove (2.7) wi th y s qU~ for an a rb i t r a ry value of i - ~ 1 . . . . . t . Le t us wri te U,----- U, let u b e a genera tor of U and a its order. L e t u ' s u . Since a u ----- 0 there is an n-dimensional co-chain, v ' , such t h a t Ov I - - a u . Let v s H n ( a ) be the

s) I t can be proved in much the s a m e w a y t h a t the sequence . . . . I~lm,l,l~n~,l~' t~l,d~Jm,/~lm,1 . . . . is exaet for any 1, m ) 0 . If l~-~0 this is the same as (2. 4), with the c o n v e n t i o n t h a t I~im,l ~ ~, /~l,o ~-~ 1 if I ~--- 0.

~) Cf. [7], pp. 218 et seq. A proof of this, which is in a n y case elementary, is included in the proof of (2.7) below.

5 2

co-homology class of v ~. Then Ao v = u. Let k u e ,,,U. Since m k u = 0 we have m k --~ O, mod. or, whence or 1 k (m , or). We define

k (m, or) Am* k U - - ~um,,, V . ( 2 . S)

or

Then it follows from (2.2) t ha t

A , . A * k u --- k (m, or) A~ ,urn,,. v or

Ic ( m , or) or ~- - - ~ v

or (m , or)

~ k u .

Therefore A ~ A ~ = 1. L e t p , q > 0 be given and let a = (q, or), b = (or, p), c - ~ ( p , q ) . Le t k u e q U and let U = k q / c . Then p U u = (p /c )qku-~ O, whence U u e ~U. I t follows from (2.8) and (2.3) tha t

k (q, or) ]l~p,q A : ]g u - - /~p,q t~q,q V

or

k a qb - - - I.Zp , cr V

(~ c a

k q b c Sl~p,,, v

k' (p, a) - - # p , a V

or

= A* U u

=A,*~ q, k~ ~ I~ q) i ' which establishes (2.7).

Since O~I H ~ = # ~ , o , O~I~ttn+I-----A*, A~l%,o-----O and A~A* = 1 (P > 0) we have

A~O,(x q- y ) = A ~ l ~ , o x q- A~A*p y

= y , (2.9)

where x e H ~, y e ~r/~+x. Also, if x e H ~, y e qH ~+~ (q > 0) it follows from (2.3), wi th r = 0, and (2.7) t ha t

53

i~ ,q Oq(x -k- Y) = #~,q ttq, o x A- t%,q A* y

_ p (p , q) ~p,o x + q t

=-o ImP q ~'t ( P , q ) x-+- ( p , q ) Yl " (2. lO)

Thus A and 1~ # are expressible in terms of A* and/tin, o. The conditions A A * = 1 and (2.7) are the same as (2.9) and (2.10) wi th x ----- 0. They m a y therefore be interpreted as necessary and sufficient conditions, which A* must sat isfy in order t h a t the operators A , / t , defined in terms of A* and #~. o by (2.9), (2.10), shall be the same as those with which we start- ed. Any operator A* : , ,H n+1 ---> H n (m), which is defined and is a homomorphism for every m > 0, n ~> 0 and satisfies the conditions A A* = 1 and (2.7), will be called an admissible right inverse of A.

Now let a system of addit ive Abelian groups, H n (m) (m , n : O, 1 . . . . ; Hn(1) = 0) be given abstractly. We assume t h a t each of them has a finite number of generators and also tha t H n (m) ~ 0 for all suffkciently large values of n . All the groups H n (m) shall have the same zero element but no two shall have a non-zero element in common. As before let H n = Ha(O) and let H n (m) ( m > 0 ) be related to H~ and ~ H n+l by the equa- tions (2.5), where A* : ,~H n+l -+ H a ( m ) is a given isomorphism (into). We admi t two possibilities. The first is t h a t A, # are given, satisfying (2.2), (2.3). In this case we require A* to satisfy A A * = 1 and (2.7), and hence (2.9) and (2.10). The second possibility is t ha t A, # are not given, in which case we define t hem by (2.9) and (2.10). In either case an admissible r ight inverse of A will mean the same as before and any two admissible r ight inverses of A will be regarded as equivalent. Thus it is A and /x which are fundamenta l , not a part icular one of the operators A*.

I t is easily verified t ha t (2.2) and (2.3) are formal consequences of (2.9) and (2.10). I t also follows formally f rom (2.9) and (2.10) tha t the sequences (2.4) and . . . . #z,~,~, g~,zm, lxt,oA~,~, # t~ , z . . . . (1, m > 0 ) are exact. For example, if x e H ~, y e tH ~+x it follows f rom (2.10) that

=- O , , (mx + l y )

~--0~

10) i. e. Aq a n d ~ , q w i t h q ~ O. I t is a lways t o be unde r s tood t h a t ~m,o is the na-

tural h o m o m o r p h i s m /~m,o = Om ] H n : H n § ~ t~ (n ~ O, 1 . . . . ).

54

since Om m H n -=- O, y e tH ~+1. Conversely, let

0 = ;x,..tmOt.,(x -f- y) = O,.(x -4- ly) ,

where x ~ H ~, y ~ tmH n+l. Then x = m u (u e H n) and l y --- O, since A* is an isomorphism. Therefore y e tH ~+1 and it follows from (2.10) that

~tt.t,t O~(u -4- Y) = Otm(mu A- Y) = Oz.,(x -4- Y) �9

Therefore #~,zm(0)-i =/~lm,~ H ~ (l). The other relations follow from similar arguments.

The system of groups H n (m), related in this way by A and/~, will be called a spectrum o/co-homology groups or simply a spectrum. As before an element in such a spectrum will be an element in an arbitrary one of the groups H~(m). Notice that the whole spectrum may be defined in terms of the groups H "~, in case we are given these groups alone. For H~(m) may be defined by (2.5), where A* is an isomorphism of ~H ~+1 onto a newly defined group A* (mH ~+1) (m>0) .

By a proper homomorphism / : H - + / ~ of a spectrum, H , into a

spec t rum, / t , we mean a transformation such that / [ H n (m) is a homo-

morphism of H n (m) into H~ (m), for all values of m, n, and /A -- A/,

f# = / ~ / . If also ] ] H~(m) is an isomorphism onto /t~(m) for all values

of m, n, then ] will be called a proper isomorphism of H onto H and H

will be described as properly isomorphic to H . If f : H -+ H is a proper

isomorphism of H onto H , then its inverse,/-1, given by /-1 [ ~ (m) =

{/]H~(m)} -1, is a proper isomorphism of / t onto H . :For /A = A / , /~ -----/~/ obviously imply /-1A = A/-1, /-1/~ = /~/-1. A proper homo-

morphism / : H -+ H is a (proper) isomorphism onto if, regarded simply

as a transformation of the set of elements in H into those o f / t , it is onto and (1 -- 1). I t is therefore a proper isomorphism if there is a transfor-

mation, / ' : H - + H , such that / ' / = 1, f / ' = 1, in which case f ' = 1-1. Since an admissible right inverse, A*, of A is not, in general, uniquely

defined, a proper homomorphism, f : H - + i t , does not necessarily

commute with A*. But let A* be given in H and let f0 : H~ -+/-Y~ be the homomorphism f l H= for each value of n. Then we have the lemma

Lemma 1. I 1 each ]o : H'~ ~ [t~ is an isomorphism onto H~, then

there is an admissible right inverse, -A*, of A in /{ , such that A* 1o -~ ] A*.

55

Let A* : ,~n+l __> ~ n ( m ) be defined by A* y ---- !A* -~ - /0 y, for each e ~ n + l . Since A / : / A : / o A , where A denotes the operator A in

/ t , and since A A * - ~ 1 in H, we have

Similarly (2.7), in H , follows from # / ~ / / ~ and (2.7) in H. Therefore

z~* is an admissible right inverse of A. Obviously A* to ~ / A * .

Lemma 2. Let / : H -~ H be a proper homomorphism such that ! I Hn

is an isomorphism onto ~ n /or every value o! n. Then / is a proper iso-

morphism o! H onto H .

I t follows from Lemma 1 that ! A* --~ A*/~ with a suitable choice of

A* in H. Let 0 mean the same as in (2.6) and let 0 be the corresponding

homomorphism in H. Since 0 H ] H 'n -~ l~H. o, OH I H Hn+l ----- A* , with

similar relations in /~ , and since //~H, o

/OH(x -~ y) : OH(/oX + / , y )

Let /~ : H --> H be defined by

/ ' ~H (~ + ~) = o~ (/~-' ~ + / 0 1 ~) (~ ~ ~ - , ~ ~ H,q~§ ~) . Then

whence / ' / = 1.

: / ~H , o/0, / A* : A*/o we have

( x e H n, y e H H ~+1) . (2.11)

/' / oHr + y) =/'oH(/oX +/oy)

= OH(X + y) ,

Similarly / ] r = 1. Therefore / is a proper isomor-

phism of H onto H, and the lemma is established.

Lemma 3. A n y set o! homomorphisms /o : Hn ~ I ~ (n -~ O, 1 , . . . )

can be extended to a proper homomorphism / : H -> H .

For / I Hn(m) (m>O) may be defined by (2.11). Then it follows from (2.9) and (2.10) that A / ~ / o A , # / ~ / / ~ . This establishes the lemma.

Let H n(m) be defined in terms of co-chain groups C n (n ~ O, 1 , . . . )

and an operator ~, as at the beginning of the section. Let Hn(m) be

similarly defined in terms of co-chain groups C -n. We shall use j~ for the natural homomorphism of the group of co-cycles, rood. m, in C n onto Hn(m). Thus if a e C n is a co-cycle, rood. m, then ?H a e Hn(m) is its

56

co-homology class. We shall also use Jm to denote the na tura l homo-

morphism of co-cycles, mod. m, in ~n onto H~ (m). By a co-chain m a p ,

or mapp ing , of C n ' in to C~ we shall mean a set of homomorphisms g :

C ~ -+C~ (n ~ 0, 1 . . . . ) such t ha t 9~ -= ~g, where 6 also denotes the

co-boundary opera tor in ~n. A co-chain map, g, obviously induces a

proper homomorphism, ~ : H -~ H , which is defined by / j,na ~ jmga ,

where a is a ny co-cycle mod. m. Subject to this conditions which we write simply as / j ~ j 9 , we say t ha t / is realized by the co-chain map g.

Le mm a 4. A n y proper homomorphism ~ : H -+ H can be realized by a

co-chain m a p g : C '~ --> C'~.

As in the t he o ry of finite complexes, there is a basis a l , . . . , a~, for each group C n, such t h a t 6ai -= a ib t (i : 1 . . . . , t <~ q), Oa t -~ 0

( i : t ~ - 1 , . . . , q ) , where a ~ > 1 and b 1 . . . . . bt are basis elements in the basis for C "+~. We write ~ a ~ = a ~ b i ( i = 1 . . . . ,q), with a i ~ - 0 , b t : 0 if i > t . A co-chain 2 1 a 1 - [ - " ' A - 2 q a q is a co-cycle mod. m (m > 0) if, and only if, 2~ g~ ~ 0, mod. m, for each i = 1 . . . . . q. This is equivalent to 2~ ~ 0, mod. 0i, wheren) Q~ ~ m / ( m , at). Therefore the group of co-cycles mod. m is genera ted by ~ a~ . . . . . Oq aq and j~ Q~ a~, . . . , ?'~ Oq aq generate H n (m). Notice tha t

?'m ~i a~ -~ tur~,~ j~ a i , (2.12)

according to the original definition of /, . I say t h a t a cochain map,

g : C ~ - + C ~, realizes / provided only t ha t

Ja, a~ = )at g a~ (i = 1 . . . . . q) , (2.13)

the analogous conditions being satisfied for all values of n . For , writ ing q~ = q, at = a, a~ = a , i t follows from the relat ions (2.12) and from

] g = # / and (2.13) t ha t

/ j m q a = /~ tm,~ j , ,a = ~,,,,~f j ~ a

= ~,~,~jag a -~ jm q g a

- - - : i , , g q a ,

or t ha t / j = j 9 . Le t N be such t h a t C ~ = 0 if r > N , let n ~< N and, s tar t ing wi th

g C r = 0 if r > N , assume tha t g: C k --> ~k has been defined for each

11) (m , o) = m , w h e n c e Qt = 1 i f i > t .

57

k > n in such a way t h a t f j = i g, g 8 = ~ g- We proceed to define g :

C n- ->C n. As before we wri te a , ~ a , a , = a , and also ~ a = a b for an a rb i t r a ry value of i = 1 . . . . . q (a ~ O, b = 0 if i > t ) . I f i > t , t h en a is an absolute co-cycle and i oa e H n. In this ease we write g a : a , when a is any (absolute) co-cycle in the co-homology class

[ i o a . Then g ~ a = - ~ga-----0 and i o g a : i 0 a : / i o a . Le t i ~ t , so t h a t q ~ 0, b r 0. Then b is an absolute co-cycle and J0 b e oH n+s,

since a b ~ Sa . Also g b is defined, since b e C '~+1 , and io g b = / io b e ,,tt n+l.

Therefore a g b ----- 5 a , for some a e ~n. According to the original defi- ni t ion of A we have Aqiqa-- - - - iob , A~i q a = i o g b . Since i o g b = [ i o b and since / A = A [ it follows t h a t

A , , i ~ a : i o g b = / i o b : / A q i q a

= d a / i ~ , a �9

Therefore A,,(/1,, a - - ]a-a) = 0 . I t follows f rom the exactness of the sequence (2.4), with m = a, t ha t

1i,7 a - - i a -a ,~ f la ,o ~ n �9

Therefore [ i a a = i ~ a - ~ i a u = i , ( a ~ - u ) , where u e C n is an absolute co-cycle. Le t g a = a ~- u and define g a i in this way for each

value of i = 1 . . . . . q, thus defining g : C n__>~n. Since i~ga---- iq( ~ ~- u) ----- f iq a i t follows f rom our pre l iminary result t h a t [ i = i g in C n. Also ( ~ g a = ~ a = a g b - - - - - g a b = g ~ a . Therefore g ~ : 5g. Repea t ing this cons t ruct ion we define g t h roughou t all the groups C ~ (n = N , N - 1 . . . . . 0) and the lemma is established.

3. T h e t o . h o m o l o g y ring. We now assume t h a t the group C ~, in the sys tem C, wi th which we s ta r t ed in w 2, is the group of n-dimensional co-chains ~) in a finite.simplicial complex K. The addi t ive group of the co-homology ring, R , of K shall be the "f ini tely gene ra t ed" direct sum of the co-homology groups H n ( m ) , for all values of m, n . T h a t is to say, the addi t ive group of R shall consist of all finite sums x~ ~- x 2 ~-" �9 ", where each x t is in an a rb i t r a ry one of the g roups la) H n (m). The product [ U g, which we wri te s imply as / g , of co-chains [ e C ~, g e C q shall be

1~) We distinguish between a chain and a co-chain, which is to be an integral valued function of chains. All our chains will have integral coefficients.

is) As in w 2, the groups Hn(ra) all have the same zero element and we regard them as imbedded in the additive group of R.

5 8

v defined by the Cech-Whitney method 14) in terms of a fixed ordering of the vertices of K. The product, x y, of elements x s Hp (m), y e Hq (on), shall be the co-homology class, mod.m, o f / g , where [ e x, g e y. Then

x y s H P + q ( m ) , x y = (--1)Pqyx . (3.1)

If x s HP(r), y ~ Hq(8) we define x y as

x y -----/zc,~(x ) tzc,s(y) , (3.2)

where c = (r, s). Thus x y is the co-homology class of /g mod.(r , s), where / s x, g e y. We then define the product, x y , of any two elements, x, y ~ R, by the condition that x y shall be bilinear. I t is easily verified tha t ( x y ) z - - - x ( y z ) for any x, y , z c R . This defines R, ex- cept for the Pontrjagin squares, which will he discussed in the next section. Clearly R(m), the co-homology ring with integral coefficients reduced mod.on, is a sub-ring of R for every on ~ 0. Also R has a unit element, e s H ~ which is the co-homology class of the co-cycle, which has the constant value, unity, at every vertex of K .

Let A ( m ) denote the additive group of R(m) . Then the homomorphisms /~r,s, As, defined in the last section, may clearly be extended through A (s) to give homomorphisms #r,8 : A (s) --> A (r), As : A (s) --> A(0). I say that, if x s R ( r ) , y s R ( s ) , then

on (m, r , s) #m,r (x) #,,,s (Y) = (m, r) (m, s) /~m,c (x y) , (3.3)

for any m > 0, where c = (r, s). Because of the bi-linearity of x y and the linearity of/~ it is sufficient to prove this for x ~ HP(r), y e Hq(s). Let this be so and let ] e x , gs y. Let a = ( m , r ) , b = ( m , s ) , d = (m, r, ~) = (m, c). Then (m/a) ~ s #~,~x , (m/b) g s/~m.sY, whence

Also

and

on2 a b [ g e I~m,r (x) #~,8 (Y) �9

(mid) / g e #~,c ( x y)

0n d m m 2

a b " --d- l g = -d-b l g "

Therefore both sides of (3.3) are represented by the same co-cycle,

14) See [8], [9], [10], [11] and [6], chap. V.

59

rood.m, and (3.3) is established. Notice that (3.3) reduces to (3.2) if m=c- - - - - ( r , s ) . For then m = ( m , r ) = ( m , s ) = ( m , r , s ) and ff~,~

1.

Out of respect for the operator A we display the formula

ffc,o A c ( x Y ) = (Ac xc) Yc + ( - - 1)P xc(A c Yc) , (3.4)

where x e H P ( r ) , y s R ( s ) , c = ( r , s ) and x c = f fc , ,x , Y c = Pr This follows without difficulty from the relations ~ ([g) = (~/) g + (-- 1)P/( J g) and (2.2), where [ ~ C p, g e C q. However we shall not need this because of the special nature of our complexes.

4. The Pontrjagin squares. Let K mean the same as in w 3. Let [ e C~, g s Cq. Following Steenrod is) we use [ Ulg to denote the ( p + q --1)- dimensional co-chain, which was introduced rood. 2 by Pontrjagin is), and which is given by

p--1 (t LJl g) d~~ = ~ ( - - 1) ('p-~}{q+i) ( l o'~) (~ o~) ,

where ap+q-1 = a o . . . a ~ q _ 1 is a given (p + q -- 1) simplex and a~ = ao. . .a~ aj+q. . .a~+q_l, 0~i = aj . . .a~+q. Steenrod le) proves that

Let

~(IU~g) = (-- 1)P~-l(lg - - (-- 1Fqg[)

+ ,~tU, g + (-- 1)"tU,,~g. (4.1)

p [ = P + l u , ~ l ( f f = l [ = l U l ) , (4.2)

where [ is any co-chain. If [ is a co-cycle, mod.2r , then p / i s a co-cycle mod.4r . For let ~[-----2ru. Then, calculating mod.4r , we have

o p l = (~1) t + t(~1) + 1(~1)- (~t)1 + OlU,~l = 2 r ( : : J : : [u + lu ) + 4r2uLJl u

~ - 0 .

Let ~ [ = 2 r u , (~g= 2 r v , where [ , g c C n. Then

p ( l + g) - p l - pg = lg + gl + lU,~g + g U l ~ /

= lg + g l + 2 r ( lU ,v + gU, u) . (4.3)

15) [1] and [2]. 16) [2], Theorem (5.1), p. 296.

60

It follows from Steenrod's co-boundary formulale), with i = 2, that g U1 u ~ u Ulg (mod. 2), both 9 and u being co-cycles mod. 2. Therefore

2rgUlu ~.~2ruUlg mod.4 r . (4.4)

I t follows from (4.1), with p = q : n , that

g / ~ (-- 1)~fg -- 2r( /Ulv =[= uUlg) . (4.5)

Calculating mod.4r , it follows from (4.3), (4.4) and (4.5) that

p(f + g) -- p / -- pg ~-~/g -{- g / + 2r(/U~v + uUlg)

(1+ (-- 1)n)/g . (4.6)

Let [1 . . . . . f, c C n be co-cycles mod 2r. Then it follows from (4.6) and induction on s that

$

p ( f l + . . . + L) ~ X p f, + ~ X f, fj (4.7) i = l i ~ ]

where v = 1 + (- -1)n. I t also follows from (4.6) that P / I~ '~P 12, mod. 4r, if 11~--I2,

mod.2r, where 11,12cC n are co-cycles mod.2r . For let g = 2rh in (4.6). Clearly p (2rh) = 4r ~ ph, whence, calculating mod. 4r, we have

p(f + 2rh) --, p / + (1 + (--1) n) 2r/h =- pf .

Secondly, let g = ~h. Then pg----- (~h) ~ = ~5(h~h)~-~ O. Therefore, ff f = 2 r u , we have, mod. 4r,

--~ pl + (1 + (--1)n) (51) h

= P/�9 2r(1 + ( - -1 ) n) uh

- P / �9

Therefore

P / ~ PO t + 2rh) ,,, P(1 + 2rh + ~h') rood. 4r .

The co-homology class, mod.4r , of p / , where I is a co-cycle, rood. 2r, is independent of the ordering of the vertices of K. This follows from (8.2) (i = 1) and (8.3) (i = 0) on p. 302 of [2] and the (easily verified) fact that, if u, v are co-cycles, mod. 2, then u V0v--~v V0u, mod. 2.

We now derive a rather cumbersome set of relations, (4.8), as a basis for the relations (4.11) between p and #. Let I be a co-cycle mod. m,

61

which m a y be even or odd, and let #.f = m u . Let a = (2r , m), e =

2r/a . Then ~[ is a co-cycle m o d . 2 r and we have

~ ' ( e t ) = e~ ( l ~ + / U , , ~ l )

= e 2 1 ~ + e 2 m / U l u �9

Since 2r ] @m it follows t h a t e2m - - 0, m o d . 4 r , if e is even, which it cer ta inly is if m, and hence a , is odd. Also (4r , 2m) = 2 (2 r , m) = 2a, whence Q = 2r/a = 4r / (4r , 2m). Also if m is odd we have (4r , m) ---- (2r , m) = (r, m), whence

4 r 2 r 4 r e 2 = (r , m) (4 r , m) = (r,m-----) ( 4 r , m ) "

Therefore, calculating m o d . 4 r , we have

4 r P(~/ ) - - - -~=P/ - - - -~ ( 4 r , 2 m ) P /

p (e 1) = # 1 ~

if m is even

if m is odd

r 4 r m'-j " (4 r , ~ ) / 3 . (4. s) (r,

We now int roduce Pontrjagin squares into the ring R , of w 3. These are a fami ly of maps P2r : Hn(2r ) --> H~n(4r), which are defined for all values of n , r / > 0 . I f x e H n ( 2 r ) , ~ e x t h e n p ~ x , or s imply p x , is the co-homology class, m o d . 4 r , of p [ . F r o m (4.2) we have

(a) ff2,,4, P2rX = x ~ (r >O) I

(b) p o X = X 2 ( x ~ H ~) . (4.9)

I f x 1, . . . . x, c H n ( 2 r ) i t follows f rom (4.7) t h a t

$

p2~(xl + " " + x~) = ~7 P2~ x~ + ~ �9 x~ x~ , (4.10) i=1 i < j

where �9 x~ xj ---- 0 or tt4r,~r(x~xj) according as n is odd or even. Let x e Hn(m) . Then i t follows f rom (4.8) t h a t

2 r P~l t2r '*ax- - ( 2 r , m) #4r,~mPmx if m is even

( r , m ) la4~'mx~ i f m is odd. (4.11)

62

I t is obvious tha t

p 0 e : e , (4.12)

where e is the unit element in R. Notice tha t the relation 2p2,x ----/~4~.,rx * (r>0), which is' an ob-

vious consequence of (4.2), may be proved formally by operating on both sides of (4.9a) with #**,2,. The result then follows from (2.3). Since s Jr 2s(s -- 1)/2 ---- s 2 it follows from this and (4.10), with x 1 . . . . x s = x e Hn(2r), tha t

p2r(SX)-----8p2rX if n is odd (4.13)

~-s2p2rx if n is even .

The ring R, complete with Pontrjagin squares, will be called the co- homology ring of the complex K. Let R(K) and R(L) be the co-homology rings of complexes K and L . By a proper homomorphism R(K) -> R(L) we shall mean one, other than the trivial homomorphism R(K)-~O, which induces a proper homomorphism, H (K) -> H (L), and which commutes with the operator p, where H(K) and H(L) are the co- homology spectra imbedded in R(K) and R(L).

5. Simple 4-dimensional co-homology rings. We now lay down the conditions on a ring, R, which is given abstractly, in order tha t it may be the co-homology ring of a finite simply-connected polyhedron of at most four dimensions. We shall describe such a polyhedron as a simple, 4-dimensional polyhedron and the corresponding ring as a simple, 4-dimensional co-homology ring. First of all the additive group of R shall be the finitely generated direct sum of the groups, Hn(m), in a spectrum of the kind discussed in w 2, such that H ~ is cyclic infinite and Hn(m) -~ 0 if n ----- 1 or if n > 4. Since H 1 (m) ----- 0 it follows from (2.5) tha t H * contains no element of finite order. I t is therefore a free Abelian group.

As regards multiplication, R shall have a unit element, which is to be a generator of H ~ Also (3.1) and (3.3), and hence (3.2), shall be satisfied.

A map P2r : Hn(2r) -~ H~n(4r) is defined, for all values of n, r ~ 0, which satisfies (4.9), (4.10), (4.11) and (4.12). We shall call p2~x the Pontr]agin square of x e Hn(2r). Sometimes we shall write P2r simply as p.

Let R and R be two such rings. A homomorphism / : R -> R will be called a proper homomorphism if, and only if,

63

(1) / is not the trivial homomorphism R - ~ 0,

(2) ] induces a proper homomorphism, as defined in w 2, of the spectrum

of groups H n ( m ) c R into the corresponding spectrum in /~.

(3) f p ~ p], where p denotes the Pontrjagin square operator, both in

R and /~.

In particular a proper homomorphism may be a proper isomorphism,

and R and /~ will be described as properly isomorphic if, and only if, there is a proper isomorphism of one onto the other. The inverse of a proper

isomorphism is obviously a proper isomorphism. If f : R - ~ / ~ is a proper homomorphism, then f ] H ~ is an isomorphism of H ~ onto the cor-

responding group H--~ R. For let e, e- be the unit elements of R, R.

Since e, e generate H ~ H ~ and since f H ~ ~ we have fe = k e for some value of k. Since e, ~ are idempotents and f is a homomorphism we have k S ~ = k ~ - e 2 = r e 2 - - - f e = k ~ . Since (k 2 - k ) ~ = 0 implies k S---k it follows that ]c-----0 or 1 . I f f ( e ) = 0 , then f ( x ) = f ( e x ) = - O for every x s R, which is excluded. Therefore f(e) = ~ and f I H~ is

an isomorphism of H ~ onto /~o. Our main theorems are:

Theorem 1. A n y simple, 4-dimensional co-homology ring can be realized geometrically by a simple, 4-dimensional polyhedron. That is to say, it is possible to construct a simple, 4-dimensional polyhedron, whose co-homology ring is properly isomorphic to a given ring of this type.

Theorem 2. Two simple, 4-dimensional polyhedra are of the same homotopy type if, and only if, their co-homology rings are properly isomorphic.

The part of Theorem 2, which is contained in the clause "on ly if" asserts that the co-homology ring is an invariant of the homotopy type. This is true of any simplicial complex since the Pontrjagin squares of co- homology classes, like the products, are independent of the ordering of the vertices. More generally, let K and K / be two simple, 4-dimensional polyhedra and let R and R I be their co-homology rings. Then a given homotopy class of maps, f : K -~ K I, determines a unique proper homomorphism f* : R ~ --> R. Theorem 2 will be proved with the help of a partial converse of this, namely:

Theorem 3. Let R and R ~ be the co-homology rings of simple 4-dimensional polyhedra, K and K ~. A n y proper homomorphism, f* : R r --> R , is the one determined by at least one homotopy class of maps f : K -~ K v.

64

The fact t h a t the h o m o t o p y class of maps f : K -~ K t is, in general, not de termined uniquely by /* : R 1 -~ R m a y be seen b y taking K to be a 3-sphere, S 3, and K I a 2-sphere, S 2. Then H n = 0 unless n = 0 or 3 and H 3 is cyclic infinite. Similarly H rn = 0 unless n = 0 or 2 and H ~ is cyclic infinite. There is on ly one proper homomorphism f* : R ~ - ~ R , which is de te rmined b y f*(e ~) = e, ] * H t2 = 0, where e, e r are the uni t e lements of R , R ~ . This is induced by an y map ~ : S a -~ S 2 . I n order to establish a (1 -- 1) dual correspondence between h o m o to p y classes of maps / : K -> K I and proper homomorphisms /* : R f -+ R one must , presumably, enrich the concept of a proper homomorphism. For examplel~), we m a y demand tha t , if u e H ~p, v e i l ~q are such t h a t /*(u) = O, uv ----- 0 t hen an element

[/*, u , v] ~ H ~ +~-1 - - (f*H '~+~-1 + tt~-ll*v)

is uniquely de te rmined by ]*, u , v. However we shall leave these possible refinements aside and shall proceed to prove our theorems.

6. Theorem 3 implies Theorem 2. Le t K and L be finite s imply connected complexes zs) of a rb i t r a ry dimensional i ty and let f : K -> L be a map which induces an isomorphism of each co-homology group H~(L) , with integral coefficients, onto the corresponding group Hn(K) . Then Theorem 2 obviously follows f rom Theorem 3 a n d :

Theorem 4. Under these conditions K and L are o] the same homotopy type and ] is a homotopy equivalence.

In proving this we m a y assume, af ter a suitable deformat ion of the map f , if necessary, t h a t ] K n c L ~ for each value of n , where K n, L n are the n-sections of K , L . Then / determines a co-chain mapping, /* : C n(L) --~ C n (K), where C n (K) and C ~(L) are the groups of n-dimensional co-chains in K and L . B y hypothesis the homomorphisms H~(L) -+ H "~ (K), which are induced b y I , and hence b y ]% are all isomorphisms onto. I t follows f rom a theorem due to Lefschetz 19) t h a t the homomorphisms, H ~ ( K ) - > H , ( L ) , of homology groups, which are in-

17) See [3]. zs) These shall be simplieial or, more generally, cell complexes of the kind discussed in

w 7 below.

zg) [6], p. 148. There is a flaw in the paragraph following (10.11). For let there be just one 2-dimensional chain, b, and just one 3-dimensional chain, d, such that F d ~-- ad = 5 b. Then the chains 2b, 2d satisfy the conditions (a) and (b), with p = 2, but are not basis elements in any canonical basis for the chains. This may be repaired by an elaboration of Lefsehetz' argument. See also the following paragraph.

5 CommentarU Mathematic[ Helvetici 65

duced by the chain mapping dual to /*, are also isomorphisms onto. Since these are the homomorphisms induced by [ the theorem follows from a theorem which is proved in [12].

We give an alternative proof of the theorem of Le[schetz 1~). Let H (K) and H(L) be the cohomology spectra of K and L and let h : H (L) -~ H ( K ) be the proper homomorphism induced by the co-chain mapping /*. Since h I Hn (L) is an isomorphism onto, for each value of n, it follows from Lemma 2 tha t h is a proper isomorphism of H(L) onto H(K) . By Lemma 4 its inverse, h -I : H ( K ) --> H(L), can be realized by a co-chain mapping, g* : Cn(K) -~ Cn(L). Then /*g* induces the identical automorphism of H ( K ) and g*]* induces the identical automorphism of H (L). I t follows from another Theorem of Lefschetz 20), with homology replaced by co-homology, tha t /*g* and g*/* are each co-chain homotopic to the identity. That is to say, there are families of homomorphisms, a* : Cn+I(K) -~ Cn(K), b* : Cn+I(L) -~ Cn(L), such tha t

/ * g * - - 1 : ( l a*-~a*~ , g * / * - - 1-~ ~b* ~-b*~ .

Therefore g . / . - - 1 : a 0 ~ 0 a , / . g . - - 1 : b 0 ~ a b ,

where / . : C,~(K) --> C.(L), g. : Cn(L) --> C . (K) are the chain mappings dual to /* ,g* and a : C n(K)-~Cn+ i(K), b : C n(L)-~C~+ I(L) are the duals of a*, b*. H e n c e / . is a chain equivalence, which establishes the theorem in question.

7. Cell complexes. Simplicial complexes are unsuitable for what follows and we shall use instead a type of complex, which we shall describe as a (finite) cell complex, or simply as a complex. Let a n be a fixed n-simplex and let ~n be its boundary. A (finite) cell complex, K , consists of a finite number of open cells, no two of which have a point in common and each of which is homeomorphic to the interior of a n for some value of ~. Moreover the closure, -~n, of each n-cell, e n e K , is the image of a" in ~ map, / : a n _._>-~n, such that

(1) / ~ n c K n - 1 , where K n-1 is the (n -- 1)-section 21) o / K , and

(2) / I ( an -- ~n) is a homeomorphism onto e n.

20) [6], Theorem (17.3), p. 155. This theorem, like t he resul ts in w 2, can equally well be s t a t ed in t e rms o f homology or co-homology.

�9 t) i. e. the aggregate of ceils in K , whoso dimensionali t ies do no t exceed n - - 1. This is a sub-complex, whose d imensional i ty does no t exceed n - - 1, bu t need no t be as great. For example , a n n-sphere is covered by a complex, K , consis t ing of one 0-cell and one n-cell, in which K r - K ~ i f 0 ~ < r ~ n .

66

Such a map will be called a characteristic m a p for e n. We do not , a t this stage, impose any o ther condition on a characterist ic map. F o r example, / ~n need not coincide, as a point set, wi th a sub-complex of K .

Examples of cell complexes a r e :

(1) the complex project ive plane, regarded as a complex, K , consisting of a 0-cell, a 2-cell and a 4-cell, e 4. The 2-section, K S, is a 2-sphere and the " h o m o t o p y b o u n d a r y " fl e 4 , of e 4 , as defined below, is a generat ing e lement 22) of z3(KS) �9

(2) the topological product , S 2 x S 2, of a 2-sphere with itself, regarded a s a c o m p l e x K = e ~ 2 4 7 4, where e ~ P0 (poeS2) , ~ - 2 = S 2 x P o , ~ 2 = p o x S 2. I n this case 23) f ie 4 = a l a 2 , w h e r e a i is a generat ing e lement of ~2(~ 2) (i = 1 ,2 ) .

A cell complex m a y be buil t up cell b y cell, s tar t ing with a finite set of points K ~ Assume tha t a complex, K , has been constructed and let E n be an n-element (i. e. a homeomorph of an), which has no point in

common with K . Le t / : ~n _+ Kn-1 be any m ap o f /~n , the boundary

o f E n , i n t o K n-1. Le t e n = E n - ~ n and let ~ : K U E n - - > K U e n be

the ma p which is given by ~ v I K U e n = 1, q p = / p if p e / ~ n . Le t K A- e n be the space which consists of the points in K U e n with the identification topology determined by ~v. Tha t is to say a set X c K A- e n is closed if, and only if, q - x X c K U E n is closed. I t follows f rom this definition t h a t ~ l K , ~b I en are homeomorphisms, i. e. t ha t K and e n re ta in their topologies in K ~ e n . Therefore K + e n is a complex, whose cells are the cells in K , toge ther with e n . I f g : a n --> E n is a homeomorphism (onto), t he n q g : a'~ ~ K ~- e n is a characterist ic m ap for e ~. We shall say t h a t K ~- e n is formed by attaching e n to K by means o] the m a p

f : ~ n -~ K~-i . Let K be a given complex and let K o = K + e o ," K 1 ~ K,-4- el", be

complexes, each of which consists of K and one o ther cell, e~ (i ~ 0, 1). Let f i : a n --> e ~ be a characterist ic map for e~ in K i.

Lemma 5. I f ]o ] ~n ~ /1 I ~n in K , then Ko and K1 are o] the same homotopy type.

Let Po be the centroid of a n and let p be a var iable point in ~n. We refer a~ to " p o l a r " coordinates r , p , such t h a t (r, p) is the point which

s2) Cf. [2], w 19, pp. 310, 311. as) Cf. [13], w 3, where ab was denoted by a.b (it is now often denoted by [a, b]). See

also a passage in the proof of Theorem 5, in w 12 below.

67

divides the segment ToP in the ratio r : ( 1 - r). Then (0, p ) = Po, (1,p)----p. Let g t : ~ n - > K b e a h o m o t o p y o f go----/o[~n into g l = f l ] a ~ and let h o:Ko--~K1 be given by hoJK----1 and

ho/o(r, p) =/~(2r, p)

---- g2_~, P

if 0 ~ 2 r ~ l

if 1 ~ 2 r ~ 2 .

Let h 1 : K 1 --> K o be given by h I I K ---- 1 and

hi h ~r, p) = lo(2r, p)

g~r-1 19

if 0 ~ 2 r ~ 1

if 1 ~ 2 r ~ 2 .

I t is easily verified that ho, h I are single-valued and hence continuous ~4). We have

h l h o / o ( r , p ) = h 1 / l (2 r ,p ) if 0 ~ 2 r ~ 1

= h l g ~ - 2 r P if 1 ~ 2r ~ 2 .

Since h l l K = 1 it follows that

]~1 no ]o( r , P) ----- 1o( 4?` , P)

= 174,-1 P

= g2-~, P

if 0 ~ 4 r ~ l

if 1 ~ 4 r ~ 2

if 1 ~ 2 r ~ 2 .

Let ~, : K o -> K o be the homotopy, which is defined by ~, I K = 1 and

~,to( r , P) = 1o {(4 -- 3t) r, p} if

= g ( 4 - 3 o , - 1 P if

- - g�89 ( 4 - a 0 ( 1 - o P i f

0 ~ r ~ 1[(4-- at)

I 2 - - t 4 - - 3 t ~ r~ '~ 4 - - 3 t

2 - - t 1

4 - 3 t " ~ " "

I t is easily verified that ~ is single-valued, and hence continuous s d), and tha t ~o-~hlho, ~ 1 = 1 . A h o m o t o p y v~ :K 1 - , K 1, such tha t 7o~- ho hi, 71 = 1, is defined by (7.1) with ]o, gA replaced by ]1, gl-A, where

= ( 4 - - 3 t ) r - - 1 or (4 -- 3 t ) (1 - - r)/2, and the lemma is therefore established.

u) s~ [15], w 5.

68

If K and K I are two complexes covering the same set of points and if every cell of K / is contained in a cell of K, then K t will be called a sub- division of K and a simplicial sub-division if K t is a simplicial complex. The following lemma is useful when considering complexes which are constructed by the method of attaching cells. Let K 0 be a complex which has a simplicial sub-division K0 ~ . Let K ~- K 0 + e n, where the cell e n

is attached to K0 by a map ~n _~ K~-~, which is simplicial with respect

to K~ and some triangulation of /~n.

Lemma 6. Under these conditions K has a simplicial sub-division, of which K~o is a sub-complex.

This will be proved elsewhere.

8. Homology and co-homology in a cell complex. It follows from [151 or [16] ihat a (finite) cell complex, K , is a locally contractible com-

v

pactum. Therefore all the standard homology theories (Cech, singular, etc.) are equivalent and all our remarks on homology are to be interpreted in terms of the singular theory. We orient each cell, e n e K, and also the fixed simplex, a n , and restrict the characteristic maps, [ : an--~'e n, to those which are of degree ~- 1 in e ~. Let e'~l , " �9 �9 , eq" be the n-cells in K. Let e~ also denote the element of the relative homology group, H~ (K ~, K ' - I ) , which is represented by a characteristic map for e~, the vertices of a s being positively ordered25). Then H~(K n, K n-l) is a free Abelian group ~6) which is freely generated by e~ , . . . , e~. The natural homomorphism H,~(K n) --> H,~(K) is onto and its kernal is

aH~+ 1 (K n+l, K n) c H~ (K n) ,

where 0 is the homology boundary operator. Also the natural homomorphism H~ (K n) --> H . (K ~, K ~-1) is an isomorphism onto the sub- group 0-~ (0) c H,, (K n, Kn-~). Therefore, identifying each element of H~(K n) with the corresponding element of H~(K n, Kn-1), it follows that the homology groups of K may be defined in terms of chain groups C~(K) ~ H~(K n, Kn-1), and the boundary operator a.

We define co-chains as integral valued functions of chains and C '~(K) will denote the group of n-dimensional co-chains in K . We shall use

25) CL [5]. H n ( K n , K n - l ) ~-. Ho(K ~ if n -~ 0. Every 0-dlmensional chain is a cycle.

~) This and other assertions in this paragraph wiU be proved in a forthcoming book by S. Eilonberg and N. E. Steenrod. I have been greatly helped by a set of notes, prepared by Eilenborg and Steenrod, for a course of lectures, which Eilenberg gave a t Pr inceton in 1945--46.

69

~1,. - . , ~0q to denote the basis for Cn(K), which is defined b y ~0~ ei ~ e i~ -~ 0 if i ~: ~. We shall describe ~ , . . . , ~ as the co-chains dual to n n e~ . . . . . %. Following Eilenberg and Steenrod we shall describe a map, f : K -~ L , of K into a cell complex L , as cellular if, and only if, f K r c L r for each va lue of r -~ 0, 1 . . . . . A cellular map / : K -~ L determines a chain map, g : C~ (K) -> C~(L), as follows. Le t h~ be a characterist ic

~* '~ C~ (L) H~ (L n, L n-l) is the ele- map for the cell e i e K . Then gei e ~- men t which is represented by the map / h~ : (a n, ~n) _> (L n, Ln-1). The dual co-chain map g* : C~(L) --> Cn(K) is defined, as usual, by (g*~) c : ~p(gc), where ~ e Cn(L), c e C~(K). We shall say t h a t g, and like- wise g*, is realized b y / : K -~ L . Any map /0 : K -> L is homotopie to a cellular map ~7) /1 : K -~ L and if two cellular maps of K into L are homotopic to each other t hen the corresponding chain maps are chain homotopic .

We now assume t h a t all our complexes have simplicial sub-divisions. Le t K I be a simplicial sub-division of K . Then the identical map i : K -~ K ~ is cellular. Le t i* : Cn(K ~) --> Cn(K) be the induced co-chain mapping. The identical map i ~ : K ~ -~ K is not cellular, unless K r ~ K . However i t is homotopic to cellular map ~" : K I -~ K , with the proper- t y ~) t ha t ]K~cKo , where K o is any sub-complex of K and Ko I is the sub-complex of K ~ which covers K o. Le t j* : C~(K) --> Cn(K ~) be the co-chain mapping which is induced by ~. Since the maps ] i : K -~ K and i ~ : K ~ -> K ~ are each homotopic to the iden t i ty it follows t h a t each of i*]* and ]*i* is co-chain homotopic to the ident i ty . Therefore i* induces a proper isomorphism, as defined in w 2, of the co-homology spec t rum H ( K I) onto the spec t rum H(K) and ~* induces its inverse.

Le t the co-homology ring, R(Kt) , be defined as in w167 3, 4 and let A (K/) be its addi t ive group. Le t A (K) be the finitely genera ted direct sum of all the (absolute and modular) co-homology groups H'~(K, m) (m,n---~ 0, 1 . . . . ). Then i* determines a proper isomorphism 2s) h: A (K I) ~ A (K). We define R(K) by making h a (proper) isomorphism of R ( K ~) onto R(K). T h a t i s t o s a y , if x, y c A ( K ) , z e H ~ ( K , 2 r ) we define x y and p z by

x y -~ h{(h-lx)(h-Xy)) , p z -: h p h - l z . (8.1)

I f K is a simple, 4-dimensional complex the ring R(KI), and hence R(K), satisfies the conditions in w 5 for a simple, 4-dimensional co-

~) See w 16 below. ~s) i.e. an isomorphism which induces a proper isomorphism H(K')-->H(K).

70

homology ring. Moreover the unit element has the same geometric inter- pretation in R ( K ) as in R(K~). For let u e C~ u ~ e C~ ~) be the co-chains with constant value 1. Then the unit element, e~e R(K~), is the co-homology class of u r. Obviously i *u ~ ~ u, whence hC, the unit element in R ( K ) , is the co-homology class of u.

Let [ : K -+ L be any map of K in a complex L. Let R (L) be defined in the same way as R (K), by means of a simplicial sub-division, L ~, of L. The map [ determines a unique proper homomorphism R ( L r) --> R ( K ~) and hence, in the obvious way, a unique proper homomorphism R (L) -+ R (g).

For purposes of calculation, and especially for the sake of (8.4) below, we need to define products and Pontrjagin squares of co-chains in K. If ?, ~v are given co-chains in K we define ~ and pq by

~ = i* {(i* ~)(i* ~ ) } , o ~ = i* pj* ~ . (8.2)

When we pass from co-cycles to co-homology classes this obviously leads to (s. l).

Let Ko be any sub-complex of K and let K~ be the sub-complex of K r which covers K o. If ~o ~ C n (K) we shall denote the function ~v, restricted to chains in Ko, by ~o I Ko. Also ~r [ Kg will have a similar meaning if ~vre C~(K~). Let products of co-chains and Pontrjagin squares in K~ be defined by the same rule as in K ~, the vertices in K 0 taking their order from the ordering in K ~. Then 29)

v' 9' 1 go = (V' I Ko')(9' i K~)

(P V')I g~ = P(V'E g o ) , ~s.3)

for any co-chains ~i, ~or in K ~. Clearly iKo -= K~ and we recall that j K g c K o . Let io ----- i [ K0,

]o = j J K~ and let i* : Cn(Kro) --+ cn(go) , f f : Cn(go) -'-" Cn(K~) be the co-chain maps induced by i o, Jo. Let products and Pontrjagin squares of co-chains in Ko be defined by (8.2), with i*, j* replaced by io*, j* . Then I say that

~o ~p I go : (q~ I go)(~ t Ko) , (s .4)

(P ~)t K0 = P(~V[ go) , t

~9) Cf. [9], condition P1, P. 403.

71

for any co-chains 9~, ~ in K . To prove this we first show tha t

(i* 90') I g o = io* (90' [ Ko ~) , (J* ~) I K0 = Jo* (~P [ go) , (8.5)

for given co-chains 90'rCn(K') , y, eC'~(K). Let u 0 = u ' [ K~, % = v ] Ko, where u t, v are any co-chains in K t, K. Then (8.5) m a y be wri t ten in the form

(i* 90')0 - - - - -

Let c o e C , (Ko) and let i , i : K --> K ' , io : Ko --+ KO. C, (K). Therefore 90' i co

io* 90o, (J* V)o = Jo* Vo �9

io also denote the chain mappings induced by Then i Co = io Co e C~ (Ko), taking C~ (Ko) c

= 900 io Co and

Therefore (8.5) and (8.3) t ha t

90~[ K o =

Similarly (P 90)o = P 90o,

(i* 90') Co -----

(i* 90')0 ----- io* 900. Similarly (j* VJ)o = Jo* ~o-

" = (io* 90o)Co 90t i c o -= 900 Zo Co

I t follows from

(i* {(j* 90)(j* )})1Ko

/* {(J* 90)(?* V) I K0} b y (8.5)

io* {(j* 90)o(J* ~)o} by (8.3)

io* {(Jo* 900)(Jo* V~o)} b y (8.5)

~o Y'o �9

which establishes (8.4).

9. A lemma on extensions of maps: Let K be a s imply connected complex and let n ~ 3. Then it is an easy extension of a theorem due to W. Hurewicz 8~ t ha t the na tura l homomorphism of the relat ive homo- t o p y group, ~ , ( K ~, K~-I), into C,~(K) = H,~(K "~, K ~-l) is an isomorphism onto. We ident i fy corresponding elements in this isomorphism and, as before, we use e ~ to s tand bo th for an n-cell in K and for the corresponding element in C,~(K) = z,~(K ~, Kn-1). Let h be the natural homomorphism h : ~r~ (K n)---.z,~ (K '~, Kn-1). I f a given element, ae~,~ (K s) is represented b y a map, [ : ~n+l _+ K ~, then ha is represented b y the same map. I t follows tha t ac = h fl c, for any c e C,~+I(K), where fl : ~+1 ( Kn+l, Kn) -~ C,~+t (K) --* zrn (K n) is the boundary homomorphism in the sense of homotopy . Since ~ t ( K n) = 1 an element in ~z,~+l (K n+l, K ~) or ~r, (Kn) . is uniquely, de termined b y a map of the form (a ~+1, ~+ l ) -~ (Kn+l , K~), or ~n+t __>K n, wi thout reference to a base point.

80) [17], p. 522.

72

Let K and L be simply connected complexes, let e n be a principal cell (i. e. one which is an open sub-set) of K and let K o = K -- e n (n ~ 3). Let g : C ~ ( K ) - - > C , ( L ) be a chain mapping such t h a t the map g I C~ (Ko) (r ----- 0, 1 , . . . ) can be realized by a cellular map [o : Ko -~ L . Let /of t en denote the element of ~n-i (Ln-1) into which ft e n is carried by the map / o : K o - ~ L .

Lemma 7. I / /ofte n = f tge n then /o can be extended to a m a p [ : K - > L , which realizes the chain m a p p i n g g.

Let /ofte n = ftge n. Let h : a n --> -~n be a characteristic map for e n and let k : ( a n, ~n)._). (L n, Ln-1) be a map which represents the element ge n e C~(L) = z ~ ( L n, Ln-1). The element fie n e z c , _ t ( K n-l) is represented by the map h lh n and /oft en is represented by [ o h ] ~ n. Also flge n is represented by k 1 ~n. Since /o fl en ~- ft g en i t follows t h a t k l ~ n ~ / 0 h ] 5 n in L n-1. A given homotopy of k l ~ n into [ o h l ~ n can be extended throughout a n and we m a y therefore assume t h a t k ] ~n ___/oh I ~n. This being so, I say t h a t the map k h -1 : ~n __> L n is single-valued. For i t is single-valued in e n, since h I ( an -- ~n) is a homeomorphism onto e n and h ~ n ___ ~n _ e n. I f p e ~n _ e n we have h -1 p c ~ n, whence k h - l p = / o h h - l p - - - - - / o p . Therefore /oh -1 is single valued, and hence continuous24). Moreover /oh -1 [ enN Ko-~ [o[ engl Ko, since e~N Ko -= ~n _ e n and /c h - ~ p = / o P if p e ~n __ e n. Also (k h-~)h = lc.

For k p e Ic h - l h p (p e ~n) and since k h -1 is single-valued it follows that k p ----- k h - l h p . Therefore the required map / : K -> L is defined

by / [ K o = / o , / l ~ n = / ~ h -~.

10. Reduced complexes. A simple, 4-dimensional complex, K , will be called a reduced complex if, and only if, it satisfies the conditions

(1) K I = K o, a single point.

(2) The closures of the 2-cells, e~,. 2 �9 . ,en, are 2-spheres, S * a t tached to K ~

(3) g 3 = g 2 - { - e ~ + . . . + e t a - t - e ~ + l + . . . + e ~ + Z , where e~ ( i = 1 . . . . .

t ~ n) is a t t ached to K 2 by a map / ~ -~ S~ of degree a~ > 0 and

e~+ i by a map of the form /~?+j --> g ~ (i ~- 1 . . . . . 1). Thus e?+i is a 3-sphere, S~, and K 3 consists of clusters of 2-spheres and of 3-spheres a t tached to K ~ together wi th the bounded 3-cells e~ , . . . , e~.

(4) Each 4-cell iZ a t tached to K 3 by a map of the form ~4 __> K2_t_ + . . . +

73

Notice t ha t each of the bounded 3-cells, e~ . . . . . e~, is a principal cell of K .

In the above conditions we m a y have n = 0, in which case K s ----- K o. Similarly we m a y have 1 -~ 0 or t ----- 0 and there need be no 4-ceUs, in which case K ----- K s.

a is a t t ached to K 2 (i 1 t), is A m a p l t : / ~ - + S ~ , b y w h i c h e , : , . . . .

homotopie in K 2 to a map of the form /~ -+ S~, which is simplicial with

respect to a given t r iangula t ion of K s and a suitable t r iangula t ion of F~. Since e ia is a principal cell of K , for each i = 1 , . . . , t , it follows from L e m m a 5 t h a t K is of the same homotopy type as a reduced complex, K1, in which each of the 3-cells e~ is a t t ached to K ~ b y a simplicial map

E ~ - + S ~ The 3-spheres a . �9 S~ . . . . S~ m a y be t r iangula ted and it follows f rom L e m m a 6 t ha t K~ a has a simplicial sub-division. On applying a similar a rgument to the 4-cells it follows t h a t K1 and hence K , is of the same h o m o t o p y t ype as a reduced complex, which has a simplicial sub- division. Therefore the condit ion t h a t each of our complexes is to have a simplicial sub-division does not restr ict the homotopy type of a reduced complex.

L e m m a 8. A n y simple, 4-dimensional complex is o I the same homotopy type as some reduced complex.

This will be p roved in w 15 below. I f t ----- 0, in which case the bounded 3-cells e~ . . . . . e~ are absent

f rom K , t he n every 2-dimensional co-chain, ~ e CS(K), is an absolute co-cycle. Therefore it follows f rom (4.2) t h a t p ~ =~0 ~. Le t

4 Ko = K -- (e~ ~ - . . . ~- e~) ----- K s g- S~ -[- . . . -+- S~ -+- e~ g- . . . -~ - e r �9

Then p~ ~ = q~ ~0, where ~ e CS(Ko) : C2(K) and p ' denotes the oper- a tor p i n K 0. Also ~ l K o = ~ if y~eC n(K) for n : 2 o r d s i n c e K 0 contains all the 2-cells and 4-cells in K . Therefore it follows f rom (8.4) t ha t

P (P - - (P (P) I g o --= p'((p l go) = p ' (p

= ~ ~ . (10 .1 )

Therefore if x e HS(K, 2r) and if ~ e x, t hen p x is the co-homology class 31) of ~ ~0, m o d . 4 r . Thus if ~ e CS(K) is a co-cycle, m o d . 2 r , then

al) N . B . q ~ e Cd(K) a n d eve ry 4-d imens iona l co-cha in is a n abso lu t e co-cycle.

74

q q has invariant significance as a co-cycle mo d .d r , as apar t from mod.2r . Notice also tha t , if ~, ~ ~ C2(K), then it follows from similar arguments t ha t q v/e Ca (K) is the same whether it is calculated in K or in Ko.

11. ~a ( K a ) , where K a is reduced. Le t K s satisfy the conditions (1), (2), (3) in w 10 and let

K ~ = K ~ - e ~ ~ - . . . + e ~ , so tha t

K 3 = g~ -~ S~ - ~ . . - + S~.

Let b i be a generating element of 3r3(S~) (i ----- 1 . . . . . 1). Then ~a(K a) is the direct sum a~)

z z ( K 3) -~ ~ 3 ( K ~ ) + (b~ . . . . . bt) ,

where (b 1 . . . . . b~) is a free Abelian group, which is freely generated by b 1 . . . . . b~. We s tudy ~a(K~). Le t a~e~r2(K 2) be the element which corresponds to a map ~ -~ S~ of degree ~- 1. Then ~r 2 (K 2) is a

3 free Abelian group, which is freely generated by a 1 . . . . . a n, and fl e~ at a i (i = 1 . . . . . t). Since fl e~ . . . . . fl e~ are l inearly independent the natural homomorphism i :~rs(K 2) - ~ s ( K ~ ) is ontoaa). Tha t is to say, each element in ua(K~) has a representative map in K 2. The group ,%(K 2) is a free Abelian group, which is freely generated u) by the n ( n - ~ 1)/2 elements e i j = ej~ (i, ] - ~ 1 . . . . . n), where ei~-----a~aj if i ~ ] and e , is the generator of ~r3(S~), which is represented by a map 54 _~ S~ with Hopf invar iant + 1. I t follows easily from the definition of the product a~ aj t ha t a~ a~ has Hopf invariant 2. Therefore a i a~ = 2 e i i .

Let us also use a~ and eij to denote the images of a t ~. ~r2(K e) and % s ~s (K~) in the injection homomorphisms i : ~r2 (K 2) -+ ~r2 (K~) , i : ~ s (K e) --> ~r3(K~). These homomorphisms do not alter the fact t h a t ell ~ - a i aj in ~a(K~), if i ~ ~', and tha t a i a i - - 2 e i i . Let a j b e defined for ~ = 1 . . . . . n by writing at+ 1 . . . . . a~-~ 0. Then g3(K~) is

33) [14], Theorem 19, p. 285. Str ic t ly speaking there is a na tu ra l i somorphism of this direct sum onto zr 3 (Ka). We have implicit ly identified corresponding e lements in th is isomorphism.

a3) [13], L e m m a 3, p. 417.

34) This is an obvious general izat ion of the special case m = n ~ 2 of Theorem 2 (p. 413) in [13], where t he p roduc t ab was wr i t t en as a . b . See also [1].

75

genera ted b y e . , since i : g3(K 2) --> g3(K~) is onto, subject to the rela- t ions ~ )

(a) a , a , a ~ = 0 ( i , ~ = 1 , . . . , n) , i (11. 1)

(b) ~ e . = 0 ,

which are a complete set. Since a t aj = ar a~ ~ eij if i ~ ] and since a i a i ---- 2 e . i t follows t h a t (11. l a ) are equivalent to (a~, %) ei~ = 0 if i # ] , 2 a ~ e . = 0 , on the unders tanding t ha t ( a ~ , a r if a ~ = a , . = 0. Therefore (11.1) are equivalent to

a~j e . = 0 , (11.2)

where a . = (a~, aj) ff i # j , a . = (2a~, a~) = 2a~ or a~ according as a i is even or odd.

L e t 7 = .~ 7 ~j ei~ (11.3)

be an a rb i t ra ry e lement in ~3 (K2) and let y be represented b y a map ] : S a --> K s, which is simplicial with respect to t r iangulat ions of S 3 and K 2. Then ~ f is the linking coefficient L (/-~(Pi), /-~(qj)}, where Pi, qi (P~ ~: qt) are inner points of 2-simplexes in S~. The ma t r ix [t ~it [] is called the (generalized) Hop/ invar ian t of any map S 3 -~ K s in the class ~, or s imply the Hopf invar ian t of 7. In exac t ly the same way we may define the H o p f invar ian t of a map of the form

s . K0 = + s : + . . . +

or the Hopf invar ian t of the corresponding e lement in ~3(K03). Le t g : (K 2, K ~ --> (L 2, L e) be a map of K 2 into a reduced complex,

L 2, which consists of a set of 2-spheres, S 1Is,. . . , S~lS, a t t ached to a point L ~ . L e t us also use g to denote the induced homomorph ism g : z . (K s) -~ x . (L 2) (n = 2, 3) and let a~ e x s (L s) be the e lement represented by a map ~3 _~ S~s of degree + 1. Then

s~) [ 13], L e m m a 4, p. 418, a n d [ 11. T h e re la t ions (11. I a) express t he fac t t h a t (qi a i )aJ ~ 0 8 8

in ~z s (K I) s ince o, i a ~ = 0 in r~= (K 1). T he re la t ions (11.1 b) express t h e fac t t h a t a map "3 i ~ "8 2

of t h e fo rm S a --> E i --~ S i (i = 1 , . . . , t) is inessent ia l in K , where E i --> S i is a m a p $ . 2

by wh ich er is a t t a c h e d t o / C a. I f S a --> E~ hem H o p f i n v a r i a n t 1, t h e n S 8 --> S i has Hopf

i nva r i an t q~. I t is p r o v e d in [13] t h a t t he se re la t ions are comple te .

76

where ~ is the degree with which g [ ~ covers $~s. I say that, if y is given by (11.3), then

! fl ! I ! where e ~ p = a ~ a if n e f f , 2 % ~ = a ~ a ~ and

n n

i = l / = I

For since ~(i __= yV, a~a~ : aja~, a~a~ = 2 e , we have

n n

2 7 , : ~ ~ i ~ a ~ a j . i = l j = l

It is obvious from the definition of the product a~ a t that g(a~ aj) = (g ai)(g aj). Also b b ! is bi-linear in b, b!c~s(L2) . Therefore, using the summation convention for i , j = 1 , . . . , n and a, f l = 1 . . . . . p, we have

2g ~ : yi~ g(a i a~) : ~ t (g a~)(g at)

: 2 X v ~'~ e '~ ,

where V~P is given by (11.5). Therefore (11.4) follows from the fact that u3(L2), being a free Abelian group, contains no element of order 2.

Now let / : K 3 -~ L a be a cellular map of K s into a reduced, simply connected complex, L 3, and let / also denote the induced homomorphism / :~(K3)->~3(L3) . Then [ i = i g , where g : z 3 ( K S ) - - > ~ 3 ( L s) is the homomorphism induced by g = I l K s and i stands for both injection homomorphisms i : za (Ks) --> ~3 (Ks) and i : ~3 (L2) -~ ~a (/,3) �9 I t follows from the relation ] i = i g and (11.4) that

/~' = X ~(~Pe:p , (11.6)

where y and ~ are given by (11.3) and (11.5) and, as in (11.2), e~, e~ (~ i e~, i e~) are interpreted as generators of i ~3(K~), i ~a(LS).

12. Homotopy and co-homology in a reduced complex. Let K be a be the 4-cells in K , and, using the same reduced complex. Let e~ . . . . . e r

notation as in w 11, let

fl e~ : ~ ~i et~ ~- b~ , (12.1)

77

where b~ e (b~ . . . . . b~) and ]1 ~ i ] ] is the Hopf invariant in K] =

g ~ - S ~ - ~ . . . + S ~ of a map, E ~ - > K o 8, by which e~ is attached to K 3. Let {~.} be the basis for C'~(K), which is dual to the basis {e~} for C,~(g). The incidence relations between {e~} and {e~} are 0e~ =

2 (i : 1, t), 0e~ = 0 if ~ > t . Therefore the co-boundaries of ~ are @ ~ = a t V ~ ( i = 1 . . . . ,t), ~ V ~ = 0 ( j = t ~ - 1 . . . . . n), whence V~ is a co-cycle rood.at, with at+l . . . . . a~ = 0. Also ~ are absolute co-cycles. Let x~ be the co-homology class, mod.a~, of F~ and y~ the (absolute) co-homology class of ~ . Then H~(K, m) is generated by Yl (m) . . . . , y~ (m), where y~(m) = #~, o y~. According to (3.2), x ix~ = (l~m,qx~)(#~,afx~), where m = (ai,a~), on the understanding that

m = O and /%,,,~ =~,~,.~ : 1 if a t : a ~ = O .

Theorem 5 36). I / 7~ ~ mean the same as in (12.1), then

(a) x~x~= ~ r~ yz(m) with m = ( a i , % ) , z=~ (12.2)

f

( b ) OXi = Z ~'~ y~(2a~) i/ at is even. ) ,=1

3 Assume that the theorem is true if t = 0, the bounded 3-cells e~ . . . . . e~ being absent from K. Then it is true if K is replaced by

K o = K - ( e ~ - ~ . . . . +e~) .

Since ~,~t in (12.1) are the elements

maps E~ -~ Ko 8, they are the same follows from the theorem, with K

f 2 2

of the Hopf invariants in Ko 3, of the

when calculated for Ko as for K. It replaced by Ko, that , in Ko,

0 . (12.3)

If ~ : ~Oo in Ko, where ~ e C4(Ko) = C4(K), ~oe C3(Ko), then ~ = ~ , where ~ e C3(K) is any extension of ~o (i. e. ~e~ . . . . , ~e~ have arbitrary values)aT). Therefore ~ 0 ~ 0 in Ko implies ~ 0 in K , whence (12.3) holds in K . I t follows from the concluding remarks in w 10 that we ob- tain (12.2 a), in K, by taking (12.3) mod. (at, at) and (12.2 b) by taking (12.3) mod.2~t, in case i = ~ and at is even. Therefore the theorem is true in K if it is true in Ko.

8*) Cf. t h e m a i n t heo rem in [1 I. 3 3 Therefore (~o)c ~) N. B. t h e group of cycles in Ca(K ) is genera ted b y et+ 1 . . . . . e t+ l .

~P(~r ~ ~po(aC) = (B~o) c , where c e C4(K), ~ c C~(K), ~Po ---~ ~P [ Ko"

78

I t r emains to p rove the t heo rem when t = 0. I n this case x i e H 2(K) 2 for each i = 1 . n . Therefore we only have to p rove and p x ~ = x i , . . ,

(12.2a). We do this b y cons t ruc t ing a complex, L , which consists of Q = $2• S 2 toge the r wi th complex projec t ive planes, P1 and P2, each of which has one of the 2-spheres S 2 x p o and p o • 2 ( P o s S 2) as a basic 2-cycle. We use the Poincard dua l i ty and intersect ion t h e o r y to prove the t heo rem in L and then p rove (12.2a) , for g iven values of i , ~, by means of the p roper h o m o m o r p h i s m , [* : R ( L ) ---> R ( K ) , which is induced b y a cer tain m a p [ : K - ~ L .

Let Q = $ 2 • ~, where S 2 is an or iented 2-sphere. Le t C ~ = PoXPo I 2 _ _ f2 (p0eS2), let S 1 - -S2• S 2 ~---PoXS 2 and let _12 r %--- - -~x--e '~ ( ~ = 1 , 2 ) .

Let M: (a 2, }2) _+ (8 2, Po) be a m a p such t h a t h ' I (a2 _ }2) is a homeomorphism onto $ 2 - - P o of degree + l . Le t E 4 = a 2 x a 2 and let h : E 4 - + Q be given b y h ( x i , x 2 ) = ( h ' x i ) x ( h ' x 2 ) (xl , x2ca2) . Then

h(a2• x2) = S~• (h' x2), h ( x l • a "~) = (M Xl)• 2, whence h E "~ = Q2 = 14 Q2. S~ 2 + S~ 2. Clearly h I (E 4 - - j~4) is a h o m e o m o r p h i s m onto e o = Q -

,2 ,2 ,4 W e orient E 4 so T h e r e f o r e Q is a complex, Q = e ' ~ 1 + e 2 + e o . tha t a 2 • x 2 intersects x, • a s wi th coefficient + 1, where x , , x 2 = a 2 - } 2,

74 and we orient e o so t h a t h l{E 4 ~4) is of degree + 1. Then i t follows , / = I where ' s ~2(Q2) is represen ted b y from [13J t h a t fl eo 4 = a 1 a 2 e12, a~ 12 and ~ ua (Q2) is defined in the same w a y a character is t ic m a p for e~ e ~ e

as e ~ e ~ 3 ( K 2) ( ~ , # = 1 ,2) . 12 12 Let u , , u 2 c C 2 ( Q ) be the co-cycles dual to e 1 , e 2 =C2(Q ) and let

i 4 = 1 Since a 2 • 2 and x i x a 2 in tersect vo e C4(Q) be defined b y v 0 e 0 with coefficient + 1 in E 4 so do 8 2 • = h(a2• and h l x , • 2 = h (x 1 • a2), and hence S~ 2 and $2 '2 in Q. Therefore i t follows f rom Poincar6 dual i ty t h a t u i u 2 = % . Similar ly u ~ = u x u z = 0 ( 2 = 1, 2).

]~4 . o 1 2 14 14 where /4 is a t t ached to S~ 2 b y a m a p ~ - - ~ x Let L = Q + e 1 + e 2 , ex (2 ~ 1 ,2) , wi th H o p f inva r i an t + 1, of such a k ind t h a t -,4ez is a complex project ive p lane 3s) P~. Then fl e~ 4 = e~z'. Le t Ul, u 2 c C2(L) = C2(Q) mean the same as before and let v 0, v 1 , v~ c C 4 (L) be the basic co-cycles dual to e o1~ , el/4, e~24c Ca (L), v 0 being the same as before. I t follows f rom (8.4) t h a t u~e~ 4 = (u~u~)e~ 4 = (u~lPz)e~ 4 = (u~lPx)2e~ 4 (2 = 1 ,2 ) .

That is to say, -2 _14 m a y be calculated in Pz ignoring the res t of L . "e~, ~, Since e 7 is so or iented t h a t f le 7 = + e~x, it follows f rom Poincard dual-

,4 1 Similar ly u~ e~ - - i ty and intersect ion t heo ry 3s) t h a t u~ e x . 4 = 0 i f = 1 , e ) . A l s o = o ,

as p roved above . Therefore u~ = v x (2 = 1 ,2) . Similar ly u t u 2 = % .

88) [2], p. 311,

79

L e t ] o : K a - - > / 2 - - - L ~ = L ~ a d - S ~ ~ be a m ap such t h a t ]oK 0 ---- L ~ and :

(1) ]o I S~ is a homeomorphism onto S~ ~ , of degree ~- 1, for a given value of i ,

(2) if n > 1, t hen ]ol S~ is a homeomorphism onto S~ ~, of degree ~- 1, for a given value of j r

(3) ]o{g s - ( S ~ + S ~ ) } = L ~ or ]o(g ~ - S ~ ) : L ~ if n : 1. Le t go : C k (K a) -> C/~ (/2) be the chain mapping which is induced b y ]o and let g : C k ( K ) - ~ C k ( L ) he defined by aa) g l C ~ ( K ) = g o [ C ~ ( K ~) if k < 4 and

e ~ = ~ ~' + ~ ' ~," + ~ ~" (~ = ~ , . . . , ~ ) . ( ~ . ~)

T he n g is a chain mapping, since ]oS~ : L ~ whence gOeS= Og e~ = O. L e t a s e ~2 (K) be the e lement which is represented b y a characteristic

]o6f s Then ]oa~ a 1, = as, ]oak-----0 if k ~ i or J, where map for e i . =

]o : ~ (K3) --> ~ (/2) is t he homomorph ism induced b y ]0 : K 3 - , / 2 . I t

' toe , , ' f o e . ' follows f rom (11.5) and (11.6) t h a t ]oe~t ----- els , - ~ - e n , ----- e~2, foe~ ~ = 0 for all o ther pairs p , q. Also ]o (bl . . . . , bt) = 0 since ]oS~ = L ~ Therefore

Y ~ q " ' , ~ i , r{~ ,

�9 . ,, ~, ,~ ~J ~;'

= ~g~ .

I t follows f rom L e m m a 7 t h a t ]o can be ex tended to a m ap ] : K -+ L, which realizes the chain m a p g. L e t ]* : R(L) -> R ( K ) be the homomorphism induced b y ] and let u~ e H 2 (L), v s e H 4 (L) also denote the co-homology classes, which consist of the single co-cycles u~, v s. Then ]*u 1 ~ x~, ]*u s = xj and i t follows f rom (12.4) t h a t

]* vo = },[J y~ 1" Vl = X y~ �9 ~.~I ~,~I

Therefore

x , x , = (1" Ul) ~ = ]* (u~) = ]* v l = ~ 7~ ~ y~ , i"

which establishes the theorem.

a0) I t is obvious what modifications m u s t be made here and in the following arguments if n = 1. I n this case we only need the sub-complex P1 C L.

80

This theorem shows the par t p layed by the Pont r jag in squares. We il lustrate this b y an example. Le t K r = e ~ § e 2 § e 3 § e 4 be the complex in Theorem S with n = t = 1, a 1 = 2 , l = 0 , r = 1 and ~ 1 =

~ 0 ' in (12.1). Then K ~ = e ~ 2 4 7 ~ § 3, where ae 3 = 2 e 2, and ~3(K 3) is genera ted b y e n and is of order (2al , a~) = 4. Since f le 4 = ~, e n it follows t ha t ~3(Kr) is genera ted by en , subject to the re la t ion 4e n = 0 and the addit ional relat ion 4~ ~, e n = 0. I t is therefore a cyclic group of order (~,, 4). Thus z3 (K0) ~ ~3 (K2). The only non-tr ivial products which occur in the co-homology ring are multiples of x 2 = y y e H 4 ( K r , 2), where x and y generate H2(Kr , 2) and H4(Kr , 2). Therefore the co-homology rings of K o and K 2 would be proper ly isomorphic to each o ther if the Pont r jag in squares were ignored. On the other hand p x = $ y and now ~, may be t aken mod. 4 and not merely mod. 2. Therefore K 0 and K 2 can be dist inguished f rom each other b y means of the Pont r jag in squares.

13. Proof of Theorem 1. Le t R be a given simple, 4-dimensional co- homology ring, as defined in w 5. Le t the group Hq = Ha (0) c R be the direct sum of a free Abelian group of rank pq and tq_ x finite cyclic groups of orders a(1 q-l) a (q-l) (t~ = 0). Le t K mean the same as in w 12 ' " " " ~ tq--1

_(2) ( i = 1 t2), 1 = P3 § t3 ' r = with n = t ~ § t = t 2, (r i = ( r i . . . . ,

t3 § P4 and 0e~, = ~(3)~3 (Z = 1 ta)

O e ~ , = 0 ( 2 = t 3 § 1 . . . . . t3 § P 4 ) .

Then the co-bounding relations are

--i r i

= o

( i = 1 . . . . . t ~ ) , ~ , = 0 ( i = t 2 + l . . . . . t 2 + p ~ ) ,

. . . . ~ 3 _ ~<2) (2 = 1 ,t2q'- Pz), cPt~+v~+z-- cp~ ( 2 = 1 . . . . . ta).

Therefore H ~ (K) ~ H n for n = 2, 3, 4 and hence for all values of n . I t follows f rom Lemmas 3 and 2, in w 2, t h a t there is a proper isomorphism of the co-homology spectrum, H , in R , onto the co-homology spectrum H ( K ) . This determines an isomorphism, h, of the addi t ive group, A , of R , onto the addi t ive group, A (K), of R ( K ) . To simplify the nota t ion we ident i fy each e lement x e A wi th h x e A (K). Th en R becomes a ring whose addi t ive group is the same as t h a t of R (K) and the

operators A, # are the same in both. Le t us wri te R ( K ) = R and let

40) [14], Theorem 18, p. 281.

Commen~ll Mathematic! Helvetici 8 1

x y deno te the p roduc t of x , y c A in R and x -y the i r p roduc t i n / ~ . The coefficients ~[~ in (12.1) are still a t our disposal and we shall show

how t o choose t h e m so t h a t R = R . Since x y is bi- l inear in x and y and since bo th rings have the same

addi t ive group, it is sufficient to consider p roduc t s of the f o r m x y , where x e H r ( a ) , y e l l s ( b ) . Moreover , if x e H r ( a ) , y e l l s ( b ) , t hen x y = /z~,a(x ) / t~ ,b(y) , where m -~ (a, b). S ince /z is the same for bo th rings we h a v e only to ensure t h a t x y ~ x y if x e H r (m), y e H 8 (m) for all va lues of m , r , s . Since x y e H r+~ (m) if x e H r (m), y e H ~ (m) and since H'*(m) ~ 0 if n = 1 or n > 4 , we have x y = x y ~ - 0 unless r ~ - 0 or s - - 0 or r = s = 2 . I n e i ther case x y = y x , x y = y x .

We first dispose of the case x e H ~ y e H 8(m). The uni t e lement

of each ring, R and /~, is one of the two genera tors of H ~ Alter ing the original ident if icat ion of H ~ wi th H~ if necessary, we assume tha t bo th r ings h a v e the same uni t e lement , e. Then /z~ , 0 e is the uni t e lement

of b o t h rings R (m), /~(m). I t is also a genera tor of H ~ (m), whence any e lement x e H ~ is of the fo rm x ----- k / ~ , o e , for some value of k. Therefore x y = x y ~ k y .

We now consider the p roduc t s x y , x y , where x, y c H 2 (m). I say tha t

x y ----- x'-'y prov ided x i x ~ ----- x i x ~ (i, j = 1 . . . . . n), where x i e H2(ai) is the co-homology class of ~ . F o r consider first the case m ~ 0. The group H 2 ~ H 2 (0) is genera ted b y X t + l , . . . , x~, whence x y ~ X y for all x , y c H 2, prov ided x~x~ ~ x~x~ for i , ?" ~ t ~ 1 . . . . . n . Secondly let m > 0 . T h e n H2(m) is genera ted b y /xm,~ x 1 . . . . . /x,~,anx~, as shown in the proof of L e m m a 4, in w 2. Therefore, if x i x j ~ x t x ~ i t follows f rom (3.3) in the f o r m

m (m, at , a~) fz,~,~ (xi xj) (/um,a i xi) (~m,, i xr --~ (m , ai) (m, aj)

where c -~ (a~, ar t h a t x y -~ x y for a n y e lements x, y c H 2 (m), since /x is t he same for b o t h r ings and p roduc t s are bi-l inear.

Assume t h a t x y ~ x--y for eve ry pa i r x , y c R and consider the

Pon t r j ag in squares, p x , p x , in R a n d / ~ . Since H ~ is genera ted by ~t~,oe (m ~> 0, Fto,o ~ 1) i t follows f r o m (4.12), (4.11) and (4.13) tha t , if x ----- s /x~,0e, t h e n p x = p x ~- s 2 #a~, 0 e- Because of d imensional i ty the on ly o the r non-zero Pon t r j ag in squares in e i ther ring, if any , are of the f o r m p x , p x , where x ~ H 2 ( 2 r ) . Le t x~ m e a n the s ame as in the preceding pa rag raph . Then it follows f r o m (4.11) a n d (4.10) t h a t p2~x ~2,x for e v e r y x e H~(2r) and eve ry va lue of r , p rov ided p x i -~ p x~ for e v e r y i such t h a t ai is even. Therefore T h e o r e m 1 will have been

82

established when we have chosen ~,~J in (12.1) in such a way tha t x~ x~ = x~x~ ( i , ~ = 1 , . . . , n ) and p x i = p x i if a i is even.

Let Yl . . . . . y, mean the same as in w 12 and let y:~(m) ----- ~ ,oY,~ . I t follows from ( 2 . 3 ) t h a t y;~(p)-------,%.~y;~(q) if p[q. Let v;~= a(~ a) for 2 = 1 . . . . , t 3 and let v a---0 for 2-----t~q- 1 . . . . , t 3 ~ p d - Then ya i s of order v~ and y;~(m) is of order (m, z~). In the given ring, R, let

r

(a) x i x s = ~ e~ J ya(m) with m --- (ai, as) ),=1

r ( 1 3 . 1 )

(b) p x t = ~ Q~ y~(2~i) if ~i is even. h = l

Since y~(m) is of order (m, v~) ----- (el, as, va) the coefficients Q~J are only determined mod. (a~, %, v~). I f i =~ 5, or if i ----- ~ and a~ is odd, we give them arbitrary numerical values in the appropriate residue classes. The coefficients ~ are determined mod. (2 ai, ~;0 and we give them arbitrary values in the appropriate residue classes. I t follows from (4.9)

2 in case a i is even. Also ~ i , ~ y a ( 2 a i ) = y,~(at). that ~ , ~,~ p x~ -= xi, Therefore, operating on both sides of (13. lb) with /~a~,2,~, we see tha t e~ ~ e~ ~ , mod. (a~, v~). Having assigned numerical values to e~, in case a~ is even, we take e~ i = e~.

Let y~1 = ~i in (12.1). Then Theorem 1 follows from Theorem 5, the products x~ x~ and the Pontrjagin squares p x~, in Theorem 5, being

formed in the ring R = R (K). This completes the proof of Theorem 1.

14. Proo| of Theorem 3. Let P , Q be given simple, 4-dimensional complexes and let /* : R(Q) -> R (P) be a proper homomorphism. We have to prove t h a t / * can be realized by a map 1 : P -~ Q. We first show that P , Q may be replaced by reduced complexes. By Lemma 8, in w 10, there is a reduced complex, K , which is of the same homotopy type as P . Let u: K-> P , v: P-> K be maps such that v u ~ 1, u v ~- 1. Let u*: R (P)

R(K) , v* : R ( K ) -~ R ( P ) be the proper homomorphisms induced by u a n d v . Since v u . ~ l , u v _ _ l it follows tha t u ' v * = 1, v ' u * = 1. Assume that the homomorphism u*l* : R(Q) --> R ( K ) can be realized by a map h : K -~ Q. Then the homomorphism induced by hv : P --> Q is v* (u* ]*) = v* u*/* = ]*. Therefore we may replace P by the reduced complex K. Similarly Q may be replaced by a reduced complex, L, and the theorem will be established when we have proved it for K, L.

By Lemma 4, in w 2, the proper homomorphism of the co-homology spectrum, H(L) , into the co-homology spectrum, H ( K ) , which is induced by f* : R (L) -> R (K), can be realized by a co-chain mapping g* : C'* (L) ~C'~(K) . Let g: Cn(K)-->C,,(L) be the chain mapping dual to t/.

83

Assume t h a t g can be reahzed by a cellular / : K -> L and let / r : R (L) -~ R (K) be the proper homomorphism induced b y / . In each co-homology group, H ~ (L, m), ] ~ is the homomorphism induced b y g*. Therefore /~ = ] in H ~ (L, m), whence [r = / . by l ineari ty. Hence the theorem will follow when we have shown t h a t the chain map g can be realized by a map / : K -~ L , using the fact t h a t the dual co-chain map g* induces a p roper homomorphism ]* : R (L) -~ R(K) .

Each of K ~ and L ~ consists of a single 0-cell, e ~ e~~ Let u e C ~ v s C ~ be the co-cycles defined b y u e ~ -=- 1, v e t O = 1 and let u , v also denote the co-homology classes u ~ H~ v ~ H~ of which the co-cycles u , v are the sol i tary members . Then u, v are the unit e lements of R(K) , R(L). Therefore /*v = u, as proved in w 5. There- fore g * v - ~ u , ge ~ I~ and glCo(K) is realized b y the map K 1 = K o -~ L o = L 1 .

Because of the na tu ra l homomorphisms ~r2(K 2) ~ H2(K2), Jr2(L 2) H~ (L ~) it follows f rom an argument , which is similar to the one at the beginning of the proof of L e m m a 8, in w 15 below, t h a t g, restr ic ted to t he chains in K a, can be realized b y a cellular map /0 : K3 -§ L3- Let ]o : zr3 (K3) -> ~r3 (L3) be the homomorphism induced by /o : K3 -~ L3. I f /o fl e4 = f lg e4, for eve ry 4-cell in K , t hen it follows f rom repeated applicat ions of L e m m a 7, in w 9, t h a t / o can be ex tended to the required map ] : K -~ L . However , this is not always possible. F o r example, let K 3 = S 2 ~- S a , where S ~ is a 2-sphere and S 3 a 3-sphere, which meets S '~ in a single point , K ~ and let K = K a ~ e 4 where e 4 is a t t ached to K 3

b y an essential map / ~ --> S 3. Le t L -= K and let /* = 1. Then the ident ical chain map g : C~ (K 3) -~ C~ (K 3) is realized by a map, /o : K3 -~ K s , such t h a t /0 [ $2 = 1 and [0 [ Ha covers S 3 with degree un i ty and has an a rb i t r a ry H o p f invar iant , ~, in S 2. Then /0 fl e4 :~ fl e4 = flg e4 unless ~ -= 0. This shows t h a t we m a y have to modi fy [o before extend- ing it.

3 3 As in w let K ~ = K s - ~ e 1 - ~ . . . ~ - e t ,

and let where ~ e ~ = a i e i

K 3 = K ~ - S ~ + . . . + S ~ .

Le~ b i be a genera tor of ~r a (S~) and, as before, let us make the identific-

a t ion ~r3 (K a) ----- ~r3 ( g l 3) + (b~ . . . . . b~) .

L e t h : ~z 3 (K s) --> H a (K a) be the na tu ra l homomorphism. Clearly h { (bl . . . . . b~) is an isomorphism on to Ha (K 3) ----- Ha (S~ + . �9 �9 + S~).

84

Also g3 (K, 3) ----- i ~3 (K~), whence ~3 (K1 a) ----- h-1 0 . Le t us ident i fy each e lement b e (b 1 . . . . . bt) with h b e H a (K3). Then

~3(K 3) = i z 3 ( g 2) ~- H3(K 3) .

Let us also ident i fy each e lement in H 3 (K 3) with the cycle, which is its unique representat ive. Then b i is an element of H 3 ( K 3 ) = C 3 ( K ) .

I f c e C4(K) = ~ , ( K , K3), then fl c = r c ~- ac, where r c e i ~3(K2), ac ---- h fl c e H a (K a) and a = h fl is the homology boundary operator . We write fl = r + 0. Similarly

~3(L a) = i ~3(L 2) + Ha(L a)

and fl, y , 0, h will mean the same in L as in K . Since g : C a (K) -~ C a (L) is induced b y ]0 : Ka -+La, we have h /0 = g h : ~3(K 3) -->H3(La)=C3(L), where ]0 now denotes /0 : z3 (Ka) -~ ~a (La) �9 Therefore �9

h ( / 0 fl - - ff g) = g a - ag = o

Since h -10 ----- i ~a (L2) it follows t ha t (/0 fl -- ff g) c e i ~3 (L2) for any ce CA(K). Therefore /off - - ffg is a homomorphism of the form (/0 ff - - ff g) : CA(K) --> i z~3(L2). Tha t is to say, [0 ff -- ff g is a 4-dimensional co-cycle in K , with coefficients in i ~a(L2). Assume th a t fo ff - - f ig ,-.~0 and let [off - - f f g = ~ f , where ~psCZ{K, izc3(L2)}. Tha t is to say, ~ is a homomorphism of the form ~ : C 3 (K) -~ i ~3 (L~) and

/off - f fg = ~v = ~0 a .

Let bj e =a (Ka) be defined by a homeomorphism hj : (S a, P0) --> (S~, K ~ (j =: 1 . . . . . 1), where S 3, wi th base point p o eS 3 , is the s tandard 3-sphere in terms of which n3(K a) and na(L 3) are defined. Le t k~: ( Sa, Po) -~ (L a, L ~ be a representa t ive map of /o b~ -- ~ b~ e n3(L 3) and let / I : K a -->L 3 be defined by /1 = / o in K~, /1tS~ = kjh~ -1. Then /~ bj e 7~3(L3 ) is represented b y the map /1hi : k~ : S a --> L a. Therefore ] l b ~ : / o b ~ - - , p b ~ . Since / 1 : / 0 in K~ and ~ b j ~ i ~ a ( L 2) it follows t h a t / , also realizes the chain mapping g : C~ (K 3) -> C . (La). I f c e C~ (K), then ac is of the form Oc = n lb 1 ~ . �9 �9 ~- n~b~, whence /~0c -----/• -- Vac or / l a : / o a _ v2~. Also /1~ : / o r , since r c c i ~ a ( K 2) and /~]K ~ = ]olK 2. Therefore

h ff - ff g = f l ( r + a) - ff g

= / o r + l o a - ~ , a - f f g

= l o f f - f f~ - ~,a

85

Therefore / t can be ex tended th roughou t K to give the required map / : K - ~ L .

We now prove t h a t ]o f l - fl 9 ~ 0. Since the coefficient group, i ~3(L2), is a direct sum of cyclic groups it is sufficient to prove that (fo fl - - fl g) e = 0, rood .m, where c e C4 (K) is an y cycle, m o d .m , for any m = 0 , 1 , 2 . . . . Since f l = ? A - O and h ~ - 0 we have ~ = h f l = h ~ . Therefore h()t o 0 - ~ g ) = g ~ - - ~ g = 0 , whence / o ~ - - is a co-cycle with coefficients in i ~r 3 (L2). Since (/o 0 -- 0g) c = / o Oc -- g 0 c = 0 , rood. m, if O c = 0 rood. m, it follows t h a t / 0 O - -O g ~ '~0 . Therefore i t is sufficient to prove t ha t

( l o , e - ,e ~,) - ( I o a - a g ) = 1o 7 - ~' ~, , - - ' o .

L e t the notat ions , wi th respect to K , be the same as in the preceding 4 be a cycle mod. m. Then sections and let c = n 1 el 4 A-" "- ~- n~ e r

where

? c = X nz?e~

--= ~ ~,~ e~ , i<~j

= Z ~i j v~ c 2 = 1

2 2 (mod. m) = (q~ ~s) c

the last step following f rom (12.3) and the fac t t h a t c is a cycle mod.m. Therefore

c ~-- ~: eij ( ~ ~ ) c (mod. m) . (14. l) i~<j

Le t e 1'2, . . . , err2 be the 2-cells in L and let ~ , . . ., ~ c C 2 (L) be the dual co-chains. Le t

p n 2 ~t e l 2

o(=I t = 1

Let e ~ ~ i ~3 (L~) be defined in the same way as e~. Then i t follows from (11.6) and (14.1) tha t , calculating rood .m,

86

n n

/ o 7 c . i o, ~ ~ 2 = X , e~,fl ~ X g~ g j (qJi 9 j ) c O~<f l i = 1 i = 1

n n f

~<fl = i=1

= ~ e'~ (g* ~ ) (g* ~ ) c . (14.2)

2 Let ~ = ~ ~ . . . . . ~ = ~o ~ , ~ § . . . . . ~ = 0, and let ~.~ = (~=, ~:~)(~' r 3 ) , ~ = = (~:%, 2 ~ ) , w i t h ~,, 3 = 1 . . . . . p , ~ + ~ =

. . . . v~ = 0 and (0 ,0) = 0. Then z=fle~fl= 0 in i ~ ( L ~ ) , as in ( 1 1 . 2 ) .

I say t h a t g* ( ~ ~ ) ~ (g* V~) (g* V~) rood. v~fl . (1~. 3)

For let ~ mean the same as in L e m m a 4, in w 2. Then (3.2) m a y be writ ten in the form

(i~ ~)(s ~') = i(~,~)(~ ~') ,

where ~ and ~ are co-cycles mod. r and mod.s , respectively. Also

P~ ~ ~ = ~4~ P W ,

i f ~ i s a co -cyc l emod .2 r . As proved in w p ~ = ~ 2 if ~ e C~ ( P ) , where P is a reduced, simple, 4-dimensional complex. Final ly ~" g* = / * j , since if* real izes/* . Therefore, if v = v~ is even, whence ~=~ = 2 v, we have, writing ~ for Y)a

= ~..Ag* ~)(a* ~) �9

This establishes (14.3) in case a = fl and ~ is even. I f a r fl or if = fl and ~a is odd it follows from a similar argument . Therefore there

are co-chains uafl e C ~ (K), v~fl s C a(K) such t h a t

( g * ~ ) (~*~) ~* ~ = ( ~ ~p,) + ~ u ~ + ~ v ~ .

Since ~ vo, fl(Oc ) 0, mod .m, it follows from z~fl %fl = 0 and (~ vc, fl) c = = (14.2) t ha t

r ? 2 2 = X e ~ ( ~ ~ ) g c .

(mod. m)

87

Hence it follows from the analogue of (14.1) in L, with c replaced by gc, tha t ] o 7 C = 7 g c , mod.m, or that ( / 0 7 - - 7 g ) c = 0 , mod.m. Therefore /o 7 -- 7 g ~ 0 and the proof is complete.

As an example of the applications of this theorem let P be a simple, 4-dimensional complex, in which there is no 2-dimensional or 3-dimensional torsion. Then P is of the same homotopy type as a reduced complex, K , without any bounded 3-cells (i. e. t = 0) and without any 4-cells which are bounded in the sense of homology. The homotopy type of such a complex is completely determined by its Betti numbers and the "mixed tensor" which is the equivalence class of the set of components 7~/under transformations of the form

~q ~ q .- = a iajT~/bg ,

where i, j , p, q = 1 . . . . . n , 4 , # = 1 . . . . . r , repeated indices imply summation and ]l aV]t, [[ b~[] are unimodular matrices of integers. The components 7~J are the coefficients of the trilinear form x y v = (x U y) N v, defined by Hassler Whitney 41), where x , y c H 2 ( K ) , v~H4(K). Thus the homotopy type of P is completely determined by its Betti numbers and this tri-linear form. That is to say, two such complexes, P and P~, are of the same homotopy type if, and only if, they have the same Betti numbers and if there are isomorphisms (onto), a : H2(P r) -+ H 2 (P), b : Ha (P) --> H4 (P~), such that

(a x ' ) (a y ' ) v = x' yr(b v)

for all elements x ~, y~cH2(P~) , v e H4(P ). In particular the problem of classifying the homotopy types of complexes of this nature, whose fourth Betti number is unity, is reduced to the classification of quadratic forms, with integral coefficients, under unimodular transformations, on the understanding tha t a quadratic form and its negative belong to the same class.

15. Proof of Lemma 8. We have to prove tha t any simple, 4-dimem sional complex, P , is of the same homotopy type as a reduced complex. First let P = P~ and let K 3 be a reduced 3-dimensional complex, whose homology groups are isomorphic to those of p3. Then it follows from Lemmas 3 and 4, interpreted in terms of homology rather than co-homology, tha t there is a chain map g : C~(P 3) --> C~(K3), which induces isomorphisms of the homology groups of P~ onto those of K 3. Clearly g l C~ (p2) can be realized by a map /0 : p2 _~ K 2, such tha t /0 p1 ~ K 0.

~) [9].

88

Because of the na tu ra l i somorphisms x2(P 2) ~ H2(P 2) and g~(K 2) H+.(K ~) it follows easi ly enough f rom L e m m a 7, in w 9, t h a t / 0 can be extended to a m a p / : p s _+ K s, which realizes g. Then / induces isomorphisms of the homology groups of p s onto those of K s and it follows from [12] t h a t / is a h o m o t o p y equivalence.

Now let P be a n y simple, 4-dimensional complex and let K s be a reduced complex, which is of the same h o m o t o p y type as p s . Le t / : p s -+ K 3 be a h o m o t o p y equivalence, which is a cellular map . Assuming that K s does no t mee t P , we ident i fy each point p ~ p s wi th / p s K s, thus forming a space K , whose points are those of K s and of P - - p s . Let + : P - + K be the map , which is g iven b y ~0 I P 3 = - / , ~ I ( P - - P S ) -- 1. Le t e 4 be a n y 4-cell in P and let h : 34 _+ p be a characteris t ic m a p for e 4 . Then T h : a 4 -+ K is a m a p such t h a t + h I (a 4 - - 54) is a homeomorphism onto e a and ~ h ~4 c K 3 . Therefore K is a complex, whose cells are the cells in K a and the 4-cells in P , the m a p ~ h : a 4 -+ K being a characteristic m a p for e 4 in K . Since / : p a _+ K s is a h o m o t o p y equivalence so is 42 ~ : P - + K .

The complex K is not necessari ly reduced, since the f ront ier of a 4-cell may mee t one or more of the bounded 3-cells, e~S,.. . , e~ c K s. Le t

K ] = K 2 + S ~ a + . . - + S s K1 a = K 2 s l ' -~- el -~" " "-~" eat ,

the nota t ions being the same as in w167 10, 11. As in w 11 we write

~ s ( g 3) = ~3(g~) + us(S1 a + . . . . +- Sat)

:= i za (K2) g- us (S~ g - - . . + Sat)

= i n3(g] ) ,

where i : n3(K 2) -~ zs(K~), i : zs(Ko 3) -~ z s ( K s) are the na tu ra l homo-

morphisms. Therefore eve ry m a p of the fo rm ~4_~ K s is homotop ie in K ~

to a m a p of the fo rm ~4 _~ K] . Therefore it follows f rom re i te ra ted applications of L e m m a 5, in w 7, t h a t each 4-cell in K m a y be replaced b y one

which is a t t a ched to K 3 b y a m a p of the fo rm /~4 _> K0a, wi thou t al ter- ing the h o m o t o p y type . This last opera t ion t r ans fo rms K into a reduced

complex and the proof is complete.

16. Note on cell complexes 4s). I t is clear f rom the definition of a (finite) cell complex, K , t h a t a sub-set of the cells in K const i tu tes a sub-

43) [15], Theorem 2. 4s) The complexes referred to in this section need not have simplieial sub-divisions.

89

complex if, and only if, the union of these cells is a closed sub-set of the space K . Thus the union 44) and intersection of a n y set of sub-complexes are themselves sub-complexes. I f X is a n y set of points in K we shall denote by K ( X ) the intersection of all the sub-complexes of K , which contain X . Thus K (X) c L if L c K is a n y sub-complex which contains

X . Clearly K ( X ) ~ K(X) , where 2t~ is the closure of X . We shall describe a homotopy, [~ : P -~ K , where P , K are complexes,

as restricted if, and only if,

[tPo c K([oPo) (16.1)

for every sub-complex P o c P . Le t ft, gt : P --> K be restricted homo- topies such t h a t go = [1- Then we h a v e :

L e m m a 9. The homotopy which consists o[ [~ [ollowed by g~ is al~o restricted.

I t follows from (16. l) t h a t

g ([tPo) c g ([oPo) , whence

gt Po c g (go Po) = g ([1 Po) c g ([o Po) ,

which proves the lemma. Le t ]o : P -~ K be a given map and let g~ : Q -> K be a restricted

deformation of the map go = [o [ Q, where Q is a sub-complex of P. Then we have :

L e m m a 10. There is a restricted homotopy, [t : P- -~ K , such that

I , IQ = g,. I f P = Q there is nothing to prove and we proceed by induction on

the number of cells in P -- Q. Le t P = Q' + e n, where e n is a principal cell in P -- Q, and assume t h a t g~ has been extended to a restrict~ ~d homotopy g't :Q' --> g (g~ = [o]Q'). Let h : (a n, ~n) _~ (p, Q,) be a characteristic map for e n and let the homotopy g'th] ~n be extended to a homotopy, ~t : a n -> K , of ~0 = [o h : ~'~ ---> K , by the method used on p. 501 of [7]. Then it follows f rom an argument , which is similar to one used in the proof of L e m m a 7, in w 9 above, t h a t a (single-valued and continuous) homotopy, It : P -~ K , is defined b y [t]Q' = gtt, [~[ " ~ "

~ h - l l ~ n. Le t P o c P be any sub-complex. Then [~PocK( foPo) if P o c Q ', since [~]Qt = g't is restricted. Let Po = Qo -Jr en, where Q ~ c Q ' . Then h is of the form h : ( a n , ; n ) - ~ ( P o , Q ~ ) and

glth~n C , , gt Qo c g ( g Q~) c K ([o Po) �9

44) N. B. there are but a finite number of sub-complexes of a (finite) complex.

90

Also it follows f rom the fo rmula a t the b o t t o m of p. 501 in [7] t h a t the set of points covered b y $~a n consists of ~0a n = / o h a n t oge the r wi th the set covered b y gr t h ~n c K ([oPo). Therefore

[~ e" ~ $~ a n c ~o a~ U K([oPo) = [o'e n U K([oPo) = g ( / o P o ) .

Since [t[Q ~ is res t r ic ted i t follows t h a t

[~Po : [~ Q' tJ /re n c g (/oQo) 0 g (/oPo) : K ([oPo) .

Therefore [~ is res t r ic ted and the L e m m a is proved. Let [o : P -+ K be a given m a p of a complex P in a complex K and

let [o]Q be cellular, where Q is a sub-complex of P. Then we h a v e :

Theorem 6. There is a restricted homotopy, / t : P - - > K , such that

]t : [o in Q and [1 is cellular.

This will follow f rom L e m m a s 9 and 10 and induct ion on the numer of cells in P - - Q when we have p roved it in case P = Q + e n . L e t P = Q + e ~ and let L = K ([0 en) �9 Le t h : a n -~ ~n be a character is t ic map for e n. Then h ~ c Q "-1 and since ]olQ is cellular it follows t h a t /oh is of the fo rm ~o = [o h : ( a~, 5n) _~ (L, L~-I) . Assume t h a t there is a homotopy , ~ : a n -->L, such t h a t ~ - - ~o in ~ , ~ l a ~ c L ~, and let /t be defined b y [t = / o in Q, [t]~n = ~th-1]~n. Then [1 e n c ~ l a n c K n

and [I[Q : ]olQ, which is cellular, Therefore [t is cellular. Le t Po c P

be any sub-complex. I f P o c Q , t hen [ , P o : [ o P o c K ( [ o P o ) , and if P o : Q o + e n , with Q o c Q , t hen

/t (Qo + en) = /~ Qo [j [~ e r i c [oQo u ~(rn c [oQo U K fro en) C K ([o Po) �9

Therefore It is res t r ic ted and the t heo rem follows. I t r emains to p rove the existence of ~t, which we do b y induct ion on

the n u m b e r of cells in L - - L n. I f L - ~ L n we m a y t a k e ~ t - - ~ o . Otherwise let e *~ be an m-cell in L , where m = d im L > n ~ 0, and let g : o m --> em be a character is t ic m a p for e ~. Le t P0 be the centroid of a m and let ~ : a ~ - P o - ~ a m - P0 be the radia l deformat ion , which is given in polar coordinates b y qt(r , P) = {t + (1 - - t ) r , p} (p ~ 5m). Then the h o m o t o p y g ~t g-~ : e~ - - qo - + e ~ - - qo (% = g Po) is single-valued, and hence continuous~a). Also g ~t g - I Ig a ~ = 1. Therefore a h o m o t o p y 0~ : L - - qo - ~ L - - qo is defined b y O~[L - - e ~ : 1, O~le 'n - - qo : g ~ g - ~ .

I f ~oO n ~ L - q 0 we define ~ = 0 ~ o . Let qo t ~o an" Then the theorem will follow f rom the preceding para-

graph when we have p roved t h a t there is a homotopy , ~ : a n - ~ L

91

(~0-----to) such t h a t t ' ~n ' t - = t o in and t l a n C L - % . L e t E b e a t r i - angulat ion of a n, whose mesh is so small t h a t to a c e" if qo e to a, where a is any (closed) simplex of E . Le t A c E be the sub-complex consisting of all the closed simplexes which meet t o 1 q0 �9 Then ~o A c em. Le t

B = E - A . Each point of t o lq0 is obviously an inner point of A , whence t o B c L - - q o . Therefore t o ( A N B ) c e m - % . Since em -- qo is arcwise connected (m > 0) and nk (em -- %) = 0 if 1 ~< k n - - l < m - - 1 t h e m a p ~ o [ A v I B can be extended to a map ~ r :A (em -- %). Le t e ~ be given a Euclidean geometry and let t~ a (a e A) be the point which divides the linear segment (toa) ( t ' a ) in the ratio t : (l --t). Then tit : A -+ e" is a deformation of to into t ' , such tba t t~t = to in A N B . We extend t~t th roughout E by taking t~t = to in B . Then t~, thus extended, is a homotopy of ~o such tha t t~ = to in 5~, t~ a " c L--qo. This completes the proof.

(Received the 17 th J a n u a r y 1948.)

REFERENCES

[1] L. Pontr]agin, C. R. (Doklady), 34 (1942) 35--7.

[2] N. E. Steenrod, Annals of Math. 48 (1947} 290--320.

[3] N. E. Steenrod, Proc. Nat. Acad. Sc. 33 (1947) 124--8.

[4] M. Bockstein, C. R. (Doklady) 37 (1942} 243--5.

[5] S. Eilenberg, Annals of Math. 45 {1944) 407--47.

[6] S. Lefsehetz, A l g e b r a i c T o p o l o g y , New York 1942.

[7] P. Alexandro U and H. Hop], T o p o l o g i e , Berlin 1935; also Ann. Arbor, U.S.A. 1945.

[8] E. Cech, Annals of Math. 37 (1936) 681--97.

[9] Hassler Whitney, Annals Math. 39 (1938) 397--432.

[10] J. W. Alexander, Ann~ls Math. 37 (1936) 698--708.

[11] P. AlexandroU, Trans. Amer. Math. Soc. 49 (1941) 41--105.

[12] J. H. C. Whitehead, On t h e h o m o t o p y t y p e o f ANR's, Bull. Amer. Math. Soc.

[13] J. H. C. Whitehead, Annals Math. 42 (1941) 409--28.

[14] J. H. C. Whitehead, Proc. L. M. S. 45 (1939) 243--327.

[151 J . H . C . Whitehead, N o t e o n a t h e o r e m d u e to K. B o r s u k , Bull. Amer. Math. 8oc.

[161 K. Borsuk, Fund. Math. 24 (1935} 249--58.

[17] W. Hurewiez, Kon. Akad. Amsterdam 38 (1935) 521--8.

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on simply connected, 4-dimensional polyhedra

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