a certain exact sequence

Annals of Mathematics

A Certain Exact SequenceAuthor(s): J. H. C. WhiteheadSource: Annals of Mathematics, Second Series, Vol. 52, No. 1 (Jul., 1950), pp. 51-110Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1969511 .

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ANNEAI ow MATEsmAncs Vol. 62, No. 1, July, 1950

A CERTAIN EXACT SEQUENCE BY J. H. C. WHITEHEAD

(Received May 31,1949)

(Revised January 10, 1950)

CONTENTS

Page INTRODUCTION ...................................................... 51 CHAPTER I. THE SEQUENCE 2(C, A) ..................... 53

1. Definition of 2(C, A) ..................................... 53 2. The secondary modular boundary operator ...................... 55 3. Induced homomorphisms of 2 .................................. 55 4. Combinatorial realizability ..................................... 57

CHAPTER II. THE GROUP r(A). . 60 5. Definition of r(A)............................................ 60 6. Induced homomorphisms of r(A) .............................. 64 7. Relation to tensor products .................................... 67 8. "A" finitely generated......................................... 67 9. Direct systems............................................... 70

CHAPTER III. THE SEQUENCE 2(K) .................................... 71 10. Definition of 2(K) ........................................... 71 11. The invariance of 2(K) ..................................... 73 12. The sufficiency of 2(K) ..................................... 75 13. Expression for r3(K) ..................................... 75 14. Geometrical realizability...................................... 77 15. q-types..................................................... 81

CHAPTER IV. THE PONTRJAGIN SQUARES ............................... 83 16. The main theorem........................................... 83 17. Secondary boundary operators ................................. 92 18. The calculation of 24(K) ..................................... 97

CHAPTER V. THE SEQUENCE OF A GENERAL SPACE . . 98 19. The complex K(X) ..................................... 98 20. The maps K and X ...................................... 101 21. The sequence 2(X) ..................................... 106

APPENDIX A. ON SPACES DOMINATED BY COMPLEXES...... . 107 APPENDIX B. ON SEPARATION COCHAINS ................... 108

Introduction

In this paper we give detailed proofs of the theorems announced in' [1]. These theorems concern, first of all, an exact sequence, 2(K), where K is a (connected)

1 Numbers in square brackets refer to the list of references at the end of the paper. The two papers in [2] will be referred to as CH I and CH II.

51



52 J. H. C. WHITEHEAD

complex.2 The sequence :2(K) is the same as z in ?1 below if

Cn+1 = 7n+l(K + Ku), An = Tn(Kn) (n > 2),

,3, j are the (homotopy) boundary and injection operators and C2 = jA2, Cn = An = 0 if n < 2. It is shown to be a homotopy invariant and a fortiori a topological invariant of K.

Various realizability theorems are proved, which show that, if wi(K) = 1 and dim K < 4, the part of Z (K) which we call 2;4(K) is an algebraic equivalent of the homotopy type of K. Thus 24(K) may be used to replace the more compli- cated cohomology ring, R(K), which was defined in [3]. Moreover 24(K), besides being simpler, is in some other ways better than R(K). For 14(K), unlike R(K), is defined for infinite complexes. Besides this, 24(K) includes r3(K) as a component part and therefore yields more information than R(K) concerning homotopy classes of maps K -* K'. Thus the theory of 24(K), unlike that of R(K), includes the homotopy classification of maps S-* S2, where Sn is an n-sphere.3 But 2A(K) does not include r4(K) and is incapable of distinguishing between the two classes of maps S4 -_ S3. A step towards including 74(K) in a purely algebraic system would be to calculate 7r4(Sl u S2), where S1 n 2 is a single point.

On replacing H4(K) by H4(K)/i7r4(K) we obtain a very simple algebraic expression for the 4-type, as defined in CH I, of a simply connected complex.4

Another set of theorems concerns a certain group, P(A), which is constructed from a given Abelian group A. We prove that r(112) r F3, where i12, r3 are taken from 2(K). When K is a finite, simply connected complex we use r(112) to express the secondary modular boundary homomorphism,

b(m) :H4(m) -* r3/mr3 in terms of the Pontrjagin square map

p :H2(K, A) -* H4(K, r(A)), which is defined in Chapter IV. Our expression for b(m) shows that, if K is given in a suitable form (e.g. as a simplicial complex, which is known to be simply connected) 24(K) can be calculated constructively.

In Chapter V we show how the domain of definition of : (K) can be extended from the category5 of CW-complexes to the category of all arcwise connected

2 All our complexes will be CW-complexes, as defined in CH I. Here we define 2 (K) for complexes which need not be simply connected.

' Consider also the group of homotopy classes of maps, 4:K -+ K, where K = S2 U SI and S2 n SI is a single point, such that I j S" is of degree +1 over S for each n = 2, 3. All such maps induce the identical automorphism of R(K). But the induced automorphism of 7rs(K) varies with the Hopf invariant of O I S8 in S2.

4An algebraic expression for the 3-type of a complex, which is not simply connected, is given in [9].

5 The term category will mean the same as in [10]. We follow Eilenberg and MacLane in recognizing categories in which the objects are "all" groups etc. Theyindicate various means by which this can be justified.



A CERTAIN EXACT SEQUENCE 53

spaces. The method used is to realize the singular complex of a space, X, by a CW-complex, K(X), which is seen to be of the same homotopy type as X in case X is itself a CW-complex. We then define 2(X) as 2 { K(X) 1, if X is any arcwise connected space.

In presenting these theorems we have, as far as possible, separated the purely algebraic part of the theory from the geometrical applications. The result is that Chapters I and II are purely algebraic. The geometrical applications are given in Chapters III, IV and V. In the latter we refer to certain "topological" and "homotopy" categories, which we define as follows. The topological category of all (topological) spaces will mean the one in which the objects are all spaces and the mappings are all maps of one space into another. The homotopy category of all spaces will mean the one in which the mappings are all homotopy classes of maps of one space into another. Similarly we define the topological and homotopy categories of all (geometrical) complexes of any specified kind. Here a complex means a pair (X, K), where X is the space which is covered by a complex K. A map of (X, K) into a complex (Y, L) means a triple (4, K, L), where 4 maps X into Y, and a homotopy class of maps, (X, K) -+ (Y, L), has a similar meaning. We shall denote (X, K) by the single letter K and 0: K -- L will stand for (4, K, L).

We shall introduce a number of standard operators, 3, j, k, 1 etc., which we shall denote by the same letters, with or without subscripts, in whatever system they occur. With the exception of the deformation operators, in ?3 for example, a subscript attached to an operator, as in On CGI - An-, , will always agree with the one attached to the group which is being operated on. All our groups, except groups of operators, will be additive and we shall denote zero homomorphisms, C -O0, and identical automorphisms C --* C, by 0 and 1.

CHAPTER I. THE SEQUENCE Z(C, A)

1. Definition of 2(C, A) Let (C, A) denote a sequence of arbitrary Abelian6 groups, C. , An , together

with a sequence of homomorphisms,

* CXX , Ceil An -4 Cn A An-1<* such that jn~n = ,3n1(0). In general 8n # jn 1(O). We assume that CR , AR are

defined for every n = 0, ?1, t2, ... . Let

dn+1 = -n*n+l Call - Cn.

Then dndn+l = 0, since #,%$, = 0. Let Zn = dn'(0). Then jAn C Zn , since dnj. = jn-lnjn = 0, and dCn+l C jAn . Let

r = j-l(0) Iin = An/GCn~l, Hn = Zn/dCs+l

It is just as easy to define 2: and to prove Theorem 1 if An , Cn are non-Abelian, provided OCn+l , dCn+l are invariant sub-groups of An , Zn .




and let

(1. 1) in rn -+An y kn:An -- Hn In: Zn -- Hn

be the identical map of rn and the natural homomorphisms of An , Zn . Notice that the sequence

(1.2) rn An 24 Cn B An-1 is (internally)7 exact. Notice also that jnk-1 (O) = j3n+lCn+l 1= n1(O). Therefore a homomorphism, in: Hn f-> Hn , is defined by fnkn = Injn *

Let z e Zn1 . Then jhn+1Z = 0, whence fn+lZ e rn . Since

On+, dn+2 = On+ljn+l On+2 = 0 it follows that 0nf1 I Zn+j induces a homomorphism bn~l Hn~+l rn F Therefore a sequence of homomorphisms,

1* b i I b 2:*1* I--> rn -> IHn n-* *n 1 r-*

is defined by (1.3) blz = (3z, t = ki, jk = lj. We describe b as the secondary boundary operator. We shall sometimes write 2 = 2(C, A).

THEOREM 1. The sequence 2 is exact. It follows from (1.3) and the exactness of (1.2) that

tblz = i;z = k3z = 0 (zeZ n+1)

ft = ki = Iji = O

bik = blj = jj= 0.

Therefore ib = 0, fi = 0, bi = 0. Let ty = ky = 0, where y e rn . Then y = O3c, for some c e Cn+1 , since kn'(O) =

f3Cn+i . Moreover dc = jfc = jy = 0. Therefore c e Zn+1 , Ic e Hn~l and we have

7Y = #c = blc e bHnl+.

Therefore 1n'(0) = bHn~l Let jka = Ija = 0, where a e An . Then ja = dc = jfc, for some c e C.+,

since 1t1(0) = dCn+l . Therefore a = y + flc, where y e rn , and ka = ky + kjc = e trn.

Therefore f1(O) = trn Let blz = O3z = 0, where z e Zn . Then z = ja, for some a e An , since fl3(0) =

jAn . Therefore Iz = Ija = jka e jfln .

Therefore b-n(0) = fn and the theorem is proved. 7 I.e. the last homomorphism need not be onto nor the first an isomorphism into.




2. The secondary modular boundary operator

Let m > 0 and assume that, for some particular value of n,

(a) {mz = 0 implies z = 0, where z e Z.-

(b) 1jA n. = Zn-1 Let

H.(m) = 1di(mZn-1)/dCn+l}m,

where Gm = G/mG if G is any additive Abelian group. We proceed to define what we call the secondary modular boundary homomorphism

(2.2) bn(m) :Hn(m) -* rn-l,m = rn-1/mrn-1. Let c* e Hn(m) and let c e Cn be any representative of c*. Then dc = mz =

mja, for some a e An,1 , by (2.1b). Therefore j(8c - ma) = 0, whence Oc - ma e rn-1. If a' e A,,- is any other element such that mja' = dc = mja it follows from (2.la) that j(a' - a) = 0. Therefore a' = y + a, where -y e r.1 , and f3c - ma' = (8c - ma) - m-y. Therefore the coset, (ic - ma)m e rn-ion , which contains f3c - ma, is uniquely determined by c, where a e An-1 is an arbitrary element such that mja = dc.

If c' e Cn is any other representative of c*, then c' = c + mc1 + dc2, where cl e C", c) e c 1 , and

dc' = dc + dmc1 = jm(a + 3c1).

Moreover Oc' - m(a + #c1) = f3c - ma. Therefore a single-valued map (2.2), which is obviously a homomorphism, is defined by

bn(M)C* = (j8c - ma)m,

where c e Cn is any representative of c* and jma = dc. As an alternative to (2.la), let j: A -1 -- Cn-i have a right inverse,8 u: Zn ->

A n- . Then j(f3 - ud) = 0, whence (I - ud)Cn C r.-1 . We define b(m): H(m) -- rn-l,m, by b(m)c* = { (3 - ud)c}m , where c e dU'(mZn-1). The homomorphism u is determined by j, modulo an arbitrary homomorphism 0: Zn~ -* rn1 . If dc = mz we have {,3- (u + 0) d} c = (p- ud)c-mz. Therefore u andu + 0 determine the same homomorphism b(m), which is therefore determined uniquely by the system (C, A). If (2.1a) is also satisfied this definition of b(m) is equivalent to the previous one.

3. Induced homomorphisms of z

Let A' be a sequence of the same sort as I. By a homomorphism (isomorphism),

F = (%, g, f):I --

8 This is always the case if C"- , and hence jAn,~ (See p. 50 of [18]) is free Abelian.




we mean a family of homomorphisms (isomorphisms)9

bfn+l:HRl - H H nl, :rn -+ r, fn: Hf -+ H' such that

(3.1) bt= b, g = fti if -t), where b:HN'+1 -+ r' etc. are the homomorphisms in 2'.

Let (C', A') be a system of the same sort as (C, A). Then a homomorphism (isomorphism)

(3.2) (h, f): (C A) -* (C', A')

will mean a family of homomorphisms (isomorphisms),

hn+I Cntel Cn/ 1 fn:An --) An' such that

(3.3) O= ffl, jf = hj.

Notice that dh = j3h = jf, = hj3 = hd in consequence of (3.3). Also fr. C rn since jfi = hji = 0, where i: rn -+ An is the identical map.

Let (3.2) be a given homomorphism and let 2' = 2(C', A'). Then kfB = k1ch = 0, Ihd = ldh = 0. Therefore h, f induce homomorphisms

b:Hn+l --* H g:Q rn -*r', f Hn --+n X

according to the rules

(3.4) blz = 1hz, ig = fi, fk = kf (z e Zn+l).

It follows from (1.3), (3.3) and (3.4) that

bblz = blhz = #hz = fBz = g0z = gblz,

ig = kig = kfi = fki = ft jfk = jkf = 1jf = lhj = blj

= ljik.

Therefore (b, g, f) X --+ V is a homomorphism. We call it the homomorphism induced by (h, f).

By a deformation operator, t:C -+ C', we mean a family of arbitrary homomorphisms, tn+l:Cn -* C'+1 . We describe two homomorphisms,

(h, f), (h*, *): (C, A) --+ (C', A'),

as homotopic, and write (h, f) c (h*, f*), if, and only if, there is a deformation operator, t:C -+ C', such that

' An isomorphism, without qualification, will always mean an isomorphism onto.




{h- hn h j = dn+ + tn dn

fn - fn = flniR+1jn

Since Id = 0, ji = 0, kB = 0 it follows from (3.5) that lh*z = Ihz, f*i fi, kf* = kf. Therefore (h, f) and (h*, f*) induce the same homomorphism 52(C, A) -) X(C', A').

Let (C, A) and (C', A') both satisfy (2.1). Let c e d;'(mZn,-) and let dc = mz. Let Z'_1 C C'-1 and H'(m) be defined in the same way as Z,,- and H"(m). Then hZn,1 C Z$-1, since dh = hd, and dhc = hdc = mhz. Therefore h. induces a homomorphism

(M):Hn(M) -Hn(m) according to the rule b(m)c* = (hc)*, where c* and (hc)* are the elements of Hn(m) and Hn(m), which correspond to c and hc. Also g induces a homomorphism

g (m): rnion -- rn'_l1m such that ?(m)-ym = (g-y)m, where yn , (g-y) are the cosets which contain y, gY. By the definition of b(m) we have b(m)c* = (c - ma)m where a e A.,, is such that dc = mja. Then dhc = hdc = mhja = mjfa and

b(m)(hc)* = (hc -mfa)m

= {f(c -ma)Jm

= {g(c -ma)}I,,

-g(m)b(m)c*.

Therefore b(m) is natural in the sense that

(3 .6) b (m) b(m ) = g (m) b(m )

Obviously (h, f) and (h*, f*) induce the same homomorphisms [(m), g(m) if (h) f) c-- (h*) f*).

Let j: A,- - C.-, and j: A' -> C'-, have right inverses, u and u', let jAl = Z,-1 and let b(m) be defined by the second method in ?2. Since j(fu - u'h) = hju - h = 0it followsthatfu - u'h = :Zn~j--> rP_1. Let dc = mz, where c e C, . Then f(# - ud)c = (fi -u 'd)hc- mz, whence g(m)b(m) = b(m)b(m).

4. Combinatorial realizability By a composite chain system we shall mean a system (C, A), of the kind intro-

duced in ?1, such that

(a) C, = A, = Oifr < 2

(b) each C. is a free Abelian group.

Let (C, A) be a composite chain system. Then X(C, A) terminates with H2 ->0,




followed by a series of homomorphisms, 0 -O 0, which we discard. Let

?': ** I Hn> r. _+iin I * 2>

be an exact sequence in which the groups are Abelian, but otherwise arbitrary. A composite chain system (C, A) will be called a combinatorial realization of ?' if, and only if, ?(C, A) l 1'*

THEOREM 2. ?' has a combinatorial realization. Assume that we have constructed a composite chain system (C, A), and

homomorphisms

Inl Zn + -*> H, gX: rn r, AA k -> An I'n for every n = 1, 2, * * , such that

(a) Jnl lt'n l = 43nlz, iXngn = k i n = ljn

(b) lIn+1Zn+1 = H n X dCn+1 = In (0), 3Cn1 C k'-(O)

where, as usual, z e Zn+j and in: rn I-+ An is the identical map. Then it follows from (4.1b) that isomorphisms and homomorphisms,

gn~l: Hnl Hn X+ fn r n -? '

(n = 1, 2, ...

are defined by bl = 1', fk = k', where H,,+,, ln are in z = (C, A). It follows from (4.1a) and (1.3) that

bbzz = bl'z = goz = gblz

tg = V'i = fki = ft

ifk = jk' = I'J = plj = Djk.

Therefore bb = gb etc. and (), g, f) X -> I' is a homomorphism. Since Anal , gn are isomorphisms (onto) for every n, so if fn , by (7.5) on p. 435 of [17]. There- fore (C, A) is a combinatorial realization of ?'.

We now construct (C, A) inductively, starting with C, = Ao = 0. Let r > 1 and assume that we have constructed the groups and homomorphisms,

Cr -4 Ari r- Cr-i --- * Ao

likewise +l XX so as to satisfy (4.1) for n = 0,. , r- 1. Let r,., B, be any groups which are isomorphic to 1r, , 3(0) and let gr: rr r Fr, u: (3r' (0) 1Br be any isomorphisms. We define A, and j. by'0

Ar = r. + Br X jr (y + b) = u-1b (y eF rr , b i Br).

Then rI. =j-'(O) and jrA, = (3'(0). Since C. is a free Abelian group so are Trl (0) and B. . Let Ibx} be a set of free generators of B. . It follows from (4.1a), 10 If X, Y are any (additive) groups X + Y will always denote their direct sum. It is

always to be assumed that x e X, y E Y are identified with (x, 0), (0, y) E X + Y.




with n = r - 1, that b,1j,. = g,.-,.j, = 0. Therefore lIjrb?, = ixs( forsome e l' by the exactness of Z'. We define k:Ar ,--)Il by

(4.2) k(y + by,) = irg/y+ I,.

Then irk'y = 0 = 1j,-y and jrkIbx = lrjrbx . Also krir = i.gr . Therefore

(4.3) ig, = k'ir X jrkr = Ijr. Let {ye} be a set of elements which generate H$+1 . Let Zr+j be a free Abelian

group with a set of free generators, { z}, in a (1-1) correspondence, z, -+,, with {ye}. Let 1+1: Zr+1 -- Hr+1 be defined by lI+rzy = . Then 1+Zr+i = H +I Let Pr+,w be any group such that v Pr+1 ior-l() C Zr and let Cr+1 = Zr,1 + Pr+1. Since Cr is free Abelian so are lh(0) Pr+, and hence Cr+1. Let {pa} be a set of free generators of Pr+?1. Since vPr+1 = lr1(0) it follows from (4.1a), with n = r - 1, that

3rVp, = gr-ibrlrpu = 0.

Since jrAr = j3i l(0) it follows that vpa = jra, for some a, e Ar and from (4.3) that jrkrao = f 1jean = lrvpa = 0. Therefore there is a zy e r' such that k'aa = t = kroya, where aY = grl Y. Also it follows from (4.3) and the exactness

of ' that

krfgrlbrlrlz = trirbilr+lz = 0.

We define 0,+1 by

3r.+(z + pa) = fir1br+jl,.+z + (a. - ya). Then drl(z + pA) -j4,r+l(Z + pO) = jaa = vpa, whence dr+1(0) Z-Zr and dr+lCr~l = lr (0). Also kr3r1 = 0 and g9rgr+1z = br+ll'+lz. Therefore (4.1) are satisfied when n = r and the induction is complete.

ADDENDUM. The combinatorial realization, (C, A), of Z' may be constructed so that (a) In: Zn a H , if H' is free Abelian. (b) The rank of Cn is finite if both H' X H'-l have finite sets of generators.

To prove this we assume, as part of the inductive hypothesis, that these conditions are satisfied for n < r and also that the rank of Z,, is finite if H' is finitely generated. We insist that the generators hYi of H'+1 shall be free if Hr+1 is free Abelian, in which case lr+:Zr+l / Hr1 and finite in number if H+1 is finitely generated. In the latter case the rank of Zr+1 is finite. If H', and hence ZrX are finitely generated, so are l-lh(0) and Pr+1 . Therefore the rank of Cr+i is finite if both H'+,, H' are finitely generated. This proves the addendum.

Let z = 2(C, A), A' = 2(C', A'), where (C, A) and (C', A') are composite chain systems. If a given homomorphism F:Z -* V' is the one induced by a homomorphism (h, f): (C, A) -- (C', A') we shall call (h, f) a combinatorial realization of F.




THEOREM 3. Any homomorphism, F:Z -+ a' has a combinatorial realization (C, A) -+ (C', A').

Let F= (j, g, f). As in the proof of Theorem 2 we have

Cn+1 = Znl + P++1 , An = rn + BnX

where Pnl+ ~ dCn +, B, jAn . Let {bx}, {z,}, {pa} be sets of free generators of B, X Zn+, X Pool ,

Let ze Fn+t1An+14n+1z C Zn+1 and let ho +1:Zn+l - Zn+Z be defined by hoz, = z' for every n _ 1. Then

(4.4) iho= bl.

Let a' e k-lfnknb), C A';. Then it follows from (1.3), (3.1) and (4.4), since jA, C Zn X that

ljax= Ika= 'fkbx= bjkbx

= bljbx = lhojbx.

Therefore ja' = h0jbx + dc', for some cA e C'l . Let f A n A be defined by

fi=ig fbx= ax - Oc2

Then kfi - kig = ig = =t= fki and kfbx = kax = fkbx . Therefore

(4.5) bi = ih0, ig = fi, f kf.

Since lcf -fk and k= 0 we have kfgpa = fkgp, = 0. Therefore fgpa = fOc: for some ca e C'+ . Let h: Cn,+1 - C'+ be defined by hz = hoz, hp, = c"" . Then Ohp, = ff3pa . Since fOz = blz it follows from (4.5) and (4.4) that

ffz = fblz = gblz = btlz

blhz =,hz.

Therefore ff3 = Oh. Also jfi = jig = 0 = hji, jfb = jax- dc = hjbx . There- fore if = hj. Thus (h, f) is a homomorphism. Since hz = h'z it follows from (4.5) that (h, f) is a combinatorial realization of (b, g, f) and the proof is complete.

CHAPTER II. THE GROUP 1(A)

5. Definition of r(A) Let A be any additive, Abelian group. We shall define F(A), additively, by

means of symbolic generators and relations. The symbolic generators shall be the elements of A. We emphasize the fact that an element a e A is not an element of r(A). The elements of r(A) are equivalence classes of words, written as sums, in the pairs (+, a), (-, a) for every a e A. We write (i, a) as ?w(a) and, frequently, +w(a) as w(a). We shall use the symbol _ to denote equivalence between words and = to indicate that two symbols, in any context, stand for the same thing. In defining r(A), or any other group, by this means we




always assume that the "trivial relations", x - x 0, are satisfied as a matter of course.

The relations for r(A) are

(a) (w-a) w(a) (5.1) (b) w(a + b + c) -w(b + c) -w(c + a) -w(a + b)

+ w(a) + w(b) + w(c) 0,

for all elements a, b, c e A. It follows from (5.1b), with a = b = c = 0, that

(5.2) w(0) 0.

Hence it follows from (5.1b), with b = 0, a + c = d, that

(5.3) w(d) - w(c) - w(d) + w(c) 0. Therefore r(A) is Abelian. It follows from (5.1a), (5.2) and (5.1b), with a = b = -c, that

w (a) -w (2a) + 3w (a) O0,

whence w(2a) - 4w(a). Let

(5.4) W (a, b) = w(a + b) - w(a) - w(b).

Then W(a, b) _ W(b, a) since A and r(A) are Abelian. Since w(2a) = 4w(a) we have

(5.5) W(a, a) 2w(a).

Given that addition is commutative, it is easily verified that (5.1b) is equivalent to (5.6) W(a, b + c) W(a, b) + W(a, c).

It follows from (5.4), (5.6) and induction on n that

(5.7) w(ai+ * +an) = w(a,) + E W(ai, aj). i<i

On taking a, = = an it follows from (5.5) and (5.7) that

(5.8) w(na) n w(a).

Let y(a) e r(A) be the element which corresponds to w(a) and [a, b] e L'(A) the element corresponding to W(a, b). Then

y(a + b) = y(a) + y(b) + [a, b],

in consequence of (5.4). Therefore - [a, b] is a factor set which measures the error made in supposing the map y :A -* r(A) to be a homomorphism. We shall deal with the generators (i.e., generating elements) 'y(a) in preference to the symbols w(a).

Let g be a map" of the set of generators y(a), indexed to A by the map 11 We admit the possibility that gy(a) # gy(b) if a 6 b, even if 7(a) = 7(b). But tte map

g9: A -+ G shall be single-valued.




a -a y(a), into an (additive) group G. We shall say that g is consistent with a relation

Elw (a,) + * + Ew (an) 0 (Ei = A= 1)

if, and only if,

Eig'y(al) + * + Engy(an) = 0.

If g is consistent with all the relations (5.1) it determines a homomorphism r(A) -BG.

We give some examples of groups r1(A). First let A be free Abelian and let {ai} be a set of free generators of A, indexed in (1-1) fashion to a set {i}.

(A) r(A) is free Abelian and is freely generated by the set of elements y(ai), [aj, a,], for every i e {i} and every pair j, k e {i} such that"2 j < k.

Let'3 be a free Abelian group, which is freely generated by a set of elements, {gi, gjk}, indexed in (1-1) fashion to the union of {i} and the totality of pairs (j, k) such that j < k. Let 0:G -> r (A) be the homomorphism which is defined by cgi = y(aj), cgjk = [aj, ak]. Let a(x) = 2xiai where {xi} is a set of integers, almost all'4 of which are zero. Let

g(x) = E x i + E jxklgjk

Then g(-x) = g(x), where -x = {-xi}. Since

(xi + yi)2 -Xi - y' = 2xsy

(Xi + yj)(Xk + yk) - XJXk - YiYlk = XfYk + Xkyj

it follows that

(5.9) g(x + y) - g(x) - g(y) = 2 xjyigi + E (xyyk + xkyj)gjk, i j<A

where y = {jy}, x + y = {xi + yt}. Since this is bilinear in x, y and since g(-x) = g(x) it follows that the correspondence y { a(x) } -> g(x) is consistent with (5.1a) and with (5.6), and hence with (5.1b). Therefore it determines a homomorphism 4/: r1(A) - G. Obviously /y (ai) = gi and it follows from (5.4) and (5.9) that 9'[aj, ak] = gjk if j < k. Therefore 0'0 = 1. Also it follows from (5.7) and (5.8), with ai in (5.7) replaced by xiai, that 44' = 1. Therefore :G -r(A), which proves the assertion. (B) Let A be cyclic of (finite) order m and let a, be a generator of A. Then r1(A)

is cyclic of order m or 2m, according as m is odd or even, and is generated by 'y(a,).

This is a corollary of Theorem 5 below. " We postulate a simple ordering of the set {i} (j 5 k and k < j if j < k) as a conveni-

ent method of indicating a summation over all unordered pairs j, k (j 5 k). It is to be assumed, when the context requires it, that any such store of indices is simply ordered.

18 This proof of (A) and its use in simplifying an earlier proof of Theorem 7 below were suggested by M. G. Barratt.

14 By almost all we mean all but a finite number.




(C) If every element in A is of finite order and also divisible by ?its order, then r(A) = 0.

Let a e A be of order m and let a = mb. Then

2,y(a) = [a, a] = [a, mb] = [ma, b] = 0.

Therefore 2'y(x) = 0 for every x e A. In particular 2'y(b) = 0. If m is even it follows that

y(a) = 'y(mb) = m2'y(b) = 0. Since m2'y(a) = y(ma) = 0 and 2'y(a) = 0 we also have y(a) = 0 if m is odd. This proves (C).

It follows from (C) that rJ(R1) = 0 if R1 is the group of rationals, mod. 1. (D) If A is the additive group of rationals, then r (A) A. First let A be the additive group of any commutative ring, R. Then a homo-

morphism, g:r(A) -> A, is defined by g-y(a) = a2. Now let R be the ring of rationals. Then it may be verified that a homomorphism, f:A -* F(A), is defined by"5 f(p/q) = pqyy(1/q), where p, q are any integers, and thatfg = 1, gf = 1.

Let A be a free Abelian group and let I ai } be a set of free generators of A. Then the following expression for r(A) is suggested by [19]. Let lo be the group of integers and let A* be the group of homomorphisms a*:A -* 1o, which are restricted by the condition that a*aj = 0 for almost all values of i. Then A* is a free Abelian group, which is freely generated by { a' }, where ai aj = 1 or 0 according as j = i or j # i. We describe a homomorphism f:A* -* A as admissible if, and only if, fa^* 0 for almost all values of i and as symmetric if, and only if,

(5.10) a*fb*= b*fa*,

for every pair a*, b* e A*. Let

(5.11) fas* = Y =fijaj 2j(a*fa*)aj. Then (5.10) is equivalent to the condition fij = fpi .

Let S be the additive group of all admissible, symmetric homomorphisms, f:A* -* A, and let X:S -* r(A) be given by

(5.12) Xf = E (as*fai*)eij,

where eii = 'y(ai), eij = [ai, aj] if i < j. It follows from (A) and (5.11) that A-'(0) = 0. An arbitrary element a e r(A) is given by

a = 1yijeij ii i

for integral values of yjy , which are zero for almost all values of i, j. Therefore a = Xf, where f is given by (5.11), with fi, = 'yij if i _ j. Therefore

(5.13) X:S r (A). 15 Since pqk2-y(1/kq) = pq-y(l/q) we need not insist that (p, q) = 1.




6. Induced homomorphisms of r(A) Let f:A -* A' be a homomorphism of A into an additive Abelian group A'.

Let r(A') be defined in the same way as P(A) and let y(a') e r(A') be defined in the same way as-y(a). Then the correspondence y(a) -> -y(fa) obviously determines a homomorphism

g:1r(A) -* r(A').

We describe g as the homomorphism induced by f. It is given by gqy = yf. Ob- viously

(6.1) g[a, b] = [fa, fb] (a, b e A). It is also obvious that gF(A) = F(A') if fA = A' and that g = 1 if A =A' and f = 1. Let 9': r(A') - P(A ") be induced by f': A' -* A". Then it is obvious that g'g: P(A) -* P(A") is induced by f'f:A -* A". Hence it follows that

g: r(A) IF(A') if f:A /A'.

Let A admit a (multiplicitive) group, W, as a group of operators. Then so does F(A), according to the rule (6.2) uwy(a) = y(wa) (w e W). That is to say, w: F (A) -* P(A) is the automorphism induced by w: A - > A. Let f: A -* A' be an operator homomorphism into a group, A', which also admits W as a group of operators. Then it is easily verified that the induced homomorphism, g: r(A) -* P(A'), is also an operator homomorphism.

On taking A to be cyclic infinite and A' = A we see that a given automorphism, P(A) -r P(A) (e.g. -y(a) -* --y(a)), is not necessarily induced by any endomorphism A -> A; also that distinct automorphisms of A (e.g. a -* > a) may induce the same automorphism of r(A).

Let g: r(A) -> r(A') be induced byf:A -* A'. Let {a4I C A be a set of generators of A and let { b} C f'(0) be a set of elements which generate f'(0). Then

g-y(bx) = g[ai, b\] = 0. THEOREM 4. Let fA = A'. Then 9-'(o) is generated'6 by the elements

(6.3) 7(i), [ai, bx], for all values of X, i.

Let ro C ' l(0) be the sub-group generated by the elements (6.3). I say that, if a e A, b e f['(0), then

(6.4) -y(a + b)--y(a)e ro.

For let b = bi, + * + bx,, Then it follows from (5.7) that

y(b) =5 Ey(b,,) + E p b[,. P p<q

1 Cf. Theorem 6 in [20].




Each bk. is a sum of generators in the set aai}. Therefore [b,, b, ] is a sum of elements of the form [al, bNJ. Therefore y(b) e Fo . Let a = ai, + * + aiq. Then

[a, b] = ZZ ,air, bxo] E ro .

r o

Therefore

,y(a + b) -y(a) = [a, b] + y(b) Fo,

which proves (6.4). Let r* = r(A)/Fo and let a* e F* be the coset containing a given element

a e r(A). Since Fo C g'-(0) it follows that g induces a homomorphism,

9*: F*-* r (A'),

which is given by

(6.5) g*ey(a)* = g'y(a) = 'y(fa).

Then g*-l(0) = g-'(o)/ro and we have to prove that g*-l(0) = 0. Let u(a') e f-la' C A be a "representative" of a', for each a' e A'. Then

f{u(fa) - a} - 0,

whence

u(fa) = a + b(a) (a e A),

where b(a) e f-'(0). Therefore it follows from (6.4) that

(6.6) y{Iu(fa)}* = ay(a)*-

Similarly

67lu(-a'))* = PyI-u(a')}* = yIu(a')}*

(6.7) |>y{u(a+ **+ a')}* =y{u(a') + *+ u(a")*. Let g': {y (a') } -> r* be the correspondence which is given by

(6.8) g'y(a') = ayu(a )}*.

Then it follows from (6.7) that g' is consistent with the relations (5.1), for r(A'). Therefore it determines a homomorphism g': r(A') -- r*. It follows from (6.5), (6.8) and (6.6) that

glg*y(a)* = g'y(fa) = y{u(fa)}* = Py(a)*.

Therefore g'g* = 1, whence g*-'(0) = 0 and the theorem is proved. Let A' = A/B, where B C A is generated by { b,}, and let g: F (A) -*r (A')

be induced by the natural homomorphism A A'. Let A be a free Abelian group, which is freely generated by {ai}. Then A' is defined by {ail}, treated as symbolic generators, and the relations b-= 0, when be is expressed as a sum of




the generators ai and their negatives. It follows from (A) in ?5 and from Theorem 4 that r* = r(A)/g-'(O) is similarly defined by the symbolic generators y(ai), [a,, ak] ( < k) and the relations

(6.9) 'Y N ,)O [ai , b>,] 07 when y(bx), [ai, bx] are expressed in terms of +y(ai), ? [aj, ak]. Let us identify each element a* e T* with g*a* e r(A'), where g*:r* Fr(A') is given by (6.5) (obviously g* is onto). Then Theorem 4 can be restated in the form:

THEOREM 5. Let A' be defined by symbolic generators { as and relations { bx 0}. Then r(A') is defined by the set of symbolic generators y(ai), [aj, ak](j < k) and the relations (6.9).

Let A be generated by a, , subject to the single relation mal- 0, where m > 0. Since y(mal) = m2 y(al) and [a,, mai] = m[al, a,] = 2miy(a1) it follows from Theorem 5 that IF(A) is generated by -y(a,), subject to the relations mn (a1) = 0, 2mi-y(a,) _ O, which reduce to the single relation (M2, 2m)iy(a,) = 0. This proves (B) in ?5 since (M2, 2m) = m or 2m according as m is odd or even.

Theorem 4 is not necessarily true if f is into, but not onto A'. For example, let A' be cyclic of order M2, where m is odd. Let a1 be a generator of A' and let A C A' be the sub-group generated by mal . Then A and likewise 17(A) are of order m. Also F (A') is of order M2. Let g: r(A) -> r(A') be the homomorphism induced by the identical map f: A -* A'. Since

y(nmal) = M y (a,) = 0,.

in consequence of the relations for r(A'), it follows that gr(A) = 0, though f-1(O) = O.

As another example let A' be the group of rationals mod. 1 and let A C A' by the cyclic sub-group, which is generated by 1//m (m > 1). Let g: 17(A) -+1 r(A') be induced by the identical map A -* A'. In this case r(A') = 0, according to (C) in ?5. Therefore Theorem 4 may break down even if g, but not f, is onto.

THEOREM 6. Let A' be such that 17 r(A'/Ao) $ 0, for each proper sub-group 0/C A'. Then a homomorphism, f:A -* A', is onto A' if the induced homomorph-

ism, g: r(A) > r(A'), is onto r (A'). Let gr(A) = r(A') and let A' = fA. Then

f= ifo:A -* A',

where fo:A -* A' is defined by foa = fa(a e A) and i:A' -+ A' is the identical map. Therefore

9 = jgo:r (A)-+r(A'),

where go: r1(A) -+ r(Ao), j: r(Ao) >-+ r(A') are induced by fo, i. Since g is onto so is j. Let g': r (A') -* F (A'/Ao) be the homomorphism induced by the natural map f' :A' -* A'/Ao. Then f' is onto and so therefore are g' and

g'j: r(A') -> r (A'/Ao). 17 In ?8 below we shall see that this condition is always satisfied if A' is finitely generated.




But g'j is induced by fi: A' -- A'/Ao. Since f'iA' = 0 it follows, obviously, that g'j r(A') = 0. Therefore r(A'/Ao) = 0 and it follows from the condition on A' that A' = A'. This proves the theorem.

7. Relation to Tensor products Let A be a weak direct sum 18 2,A, ,where {A1,} is any set of additive Abelian

groups. Let r be the weak direct sum

r = 2r(Ap)?+ Ao Ar, q<r

where A, o A, is the tensor product of A, and Ar. Since [a,, a,] e r(A) is bilinear in a, e A, and ar e A, it follows that a homomorphism, f:1r -> r(A), is defined by the correspondences

(7.1) y(ap) -> 'y'(ap), a - a, - [a9,, a,],

where fy(ap) e r(Ap) and y'(a) 61 r(A) means the same as y(a) in ?5. THEOREM 7. f: r r(A). Let Ap be defined by a set of symbolic generators, api,, and relations bp), = 0.

We assume that each api is distinct from each a9j if p $ q. Then A is defined by the combined set of generators { api } and the combined set of relations I bp), 01. Therefore it follows from Theorem 5 that r(A) is defined by the union,

Sp X qr

of all the generators and all the relations in the sets

S' ^1'(api), [ap,, ax,] (i < j)

{y'(bpx) 0, O apj bp]- 0

[aqj, ark] (q< r)

[aj , brj] _ 0, [by , ark]- 0.

But r(A.) is defined by Sp and A9 o A, by 19 S,, rwhere S.,X S,, are obtained from S pX S'r by writing zy instead of y' in S" and x -y instead of [x, y] throughout S'r (x = a9j or b9g, y = ark or br,). Therefore r is defined by the combined system {SpX S~r} and (7.1) transforms this system into { Sp , SrI}. This proves the theorem.

8. "A" finitely generated Let A have a finite number of generators. Then it is a direct sum

(8.1) A = XI + ''' + Xt + Y1 + ''' + YrX

where X, is of finite order, ax, and Y. is cyclic infinite. Moreover we may take 18 An element in A is a set of elements {ap}, with a, e A , almost all of which are zero.

If a,=O except when p = p1, -, p. we write {ap} = ap, + *- + a., . 19 This follows from two successive applications of Theorem 6 in [20].




al X ... * at to be the invariant factors of A, so that ao > 1, ax I T+, . Let this be so and let p', *--, PP be those among al, *--, at which are distinct. That is to say

'k+l = **= k+l = PA # PX+1 (k1 = Ok+il =t) for X = 1,*,p. Let nx = k\+ -k . Then we denote (a * at) by

(8.2) (P1, n1), ... , (pp, np) .

Let {Aw} = {Xx, Y#J, in ?7, and let r be identified with r(A) by means of the isomorphism f in Theorem 7. Then it is clear that the rank of r1(A) is

r(r + 1)/2.

Let PA be odd. Then it also follows that PA occurs sx times in r(A), where SA is calculated as follows Let

N., = n,, + *-- + np + r (N,+l =r)

M = nx(nx + 2Nx+1 + 1)/2.

Then PA occurs nx(nx + 1)/2 times in the summand

>E r(Xi) + E Xi ?X (kC < i, j < kx+,) i i<i

and nxNA+? times in

EzEyXi ? Xy + Xi O Ya, j ~~~~S ~a

whereA = kx+i + 1, ,t, a = 1, ,r. Therefore sx = M. In general let Ph-, be odd and Ph even, where 1 _ h _ p + 1 and h = 1,

h = p + 1 have the obvious meanings. If as p=P and X > h then r(Xi) is of order 2px. Hence it follows that px occurs M - nx times in r (A) if X = h or if X > h and pX > 2px-l. In the latter case 2PxA- occurs nx-, times. If pA = 2px-1 and X > h then pA occurs MA - nx + nx-1 times. Also 2pp occurs n, times if h < p. Therefore the invariant factors of r (A), written in the form (8.2), are

(8.3) (Pi) SO) .. * * (PP, S P),

together with

(8.4) (2p,, n,)

for every ,U such that h _ ,U < p and either 2p,, < p,+ or I = p, where

(a) 's, = Ml if 1 _l < h

(8.5) (b) J Sh = Mh - nh

(c) 1 = MsxMx-nx if X > h, pA, > 2pxl

(d) (so = Mx-no + nx-i if X > h, PA, = 2px-l.




Notice that r(A) cannot be an arbitrary group. For example its rank must be a binomial coefficient or zero. Suppose however that a given group, r, is known to be of the form r(A), where A is finitely generated. Suppose further that the rank and invariant factors of r are known.

THEOREM 8. The rank and invariant factors of A are uniquely determined by those of r.

Let r' be the rank of r. Then the rank, r, of A is the (unique) non-negative solution of the quadratic equation

x2 + x -2r' = 0. Let the invariant factors of r, written in the form (8.2) be

(pi ,

SI') * ( Pq , 8q).

We proceed to determine the sequence (8.2) for A. If pq' is odd it follows from (8.3) that pp = pq . Since

Mp = np(np + 2r + 1)/2

it follows from (8.5a) that np is the non-negative root of the quadratic

x2 + (2r + 1)x - 2sq = 0.

If pa is even then p, = pq/2, n, = sq , according to (8.4), with j- = p. Assume that (px, nx) ... , (pp, np) have been uniquely determined, where20

X _ p, and let pA, = pi. If j = 1 the sequence (8.2) is determined and we number (p, , n,,) so that X = 1. If either j = 2 or if j > 2 and Pj-2 is odd it follows from (8.3), (8.4) and (8.5a, b) that px,_ = pali and that nx-, is the non-negative root of

X2 + ax - 2s'_ = 0,

where a = 2Nx ?t 1 according as Pl-i is odd or even. If j > 2 and P'-2 is even we consider the cases

a) s' = M,.- n,

b) s M - nx.

In case (a) it follows from (8.4) and (8.5c) that px-1 = P-1/2, -1 = si- In case (b) it follows from (8.5d) that

pX-1 = p /2, nx1 = s + nx - Mx.

Therefore px,_, nail are uniquely determined in each case and the theorem follows by induction on j.

Let g: r(A) -* r(A') be the homomorphism induced by a homomorphism f:A -* A'. If A' is finitely generated so is A'/Ao, where Al C A' is any sub- group. Therefore r(A'/Ao) P 0 unless A' = A'. Therefore it follows from

'? The value of p is not determined till the induction is complete. Therefore we allow X ? 0.



70 J. Hi. C. WHITEHEAD

Theorem 6 that, if A' is finitely generated and gr(A) = r(A'), then fA = A'. We prove a kind of dual of this.

THEOREM 9. If A is finitely generated, A' being arbitrary, then g-'(0) = 0 implies f'(0) = 0.

Since g[ao , a] = 0 for any a e A, ao efl(0) this follows from: THEOREM 10. If A is finitely generated and [ao, a] = 0 for every a e A, then

ao = 0. Let A be given by (8.1) and let xi, Yx be generators of Xi, Y. . Let [ao , a] = 0

for every a e A, where

ao = kixi + + kixt + llyi + + lryr

First assume that r > 0. Then t r

[ao, yr] = E k(xiX, yr] + Em lQyx, Yr] = 0.

But [xi, Yr] generates the cyclic summand, Xi o Yr , of r(A), whose order is ai . Also [yx, yr] is a non-zero element in the free cyclic group Yx o Yr or r (Yr), according as X < r or X = r. Therefore ki O(ai), lx = 0, whence ao = 0.

Let r = 0. Then a similar argument, with Yr replaced by xi, shows that kixi = 0 if i < t and that

kIt[xt, Xt] = 2kty(xt) = 0.

Therefore a I 2kt, where u is the order of yy(xi). But u = at or 2Ut according as at is odd or even. In either case a t I. Therefore ao = 0 and the theorem follows.

As a corollary of this and Theorem 8 we have: THEOREM 11. Let both A and A' be finitely generated. Then f:A ~ A' if, and

only if, g: r (A) r r(A'), where g is induced by f. Notice that, in consequence of Theorem 10, a finitely generated group, A,

is orthogonal to itself by the pairing (A, A) -* r (A), in which (a, b) = [a, b].

9. Direct systems Let 2f be the category of all (additive) Abelian groups, with all homomorphisms as mappings. Then a functor21 r: f -? 2I is obviously defined by the correspondences A r (A), f -* rf, where rf: r (A) -* r (A') is the homomorphism induced by f:A A'. Let C = Zir be the category of direct systems of Abelian groups, defined as in [10], except that the groups are to be Abelian. Let

rT: z-+, L=LimD *I9

be the functor defined by lifting r([10], ?24) and the direct limit functor. Let (D, T) be a given system in Z, with groups T(d) (d e D) and projections T(d2 , di)

21 See [10]. There are obvious generalizations of Theorem 12 below to non-Abelian groups. The partial ordering in D, below, is not to be confused with the simple ordering of indices, which was introduced in ?5 and which is not needed here.




(di < d2). Then r1(D, T) consists of the groups rT(d) and the projectiots rT(d2, di). Let

X(d): T(d) -- L(D, T), pu(d) :rT(d) -? Lr,(D, T)

be the injections in (D, T) and rT(D, T). Then a homomorphism

cw(D, T):Lri(D, T) rL(D, T)

is given by

(9.1) w(D, T) {/(d)'y(td)} =y{X(d)td},

where td e T(d). We recall from [10] that the transformation W:Lrl -? rL, which is thus defined, is natural and that r is said to commute with L if, and only if, c, is an equivalence, meaning that cw(D, T) is an isomorphism for each system (D, T).

THEOREM 12. The functor r commutes with L. Using the same notation as before we have

, (d2) IT T(d2, d)tdl } = M(d2) { rT(d2 , di) }Y(td1)

= ,U (di)7y(tdi) -

Therefore a single-valued map, 4, of the generators, -y{X(d)td} e rL(D, T), into Lr1(D, T) is defined by

(9.2) 7f {X(d)td} = A(d)y(td).

Let a, b, c e L(D, T). Then there is a d e D such that a, b, c have representa- tives r, s, t e T(d). Therefore

4ry(-a) = qxy{X(d)(-r)} = z(d)y(-r) = 4(d)y(r)

4y(a + b + c) = y{X(d)(r + s + t)} = M(d)y(r + s + t).

Similarly t(b + c) = ,u(d)y(s + t) etc. and it follows that 4 is consistent with the relations (5.1). Therefore it determines a homomorphism

0: rL(D, T) -+Lri(D, T).-

It follows from (9.1) and (9.2) that 4xq(D, T) = 1, w(D, T)4 = 1, which proves the theorem.

CHAPTER III. THE SEQUENCE 2(K)

10. Definition of M(K) In this chapter a complex will mean a pair (K, e?), where K is a connected CW-

complex and e? e K? is a O-cell, which is to be taken as base point for all the homotopy groups which we associate with K. Nevertheless we shall denote complexes by K, K' etc., remembering that, if Ki stands for (K, es) (i = 1, 2) and eo $ eo, then K1 $ K2. A cellular map, : K -> K', will mean one which,




in addition to 4K" C K"' for every n > 0 satisfies the condition ke0 = e'? where e0, e'0 are the base points of K, K'.

Let K be a given complex, let P2 = ir2(K2 , K),

C.+, = lrn+,(Kn+l, K"), An = 7rn(K") (n _ 2)

and let fl: C.+, -* A. , j: An -* C0 be the boundary and injection operators, where C2 = jA2 C P2. Let O:P2 -* w,(Ki) be the boundary homomorphism. Then OC2 = 1 and, as proved in CH II, P2 = C2 + B*, where 22 B* is the image of O3P2 in an isomorphism, 1*:3P2 ; B*, such that f3o* = 1. We can imbed C2 isomorphically in p2 made Abelian, which is a free 7r,(K)-module. Also Cn is a free xri(K)-module if n > 2. Therefore, ignoring the operators in 71r(K), C.(n >_ 2) is a free Abelian group. Also jnAn = jl'(0). Therefore the family of groups C.,, , An , related by (3, j with (C2 = 0, is a composite chain system (C, A) = (C0 A)(K), as defined in ?4 above. We define 2:(K) = (C, A).

Let rT,- r=(K) etc. be the groups in z = :(K). It follows from CH II that there are natural isomorphisms irn 11 rn(K), Hn Hn (R) (n > 2), where Hn (K) is the nth integral homology group of the universal covering complex, 1t, of K. The homomorphisms in , In in z are equivalent under these isomorphisms to i, I r. where in An A-* 7r,,n(K) is the injection, and to the resultant of the lifting isomorphism 7rn(K) 'it, Tn(R), followed by the natural homomorphism Tn(R) Hn(R). Also rn = i.7rn(K"'-), where i,,:n nr(Kn-) -*> An is the injection. There- fore r2 = 0 and z terminates with

* *- 13 13 S- 0 I 112 + H2 -> 0.

An element w e ri(K), operating in the usual way23 on Cn+1 , An,, obviously determines an automorphism, w: (C, A) at (C, A), which induces an automorphism, w: t A . More generally, let z be any algebraic sequence, of the kind considered in ?4. Then the totality of automorphisms of z is obviously a group G(Z). Let X: W -* G(M) be a homomorphism of a given (multiplicative) group W into G(1). Let V', W', X' be similarly defined. Then a homomorphism

(10.1) (F, W): (M, W.V X) (Z', WV', X')

will mean a pair of homomorphisms, F: Z 2', lu: W -- W', such that FX (w) X'{D(w)J F for each w e W. Since

X(wo)X(w) = X(WoW) = X(WoWWjO)X(wo)

it follows that (X(wo), [wo]) is a homomorphism, where [wo] is the inner automorphism, w -* wowwol, of W. We shall use X(K):7r,(K) > G(2) to denote the homomorphism which describes how 7r,(K) operates on 2 = 2(K).

We shall say that homomorphisms (F, ro), (F*, ro*), of the form (10.1), are in

22 In defining 2(K) we ignore B* and hence lose sight of the invariant k3(K) (Cf. [11], [13], [9]).

23 ILe. through the inverses of the injection isomorphisms 7r,(Kn) wi7r(K) (n 2).




the same operator class, {F, ro}, if, and only if,

(F, r) = (X'(wQ), [wo])(F*, l*) = (X'(wI)F*, [woJIv*), for some w' e W'. Let

(10.2) (F', t,'): (2;', W', X') -* (2;", wJ", XA)

be a homomorphism. Then, writing ro'(wQ) = wo, we have

F'X'(wo) - X"(wg)F', 0'[wo] = [w '.

Hence it follows that a single-valued product of operator classes is defined by

{F',1 V'} {Fo>I = FF1r'rII for all pairs of homomorphisms of the form (10.1), (10.2). It may be verified that all triples (2;, W, X) (the objects), together with all operator classes of homomorphisms (the mappings), constitute a category, A.

The usefulness of 2;(K) is limited by our ignorance concerning 7r,(K) for large values of n. Therefore we shall often want to confine ourselves to a finite part of 2. It will be convenient to start with Hq , and 2;q will denote the part

;q: Hq -+rq-, -- gq-1 *

of Z. We write ZOO = 2, thus defining 2;q for q < Oo. A homomorphism or isomorphism

(t,?,f):Vq -) (q < oo)

will mean the same as when q = x, except that Aa+1 , go, , fn will only be defined for n = 1,* *, q-1.

11. The invaiance of 2;(K)

Let K = (K, eo), K' = (K', e'0) be given complexes. It follows from proposition (J) in ?5 of CH I, on homotopy extension, that any map K -- K' is homotopic to one in which eo e'0. Hence it follows from (L) in ?5 of CH II that

a) any map, K - K', is homotopic to a cellular map. (11.1)

b) if 4)o - 41:K -+ K', where 4o, 41 are cellular, then To, 41 are related by a cellular homotopy, 4t:K -- K' (i.e. ,OK" C K'I'+).

Therefore, in discussing the invariance of 2(K), we may confine ourselves to cellular maps and homotopies.

A cellular map, O:K -+ K', induces a family of homomorphisms24

(11.2) hell: pr,+(K) -+pr+,(K'), fir 7r (Kr) -- r,(K") (r > 1),

such that Ah. = f if = hj. Since h2&2 = j2f2 we have h2C2 C = C2(K'). The induced homomorphism h2: C2 -+ C2 together with hn+1 , fi for n = 2, 3, **, obviously constitute a homomorphism

(h, f): (C A) -+ (C', A') = (C, A)(K'). t4 See [17] in CH II. pn(L) = 7rw(Ln, Ln-') (n > 2), pi(L) = ri(L').




This induces a homomorphism F: : (K) --+ (K'). Let tv :rl(K) -* 7r1(K') be the homomorphism, which is induced by 4) and is given by teal = tifi , where tl:7r(Ll)

7ri(L) is the injection (L = K or K'). Then h.+, , fn X h2: C2 -+ C2 are operator homomorphisms when ri(K) operates on (C', A') through Wo. Hence it follows that (F, ro) is a homomorphism of the form (10.1) where z = :(K), W = 7r1(K), X = X(K), 2' = 2(K'), etc. We describe (F, r) as the homomorphism induced by 4.

Let ) q*: K -+ K' and let (h, f), (h*, f*) be the families of homomorphisms of the form (11.2), which are induced by X, ,*. Then24

(11.3) xIh* - h = di + {d, xof*-f =

where x' e 7r1(K"), which operates on C'+, A' through the injection 7r1(K") 7ri(K'), and t: p(K) -+ p(K') is a deformation operator as defined in ?4 of CH II. The homomorphisms 03 l C2 24, 6 ..-. constitute a deformation operator i the sense of ?3 above. Also dC2 = 0 and di3C2 C C , since dCd C C' . Therefore

(h, f) - (wh*, wof*) :(C, A) -> (C', A'),

in the sense of ?3, where w' is the image of x' in the injection 7r1(K'1) -+ W'. Hence it follows that F = X'(w')F*, where F, F*: z - 2' are induced by (h, f), (h*, f*). Moreover ro = [w'] to* in consequence of the relation

xo (fi x)X 'C(fix)-l = 3262x,

which is included, additively, in (11.3). Therefore (F, to) and (F*, ro*) are in the same operator class, where (F, r), (F*, r*) are induced by 4), 4)*. Therefore a homotopy class, a K -+ K', of maps induces a unique operator class, 2a = F, t }, of homomorphisms. Let R be the homotopy category of all complexes (i.e. connected, CW-com-

plexes with base points). Then it may be verified that the correspondences

K-> (I(K) ,ri(K), X(K)), a- a

determine a functor :t -> Ad. We express this by saying that (z2(K), iri(K), X(K)), or simply that :(K) is a homotopy invariant of K.

Similarly 2q(K) is a homotopy invariant of K, for any q < oo. Also, for a particular value of n, the secondary modular boundary homomorphism

bn(m) :Hn(K, m) -+r._, m(K),

is a homotopy invariant within the category of complexes such that every (n - 1)- cycle in K is spherical. Notice that b4 (m) is defined for every complex.

The definition of (K) can be generalized as follows. Let r > 0 let A. = C. = 0 if n < r + 1 and if n > r + 1 let

Cn1 = 7rnl(K n'r+1 Kn) An = rn(K ).

Let fi:Cn+2 -*An , j An -* Cn(Cr+2 = jAr+2) be the boundary and injection operators. Then the groups Cn+1 X An , related by A, j, constitute a system, (C, A),




of the sort introduced in ?1. We define VW(K) = (C, A). Then it may be verified, in consequence of ?3 and (11.1), that 2T(K) is a homotopy invariant of K.

The groups which appear in 2;(K) are naturally isomorphic to groups which belong to a larger class of "injected" invariants. Let 0 ? p _ q < r, with q > p if p > 0, let m < n and let

,rt(Kn, K'; m, p) = i7r(Kih, Kr),

where i:7rr(K, K") -+ 7rr(K n, K') is the injection and 7rr(K8, K0) = xrr(K8) (s = m or n). Then it follows from (11.1)25 that 7rr(Kn, K'; m, p) is a homotopy invariant, and indeed an invariant of the n-type of K. In Chapter V below we shall see how these invariants may be defined for any arcwise connected space.

12. The sufficiency of Z(K) Let K, K' be given complexes, whose dimensionalities do not exceed q, where

q ? o.

THEOREM 13. If a map 4:K -+ K' induces isomorphisms !q(K) Y q(K') and 7r,(K) t 7r1(K'), then26 4 :K K'.

Since the homomorphisms Hn Hn(K) are natural this follows from Theorem 3 in Chapter I.

This is what we call a sufficiency27 theorem. We shall prove the corresponding realizability theorem, subject to the restrictions q = 4 and 7r1(K) = 1. But first we must prove a theorem concerning r3.

13. Expression for r3(K) Let u: S3 -+ S2 be a fixed map, which represents a generator of 7r3(S2). Let

v: S2 - K2 be a map which represents a given element28 x E II2 . Then vu: S3 - K2 represents an element u(x) e r3 . We have 29

(13.1) u(x + y) - u(x) - u(y) = [x, y]*, where [x, y]* is the product, or commutator (cf. [22]), of x, y e 112 . Also u(-x) =

u(x) and [x, y]* is bilinear in x, y. Therefore the map y(x) -* u(x) is consistent with the relations (5.1a) and (5.6), for r(I12). Therefore it determines a homomorphism, 0: r(112) -* r3 , and 0[x, y] = [x, y]*. Obviously30 0 is an operator homomorphism with respect to the operators in 7ri(K).

25 Cf. p. 220 in CH I. 26 :K _ K' means that 0 is a homotopy equivalence. 27 Cf. ?14 below and ?5 in [9]. 28 112 = H2(K), r3 = r3(K) and, in the following paragraph, H' = II2(K)', r' = r3(K

where r. (L) etc. are the groups in 2 (L). 29 See (7.6) in [211. Alternatively let P = S2 U S2 be formed from S2 by pinching an

equator into a point. Leto: S2 -- P be the identification map and p,: P -+ K a map such that l S2, * l S2 represent x, y, when S, S2 take their orientations from S2. Let v =

Then (13.1) follows from (11.5) in [3]. Similarly u(-x) = u(x). 30 This is obvious if the operators are defined by means of homotopies ? t: S- -* K, in

which c1tp0 varies, where po e Sn is the base point. If the operators are defined by means of the covering transformations, K -K 1?, it may be deduced from (13.2) below.




Let ( :, g, J):2(K) -* 2(K') be the homomorphism which is induced by a cellular map, q:K -* K', into a complex K'. Since 4.(vu) = (,Ov)u we have Gau(x) = u(fbx). Let g: r(ii2) -* r (12;) be the homomorphism induced by f2:II2 -, ]I' and let 0 mean the same in K' as in K. Since 0y(x) = u(x), g3U(X) = U(b2X), Y(f2X) = gy(x) we have

930'(X) = 98u(x) = u(f2x) = 6'y(f2x)=0y(x).

Therefore 0 is natural, in the sense that

(13.2) ?30 = g: r(112) -r3

THEOREM 14. 0 r(12) i r3 . Let R be the universal covering complex of K and let p :R -* K be the covering

map. Then it follows from the standard lifting theorems that

f2:1112(K) 11_ 112 , ,: r,() r. , where f2 , g3 are induced by p. Therefore g: r1{ 112(R) } F 112), where g is induced by f2 , and the theorem follows from (13.2) if it is true when K is replaced by K. therefore we may assume that iri(K) = 1.

Let ir,(K) = 1 and let {ai} be a set of free generators of A2, which is free Abelian since j:A2 C2 . Let {ex,} be the 3-cells in K and let ca e C3 be the element which is represented by a characteristic map3l for en . Then { cX} is a set of free generators of C3 and II2 is defined by the generators ai and the relations be 0, where b\ = Ocx . By Theorem 5, r(112) is defined by the generators y(ai), [aj, ak] (j < k) and the relations

(13.3) -y (N.)-02 [a; , NJ 0-.

Let K 2 = e0u{e~)}, where {e~} is a set of 2-cells in a (1-1) correspondence, ei - a , with {ai}. Thus K' = e0 and ei = u e; is a 2-sphere. Moreover w2(K2) is freely generated by the set of elements {a?), where as is represented by a homeomorphism i:S2 -?+ . Let 0: K2 -? K2 be a map such that (e I e)Oj represents ai and let 4,2:w2(Kl) A2 be the homomorphism induced by A. Then 2a? = ai .

Therefore 4b2: 112(KO) A2 and it follows from Theorem 1 in CHII that #2:K -K2. By (D) in ?5 of CH I any compact subset of K2 is contained in a finite sub-' complex. Therefore it follows from arguments similar to those used in the proof of Theorem 2 in [4] that 7r3(Ko) is freely generated by u(a?), [a? , a4]* (j < k). Therefore 7r3(K2) is freely generated by

(13.4) u(ai), [as , ak]* (j < k).

Notice that 0y(a,) = u(ai), 0[aj , ak] = [a, , ak]*.

Since any compact subset of K3 is contained in a finite sub-complex it follows from the proof of Lemma 4 on p. 418 of [4] that r3 is defined by the generators (13.4) and the relations u(bN) -0, [as , bej]* 0. It follows from (13.1) and (5.4) and the bilinearity of [a, b], [a, b]* that these relations, when expressed in terms of

1 I.e. a map, 3:I -+ e, such that 4'a C K2 and 4 | I- - I3 is a homeomorphism onto e3.




the generators (13.4), are the images under 0 of the relations (13.3). Therefore :1r(112) r3 and the theorem is proved.

14. Geometrical realizability Let q < ?o and let

24 : Hqg qi* 4H2 ' ?

be a sequence in which the (Abelian) groups are arbitrary except that r2 = 0, if q > 2, and (14.1) :1r(112) r 13

if q > 3. In this case 0, like b, i, I is to be a component part of Zq . Let 1q be a sequence which also satisfies these conditions. We shall describe (b, , f) 2 q 2 q as a proper homomorphism if, and only if, either q < 3 or q > 3 and

(14.2) g33 = g: r (I2) -fr

where g 1r(112) -* F(II') is induced by f2 . We shall describe a complex K as a geometrical realization of 2; if, and only if, 2q(K) is properly isomorphic to 2q .

The symbol (C, A)q will denote a composite chain system, with the groups Cr, Ari discarded if r > q and 0:r(112) r 17 if q > 3, where H2 = A2/#C3 . A proper homomorphism (isomorphism), (h, f)q, between two such systems, will consist of homomorphisms, (isomorphisms) hi, - - , hq and fi, * - *, fe_-, such that f = $lh, hj = if and (14.2) is satisfied if q > 3, where g3 is the homomorphism induced by f3 . That is to say i3g3 = f3i3, as in (3.4). We describe (C, A)q as a combinatorial realization32 of 2 if, and only if, there are homomorphisms (onto)

(14.3 ) InA+1 Zean+l X- sH rn t 'rn kn An -- H"

for n = 1, * , q - 1, such that k'-1(O) = #Cn+l , In-l(0) = dCn+l

n~ll'n~l= g n,1Z, ingn = knin Ik = lnke

and (14.2) is satisfied if q > 3. If (C, A)q is a combinatorial realization of ' so5

obviously, is any system (C', A')q which is properly isomorphic to (C, A)q . The existence of a combinatorial realization of 2q follows from the proof of Theorem 2, with g3 chosen so as to satisfy (14.2) if q > 3.

Let n < q and let 1n be the part of 1' which begins with H' . We shall say that (C, A)n is part of (C', A')q if, and only if, Cr+i = Cr+ X Ar = A' and 3ri,+X jr are the same in both systems, for every r < n. By an n-dimensional partial realization of 2 we shall mean a complex, K , of at most n dimensions, such that the part, (C, A)n , of (C, A) (Kn) is a combinatorial realization of 1t . Notice that a q-dimensional partial realization, K', is a geometrical realization of 2q if, and only if, l' : Zq , HQ' .

32 We take 2; to consist of a single homomorphism, 0 - 0, and we admit that (C, A), (C, = A. = 0) is a combinatorial realization of 2; .




LEMMA 1. Let n < q and let K' be a simply connected, n-dimensional partial realization of 2Q . If Fn'(K') r J the complex K' can be imbedded in an (n + 1)- dimensional partial realization of 2q .

Let (C, A). be part of (C, A) (Ks) and let (C, A). be extended by the construction in the proof of Theorem 2 to a combinatorial realization, (C', A')n+l , of 1+, . In order to simplify the notation we take J' in An to be the same as J'n, in Zn and the isomorphism F' F J., analogous to the one in (14.3), to be the identity. Let n Fn F n , where In = In'(Kn) and 9n is defined by (14.2) if n = 3. Then

An = n n+Bn y An = n r+ B'

as in the proofs of Theorems 2 and 3, where u :jAn Bn , u' :jAn BI and ju = 1, ju' = 1. Since An is the same in (C', A')n+l and in (C, A)n we have jAn = f3sl(O) = jA' . Therefore an isomorphism, f: A n A I is defined by

f(& + b) = gny + u'jb (y e Fn, b e Bn). Since jy = 0, jgny = 0 ju' = 1 we have jn f = jn .

Let { cA} be a set of free generators of C'+1 . Let

K n+1 = K n u { eA+1 }, n u

where the (n + 1)-cell e`n'l is attached to Kn by a map, 0 EA ' -+1 Kn, such that Vx: I * Kc represents f'c, where vx: E Pl is a homeomorphism.

Then en+l has a characteristic map, h: I n+I - ex'+ which agrees with 4xvx in In+1. Let (C, A) n+1 be part of (C, A) (K n+) and let CX E Cn+j be the element which is represented by f . Then { cx} is a set of free generators of Cn+j and an isomorphism, h Cn+,l Cn+1, Xis defined by hc) = c' . Moreover oxv represents both Ocx and f'fc . Therefore ff3 = flh. Also jn f = , fy = gncy and 9n satisfies (14.2) if n = 3. Therefore the identical maps of Cr,+ , A, (r < n), together with h, f, constitute a proper isomorphism (C, A)n+1 I (C', A')n+1 . Therefore (C, A)n+1 is a combinatorial realization of 2n+i and the lemma is proved.

THEOREM 15. 24 has a (simply connected) geometrical realization K, which is a) at most 4-dimensional if H' is free Abelian, b) a finite complex if each of H , H', H is finitely generated. Let K1 consist of a single 0-cell. Then K1 is a partial realization of 21 . Since

F,(K) = 0, F2(K) = 0 and F3(K) 1P{I12(K)}, where K is any complex, it follows from three successive applications of Lemma 1 that 2; has a 4-dimensional partial realization, K4.

Let (C, A)4 be part of (C, A) (K4) and let 1', g mean the same as in (14.3). Let z e lh1 (0). Then ,Bz = gjlbl'z = 0. Therefore z E j47r4(K4). Let { z, I be a set of elements which generate 1> -(0) and let a, e J41z. Let K5 = K4u {Ue}, where ei is attached to K4 by a map which represents a, , and let { c, I be the corresponding basis for C5 = 7r5(K5, K4). Then fc, = a, and dc, = z,. Therefore dC5 = lh(0) and it follows that 1 induces an isomorphism 4: H4(K5) H . Therefore K5 is a full realization of Z.




If H4 is free Abelian we may assume that 14-l(O) = 0, as in the addendum to Theorem 2. In this case 24 is realized by K4. Also we may assume that, if H2' H3, H' are finitely generated, so are C2, C3 , C4 and hence l41(0) and C5 . In this case K5 is a finite complex and the theorem is proved.

We now consider the realizability of a proper homomorphism, Fq:2 q -+ 2 by a map :K - K'g, where K, K' are given complexes and 2 = 2(K), 2' =

2I(K'). Let (C, A)q, (C', A')q be parts of (C, A) (K), (C, A) (K'). Then it follows from the proof of Theorem 3 that Fq can be realized combinatorially, in the same way as when q = co, by a (proper) homomorphism (h, f)q: (C, A)q - (C', A')q . We shall describe a cellular map, : K' - K" as a (geometrical) realization of both (h, f)q and Fq if, and only if, the homomorphisms h, f are those induced by 4. Notice that hqdCq+ C dCq+, , since lqhq = bqlq ; also that a given map, 4 :K' --

K induces a homomorphism Mq - 2 if, and only if, hq:Cq g Cq satisfies this condition, where hq is induced by 4. This is certainly the case if K = K', for then Cq+1 = 0.

Let (h, f)q be a combinatorial realization of a given proper homomorphism Fq: 2Mq *q . Let 1 < n < q, let (C, A), , (C', A'), be parts of (C, A)q , (C', A')q and let (h, f)n consist of the homomorphisms hi, * *, h. and fi , *--, f,- Let Fn: 2 2n be the homomorphism which is similarly induced by Fq. Then Fn is obviously a proper homomorphism and (h, f). is a combinatorial realization of F. . Let 4)n :K' -* K'" be a realization of (h, f)n . We assume that K, but not necessarily K', is simply connected and also that 4) K1 = e'?, the base point in K'. Let g?, gn: Fn -* Fn be the homomorphisms induced by c' and by fn in (h, f)q.

LEMMA 2. If g? = gn and if inAn is a direct summand of Cn , then O ? I K" can be extended to a realization, Kn+l -> K n+i1 of (h, fXn+i .

First let n = 1 and let P2 = 7r2(K2, K'). Then p2 = C2 + B*, as in ?10, where 3*: 3P2 B* and 33* = 1. Also fP2 = 7r,(K'), since 7ri (K) = 1. If aE P2, b* e B* we

have (fa)b* = a + b* - a, by (2.1c) in CH II, and a + b* - a e B* since P2 is the direct sum of C2 and B*. Therefore B* is invariant under the operators in 7ri(K'). Also /3P2 operates identically on C2 . Therefore h2: C2 -* C2 can be extended to an operator homomorphism, h* P2 -4 r2(K'2, K"), by taking h*B* = 0. Since f3C2 = 1 and O0K1 = e'? we have /3h*p2 = fP2= 1 where f0 :7r, (K') -> (K") is induced by 4) . Therefore it follows from Lemma 4 in CH II that 40 can be extended to a realization, K2 K K'2, of (h) f)2

Let n > 1 and let f? An A' be the homomorphism induced by 40 . If n= fn we have f n3n+1 = ,n+lhn+l since (h, f)n+l is a homomorphism. Therefore

it follows from33 Lemma 4 in CH II that 4) can be extended to a realization, Kn+l = K n+l1 of (h, f)n+l. Therefore the lemma will follow when we have extended 4? I K'n- to another realization, On: Kn -4 K'n, of (h, f)n , which induces fn

Since fn , hn are both induced by 4)n we have jnf ? = hnjn = jnfn Since j?'(0) =

33 If 7r1(K) # 1 this argument fails unless hn+1 , fn are operator homomorphisms asso- ciated with the homomorphism, 7r, (K) -+ 7r1(K'), which is induced by n -




0 it follows thatf? = fn if n = 2. Let n > 2, let go = and let Cn = A* + B*, where A* = ijAn . Let u:A* An be a right inverse of jn . Then a homomorphism, A:CX An X or cochain A E Cn(K', An), is defined by

A(a* + b*) = (fn - f )ua* (a* e A*, b* e B*). Since jn(jfn - fn) = 0 we have jnA = 0, whence ACn C 2n .

Let 'On: Kn -- K"' be an extension of 4 I K'n-, which realizes34 the separation cochain d(4n , 4) = A. Let f' An -- A ' and h'n: Cn -* C' be the homomorphisms induced by On . Then35

h'n - hn = jnA = 0, f-fn =Ajn . Therefore On is a realization of (h, f)n , because hn = hn . Also

(f - fn)ua* = Ajnua* = Aa* = (fn - fn)ua*. Therefore flnua* = fnua*. Also fon- = gon'Y = =nt fny, if I Fn Iand f'nY = fon since 'On I Kn-1 = 4?n I K n-. Therefore fln- = fny. Since jnu = 1 we have A' = rn + uA*. Therefore fl = fn and the proof is complete.

THEOREM .16. If wi(K) = 1 and q < 4 any proper homomorphism, Fq: 2Z(K) 2q(K'), has a geometrical realization K' K- .

Let (h, f)q: (C, A)q -+ (C', A')q be a combinatorial realization of Fq, where (C, A)q, (C', A')q are parts of (C, A) (K), (C, A) (K'). Since r2 = 0 the theorem follows from two successive applications of Lemma 2 if q _ 3. Let q = 4 and let : K3 - K'3 be a geometrical realization of (h, f)3 . Then 46 induces the homomorphism f2: 12 -* HI in F4 , and it follows from (14.2) that go = g3 , where

0 S3 X S3 are induced by 43 , f3. Therefore the theorem follows from another applica- tion of Lemma 2.

Let Q3 be any category. We describe two objects in e as equivalent if, and only if, they are related by an equivalence in Z. Let T:U -* Z} be a given functor, where U is any category. By the sufficiency and the realizability conditions, with respect to T, we mean the following, Sufficiency: if Ta is an equivalence, so is a, where a is a given mapping in U. Realizability: a) any object in Q3 is equivalent to the image, TU, of some object in U, and

b) any mapping. TU -* TU', in Z8, is the image, Ta, of at least one mapping, a: U -* U', for every pair of objects, U, U', in U.

Let S4 be the homotopy category of all simply connected complexes of at most four dimensions. Then a mapping (i.e. homotopy class), a:K -* K', in S0 induces a unique homomorphism 24a: 24(K) -* 24(K'), because 7rw(K') = 1. Let c54 be the category in which the objects are all sequences 24 , which satisfy the conditions r2 = 0 and (14.1), and in which H4 is free Abelian, with all proper homomorphisms as mappings. Then a functor, 24:Ko -* 4 is obviously determined by the correspondences K -* 24(K), a -* 24a.

34 See [15]. In order to apply the existence theorem (10.5) in [15] we can take sn = 4? in e , where 4 C e, is an n-simplex, for each n-cell en e K.

36 See Appendix B below.




THEOREM 17. The functor 24 satisfies both the sufficiency and the realizability conditions.

This follows from Theorems 13, 15, 16. As pointed out in the introduction, both homotopy classes of maps S4 _+ S3

induce the same homomorphism 24(S4) 4 2(S3). Therefore the function a -*

24a is not (1-1). By taking N = S3 U S4, where S3 n S4= e, we see that 24 does not even induce an isomorphism of the group of equivalences a:K = K.

Sequences of the form 24(K) can be simplified algebraically by identifying F3 H3 , H2 with r(I12), I13/tr3, 112 so as to make 6 = 1, I3 the natural homomorphism and f2 = 1. When 24 is thus simplified it is completely determined by 112 and

(14.4) H4--+ r(112) -113.

Let I12 be finitely generated. Then it follows from Theorem 8 that 112 is determined, up to an isomorphism, by r(112). Let F4: 24 -+ 2 be a proper homomorphism which determines an ismorphism of the sequence (14.4). Then it follows from Theorem 11 that F4:z4 2 :4 (obviously 43:13 H Hf if g3 , f3 are isomorphisms, where F4 = (t, g, f)).

Theorem 16 is analogous to Theorems 1, 2, 3 in [5], concerning (n-l)-connected complexes (i.e. those with Tr(K) = 0 for r = 1, - - , n - 1) when n > 2. Let 2 = Z(K), where K is (n - 1)-connected (n > 2), and if x e IL, let u(x) e r, + be defined in the same way as u(x) e r3, in ?13 above, when x e 112 . Then u: II. - r-- is a homomorphism since n > 2. It is shown in [51 that 6: Hn,2 r ,n~l where J1n,2 = HI,/2II. and 6 is induced by u. The argument leading to (13.2), with :12 -+ r(I12) replaced by the natural homomorphism HI -+ 11n,2X shows that 0 is natural. Therefore Lemmas 1 and 2 yield realizability theorems for 2+2 , with rr = iir = Hr = 0 if r < n and (14.1) replaced by :11n.2 r 1flX+,

which are analogous to Theorems 15, 16. Many of these facts have been recorded by G. W. Whitehead in f23]. In

particular his results may be used to show that, if 2 = Z(K), where K is (n - 1)- connected, then Zn+2 is internally exact (n > 2) and that the sequence 2? +2 which is defined in ?15 below, is exact if n > 3.

Theorem 16 is also analogous to Theorems 1, 2, 3 in [9], concerning the 3-type of an arbitrary (connected) CW-complex. In the following section we prove a theorem which leads to analogous results concerning the 4-type of a simply connected complex.

15. q-types

Let 2 ' q < m. We recall from CH I that complexes K, K' are of the same q-type if, and only if, there are maps,

s h:tKq --' K - 1If, o: K o ifts ntn Kq

such that,0'0 I Kq-1 1, A' I K'q-1 1. If, and only if, these conditions are




satisfied, we write 4: qKKIq. By Theorem 2 in CH I this is so if, and only if, 0 induces isomorphisms -rn(K) - xrn(K') for n = 1, --, q - 1. Obviously K and Kg have the same q-type. Therefore, when studying q-types, we may confine ourselves to complexes of at most q dimensions.

Let Iq = 2q(K), where, to begin with, dim K may exceed q. Let

H' = Hq(K) = Hq/jlIq

Thus Hg ~ Hq(fK)/Sq(fK), where K is the universal covering complex of K and Sq(fK) consists of the "spherical" homology classes. Since jIlq = bV1(0) an isomorphism

b?: H, bqHq = I l(o)

is induced by bq . Let 1q = 2(K) be the exact sequence

19q O Hgo - 0 q--* *- *I H2 0.

Since lqZq = Hq X lqjqAq = jqIIq , H(K) = Zq/jqAq it follows that lq induces an isomorphism H (Kg) H , by means of which we identify TO(Kg) with f (K). From the purely algebraic point of view 2q may be regarded as a sequence X, in which every group preceding Hq is zero. Therefore we need not redefine the terms homomorphism etc.

Let 1q = 2iq(K') and let (b, g, f) 2q g 1q be any homomorphism. Since bqgbq = gq-lbq it follows that

b2b-91(O) C b-1(O) I l'

Therefore fb induces a homomorphism

boq Ho __ It Obviously bobo = gqilbq . Therefore a homomorphism, F: --* Ito, which consists of ij' and of g. X f, X fbn for n = 2, , q - 1, is induced by (ax g, f). If (1b, g, f) is induced by a map 4: Kg K'- we shall say that 4 induces, or realizes, Fg.

THEOREM 18. a) If q1:1r1(K) - xr1(K') and F?:? ; ?, where 4) and F. are induced by : K-* K'", theno: K -_K'q .

b) Any proper homomorphism. FO: ) 2; ? has a realization K4 -* K'4.

Part (a) follows from Theorem 2 in CH I. Let {zI be a set of free generators of H4(K4). Let

e b 1g3b4z, C H4(K'4),

where 93: J'* r' is in F' , and let b4:H4(K4) -* H4(K'4) be the homomorphism which is defined by 4z, = z4 . Then b4b4 = 93b4 and it follows that b4 induces a homomorphism b'*:HO(K4) -* Ho(K'4), such that bo4b* = 93bo = .b4 - Since (b2o)- (0) = 0 it follows that b* = bo. On replacing b' by f4 we have a proper




homomorphism, F4:24 - 24, which induces FM . Part (b) now follows from Theo- rem 16.

An algebraic sequence 1' is a special kind of 4 , namely one such that 41(0) = 0. Therefore Theorem 15 applies unchanged to the geometrical realization of I' . Notice, however, that any sequence 24 is realized by a complex K4, since 2;(K) = 24(K4), even if dim K > 4.

Notice that, by analogy with (14.4), :? may be replaced by a pair of arbitrary Abelian groups, II2, II3, and a homomorphism t r(112) -* I13, which may be arbitrary. Therefore the "algebraic 4-type" of a simply connected complex is a comparatively simple affair. The algebraic (n + 2)-type of an (n - 1)-connected complex (n > 2) may be similarly defined, with r(II2) replaced by 1In,2-

CHAPTER IV. THE PONTRJAGIN SQUARES

16. The main theorem We give the following definition of a net of finite, simplicial complexes, which

differs slightly from the one in [18]. Let {K(d) be a set of such complexes, which is indexed to a directed set D. Instead of taking a projection, K(d2) -* K(di), to be a single map, where d1 < d2 , we take it to be a homotopy class of simplicial maps. We then assume that, if di < d2, there is a single projection,

K(di, d2) : K(d2) -K(d), such that K(d, d) = 1 and K(d1, d2)K(d2, d3) = K(di , d3) if di < d2 < d3. Let (D, K) denote this net and let (D', K') be a similar net. By a homomorphism, (16.1) (R*, p): (D, K) -* (D', K'),

we shall mean an order preserving map, R*:D' -* D, together with a family of homotopy classes,

p(d'):K(R*d') -> K'(d'), where p(d') is defined for every d' E D' and

p(d')K(R*d', R*df) = K'(d , df)p(df) (d d1) It is easily verified that all nets, with all homomorphisms as mappings, is a category I . We shall sometimes denote nets by X, X' etc.

Let 2{ and Z mean the same as in ?9 above and let (1, 21) be the Cartesian product of I and W, in which the mappings, (%, a), are pairs of homomorphisms, t:X -* X', a: A -* A'. Let (St, 21) be similarly defined, where 9, is the homotopy category of finite, simplicial complexes. Let P: (t, , 21) )-* W be a functor, which is contravariant in S, and covariant in 2. Let (p, a): (K, A) -* (K', A') be any mapping in (ft, %). Then P(p, a) is a homomorphism

P(p, a) :P(K', A) -* P(K, A').

Let (D, K) be a given net and let P{K, A} denote the family of groups P(K(d), A),




for every d e D. Then it follows without difficulty that (D, P{K, A}), with the homomorphisms3

(16.2) T(d2, di) = PK(d1, d2):P(K(dl), A) -* P(K(d2), A), is a direct system of groups; also that a "lifted" functor,

PI: (X) W) @, is defined by

(16-3) fP,((D, K), A)= (DP{K,A}) (P,((R*, p), a) = (R*, P{p, a}),

where (R*, p) means the same as in (16.1) and P{p, a} denotes the family of homomorphisms

P(p(d'), a) :P(K'(d'), A) P (K(R*d'), A').

Therefore LPI is a functor, LP I: (Y.) 2) 9A,

where L:Z a is the direct limit functor. Let r :P > Q be a natural transformation, where Q: (t, 2 1) -* 21 is a functor

of the same variance as P. Let Qz: (X, 2I) -* Z be defined in the same way as P, . Let (X, A) mean the same as before and let r {K, A } denote the family of homomorphisms

'r(K(d), A):P(K(d), A) -* Q(K(d), A).

Since r is natural, and since P, Q are contravariant in f, we have

'r(K(d2), A)PK(di, d2) = QK(di , d2)'r(K(di), A), or

r(K(d2), A)T(d2, di) = U(d2 , di)'r(K(di), A),

where T(d2, di) is given by (16.2) and U(d2, di) = QK(di, d2). Therefore a homomorphism,

Tz(X, A):P1(X, A) -* Q1(X, A),

is defined by 'r(X, A) = (1, r4K, A}). Let (R*, p) and a mean the same as in (16.3) and let S = P or Q. Then

S(p(d'), a) is a homomorphism

S(p(d'), a):S(K'(d'), A) -* S(K(R*d'), A'). Since r is natural we have

,r(K(R*d%), A')P(p(d'), a) = Q(p(d'), a)'r(K'(d'), A). 36 PK(d1 , d2) stands for P(K(di , d2), 1).




Therefore, writing t = (R*, p), it follows from (16.3) that

'rj(X, A')PL(, a) = (1, r{K, A'1)(R*, P{p, a})

= (R*, 'r{K, A'}P{p, a}) = (R*, Q{p, a}r{K', A})

= (R*, Q{P, a})(1, r{K', A})

(16.4) = Q(%, a)ri(X', A).

Therefore Tr is natural. Let Lr1:LP- LQ1 be the transformation which is defined by

(LTr)(X, A') = L(rz(X, A').

On applying the functor L to both sides of (16.4) we see that Lrj is natural.37 The rth cohomology functor HI: (2a, X1) -* 2l is contravariant in 9a and

covariant in 2. We define H' in the same way as PI and write

HI = LH":(^, 21) -* 21.

Then H' is the Cech cohomology functor. We define the cup-product, YUZHn (x, r(A)),

of elements y, z e Hn(X, A), by means of the pairing (a, b) = [a, b] e 1(A), where

a, b e A.

It is obvious that a covariant functor r: (XX, 2) (X,+ 2i)

is defined by r (X, A) = (X, r1(A)), r(t, ca) =(, ra). Thus we have functors

(16.5) ratr, Iir: (XX W) -- W

which are contravariant in I and covariant in 21. THEOREM 19. Let n be even. Then there is a natural transformation,38

(16.6) Aj: ran --+Humr, such that

(16.7) n(X, A)@, z] = y u z.

for every pair y, z e H"(X, A). Assume that the analogous theorem has been proved for the category (St, 21) 37Cf. the concluding remarks in ?9 of [10]. 31 We do not assert that v is uniquely determined by naturalness and (16.7). When this

theorem is quoted it is to be understood that v is the transformation which is defined in the course of the proof.




and let T: rIP - H2nr be a natural transformation which satisfies (16.7), where rIF, Hr: (2a, 21) --* 2[ are defined in the same way as rHT, Hr in (16.5).

It follows from (16.3) and the definitions of rF:S -* Z and r: (ft, 2 w) -* (32a, X ) that

(rIf) = r1111, (r r)f) = Hr r. Therefore

LTI:Lrjj n -- LH Inr

is a natural transformation. It follows from Theorem 12 that W-1: rL - Lr5 is a natural equivalence, where co is defined by (9.1). Therefore37

-'H n: rLHn -* Lr Hn is a natural transformation. Therefore, writing LH' = Ir, it follows that

= (LTz)(co-hH n): rHn - H- nr

is a natural transformation. We now verify (16.7.). Let X = (D, K). Then it follows from (16.3) that

Hr (X, A) = (D, Ir {K,A , rIr (X, A) = (D, rHr{K, A}). Let

X(d):H (K(d), A) -H (X, A)

(d):rHn(K(d), A) , Lr H n(X, A)

v(d):H2 (K(d), A) H HI (X, A)

be the injections. Let (1, p): I1Hn(X, A) H HIn(X, A) be a homomorphism of the direct system rBn (X, A) into the direct system HIn (X, A). Then

L(1, p):Lr1Hn,(X, A) -f H2n(X, A) is given by

L(1, p)p(d)g(d) = v(d)p(d)g(d),

where g(d) e rHn(K(d), A). Let

y = X(d)y(d), Z = X(d)z(d),

where y(d), z(d) e Hn(K(d), A). Since

[y(d), z(d)] = y(y(d) + z(d)) -7(y(d)) - y(z(d))

it follows from (9.1) that

w(Hl(X, A))M(d)[y(d), z(d)] = [y, z]. Therefore

(co-'H n)(X, A)[y, z] = c(Htn(X, A))-1[y, z]

= 1(d)[y(d), z(d)I




By hypothesis T(K(d), A)[y(d), z(d)] = y(d) u z(d). Therefore

-q(X, A)y, z] = (LTIr)(X, A){( Ic&1Hn%)(X, A) [y, z]} = L(1, T4K, A})ji(d)[y(d), z(d)] = v(d)T(K(d),A)[y(d), z(d)]

= v(d)(y(d) u z(d))

= yU Z.

Therefore it only remains to prove the theorem for the category (f, L). Let K be a finite simplicial complex, let CT = CT(K) be the group of integral,

r-dimensional cochains in K and let cl, ***, cq be a canonical basis for Cn. Then

(16.8) 6ci = aidi (i = 1, ... ,;aisi)) where oa di = 0 if i > t and (di, ... , dt) is part of a canonical basis for Cn+l. We recall from [3] the definition of the Pontrjagin Square (16.9) pC = C U C + C Ui 6C (c C Cn).

The cochain group, Cn(A), is the tensor product, C (A) = A o C,

and the group of cocycles, Zn(A) C Cn(A), consists of those, and only those, cochains,

x =alcl + * +aqcq,

such that aoa2 = = uqaq = 0, where i = 0 if i > t. The cup-product of cocycles, (16.10) y = a1*c, + + aqcq, z = b1*cl + + b.c.

is defined by y u z = Ei Ej [a , bil , ci u c;.

If i < j then ao I aj and, since n is even, Ci U Cj Cj U Ci mod. ai.

Also aiai = aibi = 0. Therefore

(16.11) y u z [ai, bi]*ci u ci + ([ai, bj] + [aj, bi]) ciu ci. t t~~~~<j

I say that a homomorphism (16.12) v:1r(Z (A)) Z2 (r(A))

is defined by (16.13) v-y(y) = Z(ai)pci + 1I [ai, a ci u cjj,

i<i



88 3. H. C. WHITEHEAD

where y e Z'(A) is given by (16.10). For since aiaj = 0 we have

a (ai) = y(aiai) = 0

2,Toy(ai) = o$[ai, ai] = [ajaj, ai] = 0.

Therefore (a', 2ai)y(aj) = 0. That is to say acy(ai) = 0 if ao is odd and 2aury(ai) = 0 if au is even. Obviously 5pci 5 0, mod. ai, and it is proved in [3] that bpci _ 0, mod. 2ai, if oai is even. Therefore, and since o4[a , a;] =

qi[ai, a,] = 0, it follows that bvy(y) = 0. That is to say, vy(y) e z,(r(A)). Obviously Pry(-y) = vy(y). Therefore v is consistent with (2.la). Let z e ZN(A)

be given by (16.10). Then

y + z = (a, + bi)ci + *. + (aq + bq)Cq

Since y(ai + bi) - -y(a,) -y (bi) = [ai, bi] and

[ai + bi, aj + bj] - [ai, a,] - [bi, bA] = [ai, bil + [a,, b,]

we have

v[y, z] = v(y + z) - v(y) - vY(z)

(16.14) = Z[aj, b]i.ci + E ([a., b,] + [aj, bi]) ci u c,.

The right hand side of (16.14) is bilinear with respect to (ai, ***, a,) and (b1, ** *, bq). Therefore v is consistent with (5.6). Therefore (16.12) is a homomorphism.

Since aj[ai, bi] = [o-ai, bi] = 0 it follows from (16.8) and (16.9) that

[ai, bi] * pci = [aS , bs] .ci u ci + i[a, bi] Jci ul di = [ai, bi] cs u ci.

Therefore it follows from (16.14) and (16.11) that

(16.15) v[y, z] -' y u z.

Let y = 5w where w e C'-'(A). Let a

w = -E ax a, X-1

where (el a. * ) is part of a canonical basis for C'-' and &Wx = rxct+s. Then

y = Ex ax * bax = ExTx ax * ct+x.

If SC = 0 we have pc = c u c. Therefore

y (Txax) - PC u4 = y (ax) *-(Ctx U Ctx)

= Py (ax) Sacx u Sax

= 5'y (ax).-xu cxI

[max, TaI]c-Ct U Ctx = [ax, a,]- Sax u be, = 5([ax, a,].ax u 8c)




Therefore it follows from (16.13) that v-y(bw) - 0. Also it follows from (16.15) that

v[&w, z] (5w) u z O 0. Therefore it follows from Theorem 4 in ?6 above that the kernel of the homomorphism

F(Z'(A)) - F(HN(K, A)), which is induced by the natural homomorphism, Z'(A) -* H'(K, A), is carried by v into the group of coboundaries in Z2,(F(A)). Therefore v induces a homomorphism

T(K, A):r(H (K, A)) -H (K, r7(A)).

Also it follows from (16.15) that

T(K, A)[y*, z*] = y* U Z*,

where y*, z*e-HS(K, A). Let (p, a): (K, A) -* (K', A') be any mapping in the category (a, , 9). Then

(p, a) is the resultant of (1, a), followed by (p, 1), where each 1 denotes the appropriate identity. It is obvious that not only T, but even v is natural with respect to the homomorphisms a:A A'. It remains to prove that r is natural with respect to maps K -- K'.

Let f': Zn(K', A) Z Z'(K, A)

~n: Z2n (K', r (A)) _+Z2n} (K, r (A))

be the homomorphisms induced by a simplicial map 4:K -> K'. Let

gn :r(Zn(K', A))-- rP(Zn (K, A))

be the homomorphism induced by fn and let vI:Ir(Z (K', A)) __ Z2 (K', r1(A))

be defined in the same way as v, by means of a canonical basis,39 (el, * *, for Cn(K). We have to prove that

f2nv'y(y') _ v'g7Y(y') = (fy,)

for any y' e Z'(K', A). Let Se, = a' d,(i = 1, ... , t'),c bc = 0(j = t' + 1, * * ,q'),where (di', d)

is part of a basis for C+1(K'). Then Zn(K', A) is generated by cocycles of the form arc' , where ma = 0 and a' = 0 if i > t'. Therefore it follows from The- orem 5 that r(Zn(K', A)) is generated by the elements y(a-c'), where a'a = 0, together with [y', z'], for every pair y', z' e Zn(K', A). Since

f2v'[y I Zi f_ 2(y, u z') fly U fZ'

'-N v~fny', fnZ] = Vgn[y', z1I

39 Possibly K' = K. In this case the following argument shows that T is independent of the choice of the canonical basis (cl , *-- --)C




it only remains to prove that, if a'a = 0, then

(16.16) f v'y(a-c') -' v'tyf (a-c,)) = vy(a onc'), where 4':Cf(K') -? CT(K) is the (integral) cochain mapping induced by k.

Let &ic' = nic, + * + nqc, for a fixed, but arbitrary value of i. Let ac'a = 0. Then it follows from one of the preceding arguments that

(16.17) (2 2')y (a) = 0. If a' is odd, then

PC2n 2(c u C mod. a' nC5 U f)nC' ni ni

If a' is even then

(16.18) 4)2nl P nC' mod. 2aX

as shown in [3]. In either case

(16.19) 4)~2npC _ p'n-' nqcq) a1 (16.19) (g)B~np~t ~> P~t~nCI = Nnici + *. + nCq) mod. ('2 2a').

Since Sc= 0, mod. a' , and since Scj = aj di, where (di, ..* , dt) is part of a basis for Cn+l(K), it follows that a' njoj.. Therefore njcj is a cocycle, mod. a', for each j = 1, * , q. Let ua be even. Then it follows from (16.17,) (16.19) and (4.7) in [31 that

f2 v-y(a ca) = f2n(^y(a) * pc') = 7(a) 2

'.y(a) Kp(nic, + *. + nqCq)

7(a) (ip(nici) + 2 E nici u nci) i<i

= nZ~4y(a) - c + E [a, a] - ic u njci i<j = iy(nia)-pci + Z [nia,njal-ciu c

i<i

= vy (nia * C, + * + nqa - Cq)

= v(a )c.).

If ua is odd we have the same result on replacing each Pontrjagin square, pc, by c u c. This proves (16.16) and hence the theorem.

Let n be even, let y e Hn(X, A) and let

py = q(X, A)-y(y). We call py the Pontrjagin square of y. It follows from (16.7) that

(16.20) (y + Z) = Py + Pz + y u z.




Thus -y u z appears as a factor set, which measures the error made in supposing

p:Hn(X, A) __ H2n(X, r(A))

to be a homomorphism. We also have p(ry) = r2Py, where r is any integer. Therefore (16.20), with y = z, gives

(16.21) 2py = y u y.

Let (Q, a): (XI A) -* (X', A') be any mapping in the category (X, [). Since 'O is natural we have

pH'% a)y = 7(X, A')'y(Hn(', a)y) = (X, A')FH (M , a)y(y) = H2n(r, a)n(X', A)'y(y) = H 2(, rF),y

or, writingf = H n% a), g = H 2(n, rP),

(16.22) pf = gp:Hn(X',A) -* H2n (XI r(A')).

Let I X I be an arbitrary topological space and let D be the directed set, which consists of all finite coverings of I X I by open40 sets. Let K(d) be the nerve of the covering d. Then (D, K) is a net. Let (D', K') be similarly defined in terms of a space I X' 1. Then a map 4: I X I I X' I induces the homomorphism.

(R*, p): (D, K) -* (D', K')

in which R* d' is the covering {-1U' }, where d' = { U' }, and p(d') is determined by the transformation, F-1U' - U', of the vertices of K(R* d') into those of K'(d'). Therefore q, and likewise p, are topological invariants of I X 1.

Let K be a finite cell complex, which need not be a polyhedron. We take

J2Cr(K) = Hr(Kr, Kr-)

(C3(K) = Hom {Cr(K), Io }

to be the groups of r-dimensional, integral chains and cochains in K, where Io is the group of integers. By Theorem 13 in CH I there is a finite, simplicial complex, L, which is of the same homotopy type as K. Let : K -* L, 4.t' L -, K be cellular maps such that ikq - 1, 0 - 1. Let

or: CT(L) -* CT(K), XpT:( C(K) -3 C (L)

be the cochain equivalences induced by 4), 4t. We define

(16.24) c u c = t2 (4'Cc ) c = nnC

where c, c' are any elements of Cn(K). If c, c' are cocycles mod. o, so are nC, pnC ' and

Jn(C U C') = eno>&n(4knC U pnC') '-' 4U u ' mod. a.

40 We could equally well take closed sets




Similarly i'2n pr -' O'c, mod. 2a, if a is even. Notice that this relation is analogous to (16.18).

Let X, Y be the nets which are defined by means of all the finite, open coverings of the spaces, K, L. Since K, L are compacta it follows from the Cech cohomology theory that O:K -* L induces isomorphisms HI(Y, G) - Hr(X, G), for every r _ 0 and every coefficient group G. Also the cohomology group H?(L, G), which is calculated in terms of cochains in Cr(L), may be identified with H(Y, G). When this is done 7(Y, A) becomes the homomorphism, r(L, A), which is defined by means of (16.13). It follows from the final arguments in the proof of Theorem 18 that H(X, G) and -I(X, G) may be similarly identified with Hr(K, G) and r(K, G), which are defined in terms of CT(K), when cup- products and Pontrjagin squares of cochains are given by (16.24).

If K has a simplicial sub-division we take this to be L and, as in [3], we take -? K L to be the identity and 4.':L -* K to be such that 4pLo C Ko, for every

subcomplex Ko C K, where Lo is the subcomplex of L, which covers Ko .

17. Secondary boundary operators Let K be a finite cell complex, let Cr(K), C(K) be defined by (16.23) and let

cup-products and Pontrjagin squares of integral cochains be defined by (16.24). Let (c1 , - , Cq) be a canonical basis for C'(K), with &ci = aidi, where (di, , dt) is part of a basis for C'+1(K) and aidi = 0 if i > t. Let m > 0 and let

Hn(A) = Hn(K, A), Hn(m) = Hn(K, Im)I

where Im is the group of integers reduced mod.m. Let

y* e Hn(A), z* e Hn(m), a(m) e Am = A/mA

be the cohomology, homology and residue classes of y e Zn(A), z e ZVI(K, I.), a e A. Then a homomorphism4'

UM = u,:Hn(A) -- Hom {H,(m), A}

is defined by

(17.1) (umy*)z* = (cjz)aj(m) + ... + (cqz)aq(m),

where y = a, ci + *** aq-cq, with o-ai = 0. If K is without (n- 1)-dimensional torsion, then

uo:Hn(A) Hom (Hn , A),

where Hn = Hn(O). Let K be simply connected. We make the natural identification 112(K) = H2

and we also identify r(112(K)) with r3 = r3(K) by means of 0, in Theorem 14. Also K has no 1-dimensional torsion and we identify each y* e H2(A) with

"1 Cf. (12.2) in [141.




uoy*. Moreover we take A = H2, so that H2(H2) is the additive group of the ring of endomorphisms of H2. Then we have maps

Hom (H2, H2) iP H4(r3) Hom {H4(m), r3.m}.

Since 7r,(K) = 1 the group Cn(K) (n > 3) may be identified with Cn in the system (C, A)(K). Let b(m):H4(m) -+ r3,m be the secondary modular boundary homomorphism. Let 1 e H2 (H2) be the identity 1:1H2 --+ 2.

THEOREM 20. b(m) = um p(l). First let K be any finite cell complex, which need not be simply connected,

and let the notations be the same as in (17.1). Let a E A and let a c e Zn(A), where c = nic, + + nqcq. Then

um(a c)* z* = { ur (I nia ci)} z* q

= E (ciz) (nia) (m) i-1

q

= E (niciz)a(m) i-1

(17.2) = (cz)a(m). Let : K > K' be a cellular map into a finite cell complex, K', and let

a:Ad A' be a given homomorphism. Let Hn (m) = Hn (K', Im) and let

(17.3) f: Hn(m)-+ H(m) a Am-* A'm

be the homomorphisms induced by 0 and a. Consider the diagram

H (K', A') - H (K, A') a H (K, A)

Um 1 1 Um I Um

{Hn(m), Am} m* {Hn(m) Am} * { Hn(m) Am}

in which IH, A } denotes Hom (H, A) and *, f*, a* , &* are induced by+, f, a, a. Thus

x*(a~c)* = (aa.c)*, a&h = &h, f*h' = h'f,

where (arc) e Zn(K, A), h e IHn(m), Am}I h' e IHn(m), Am'}. Let (cl, *** , c,) be the canonical basis for Cn(K'), in terms of which urn, operating on Hn(K', A), is defined and let f, O* etc. also denote the corresponding maps of chains and cochains. Let

y' =al c1 + + aqCq, e (K' Al).

Then O*y' = al.q*c + *. + a4,4*c', . Let y = a,*c1 + a, aqcq . Then it follows from (17.2) that




{Um(('O*y)*)}Z* = {Um(4*yY)*} Z*

- z{j(O*C')Z}a'(m)

- Ti(c'fz)a,(m)

(Umy *)fz*

-tf* {(Uy*)}Z*

Um(a!*y*) Z* = Zi(ciz) (aaj) (m)

= 2;j(cz)aaj(m)

- { (Umy*)z* }

- {&*(Umy*)}z*.

Therefore

(17.3) u.0* = f*um, Uma* = a*Um.

Now let K, K' be simply connected and, as before, let

112 = H2, r(H2) = r3, H2(H2) = I1H2, H2},

with the analogous identifications in K'. Let 4: K --> K' and a: A - > A' mean the same as before and, returning to the notation used in Chapter I, let

b:H > H2 '= H (0), 1(m):H4(m)-+ H(m)

g:r3 ' = r(H2), g(m):r3,m r3,m be the homomorphisms induced by 4. It follows from (17.3) and (16.22) that the diagram

I H) H'I * H2, H') I H2; H21

U4 4 Um H(K', r3) H ->H(K, r') < * H (K, r,)

{H'(m), rPm} 1,(m)* {H4(m), rPm} g(m)* IH4(m), rs,m} is commutative, where 1*e = be:H2 -> H, *e' = e'1 for every e E IH2, H21

E I H', H'I and the two bottom layers are the same as before, with n = 4, A = r3 and a = . Since b*(1) = = t*(1) it follows that

{ump(l)}b(m) = 1(m)* UnP(1) = UmP)*(1)

= UmOO*(1) = g(m)*umP(1)

= g(m)umP(1).

We also have b(m)b(m) = g(m)b(m), according to (3.6).




Therefore

(17.4) {b (m) - ump(1) }1(m) = g(m) { b(m) -umnp(1)}.

Let K' be any simply connected complex, let K = K'4 and let O:K -* K' be the identical map. Then t(m) is onto and it follows from (17.4) that Theorem 19 is true of K' if it is true of K. Therefore we need only consider complexes of at most four dimensions.

Let K' be any complex of the same homotopy type as K and let O: K -* K' be a homotopy equivalence. Then g(m) is an isomorphism and it follows from (17.4) that Theorem 19 is true of K if it is true of K'. Therefore we may replace K by any complex of the same homotopy type. Therefore we may take K to be a reduced, 4-dimensional complex, as defined in [3].

Let K be a reduced, 4-dimensional complex. Then

K = eoueju .. ueq, K3=K2ueju ...ueue+,u ... uet+l,

where e3 (i = 1, t) is attached to K2 by a map, M - ,of degree oiX and j3+ (X = 1, * , 1) is a 3-sphere attached to eO. Obviously Z2(K) = C2(K). Moreover we may assume that oi I oi+l in which case (cl * , cq) is a canonical basis for C2(K), where ci is represented by a homeomorphism S2 -* 2. Let z E Z2(K) and let (ci, * *, cq) be the basis for C2(K), which is dual to (c', * , cq). Then

Z* = (clz)cl* + .. + (CqZ)c*

Therefore it follows from (17.1), with m = 0, n = 2, HI2(H2) = {H2, H2}, uo = 1 and ai = c'*, that

(cl*c1 + * + Cq *c)*z* = Z*.

Therefore (cl*cl + *. + c*'Cq)* = 1. Since K is reduced we have pr = c u c for any c e C2(K), according to (10.1)

in [3]. Therefore it follows from (16.13) that

p (1) = (Zie pci + E eCiic uc,) i<i

= (Zei"ciuc)* i~i

where ei' = y(c'*), et' = [c'*, ci]. Also it follows from (14.1) in [3] and (17.2) above, with n = 4, A = r3 that

(17 .5) b (m)z* = Z { (Ci u c c)Z}jet(m)

= {UmZ (et3.ci u )** i~i

={UmP(1)}Z* .

Therefore b(m) = ump(1) and the proof is complete. Let K be without 2-dimensional torsion, so that H2 and H2 are free Abelian




groups of the same rank. Subject to this condition, G. Hirsch ([19]) has given a very elegant expression for the kernel, G, of the natural homomorphism

7r3(K) -* H3 .

Let S be the group of symmetric homomorphisms, f :H2 - H2, which is defined at the end of ?5 above, with A = H2, A* = H2. Let z* e H4 and let f,:H2 -* H2 be given by

f -c c* n z* (c* e H2).

Then f2 E S and a homomorphism, i:H4 .-> , is defined by Ipz* = f . Hirsch's theorem states that G S/,iH4 . We give an alternative proof of this.

Let cl, * *, c* be a basis for H 2. Then it follows from (5.13) and (5.12) that X s r3, where

Xf (c ,*fc e'j, i<i

and from (17.5), with m = 0, that

bz* = E c {c*(c* n z*)}e" c Z (c*f~c*)e" jEi iii

- Xfz = ?~z

Therefore b = Xyt and it follows that X induces an isomorphism

s1AH4 r3/bH4. Therefore Hirsch's theorem follows from the exactness of 2 (K).

Let us discard the condition that 7r,(K) = 1 but let K be without (n - 1)- dimensional torsion for some n > 2. We take A = Hn = Hn(O) and use une to identify H"(H,,) with the additive group of the ring of endomorphisms of H,,. It follows from (17.1), with m = 0 and ai = z*, that H'(H,,) operates on Hn according to the rule

ez*= (ciz)z, + ... + (cqz)z;?,

where e = (z'* * c1 + * + z*Cq)*. Let e': H"(HH) -*+ H'(H,,) be the endomorphism, which is induced by a given endomorphism e' Hn -* Hn . Then

(j'e)z* = (e'zl. cl + * + e'ze**c)*z* = (ciz)e'zl + ... + (cqz)etz'

= e'(ez*).

Therefore Z'e = e'e. Let g(e) be the endomorphism of H2In I r(Hn)} which is induced by re: r(Hn) -> r(Hn). Then it follows from (16.22), with f = e, g = g(e), that

pe = pe(1) = g(e)p(1).

Therefore p: Hn(Hn) -* H2" { F (Hn) } is determined by the correspondence e -+ g(e)

together with p(l).




18. The calculation of 2;4(K)

We return to the sequence b i i

44: H4 -+ 3 - 113 ->1 H3 -0 .

We make the same identifications, 112 = H2 and r(H2) = I3 as before. We also identify each z e r3/bH4 with !jT e i3r3 = G, say, where

i: F3/ bH4 f-/G

is the isomorphism induced by i3 . Then i3 becomes the natural homomorphism r3 -* G.

The group H3 is an extension of H3 by G. Let H' be an equivalent extension. Then there is a homomorphism, 3: H3 -+ H3, and an isomorphism, 3: 13 3, such that

fg = g, 3, (g e G),

whence j3-(0) = ji-'(0) = G. Let 24 be the-sequence which is obtained from 24 on replacing j3: H3 -+ H3 by i3: H3 -* H3 . Then F: 24 f 4 is a proper isomorphism, where F consists of f : I13 H and the identical automorphisms of the other groups. Therefore 24 is determined, up to a proper isomorphism, by the groups H2, H3, H4, the homomorphism b:H4 ,-* (H2) and the cohomology class in H2(H3, G) which determines the equivalence class of the extension II3.

Now let 7r1(K) = 1 and let K be a finite complex. Then H3 = T + B, where T is the torsion group and B is free Abelian. Let T1, * - - , T, be cyclic summands of T, whose orders, rT, ..-, rp, are the coefficients of 3-dimensional torsion. Since B is free there is a homomorphism, i*: B -+ 113 , such that j3j*b = b for each b e B. Therefore 113 is the direct sum

3 = II13 + i*B,

where HIO = '-1T. Thus Ho is an extension of T by G and the equivalence class of 113 is obviously determined by that of 11.

Let (cl, * , cp) y', ., y)) be a canonical basis for C4 such that dc = mzk, dyl = 0 where 13z' e H3 generates T7, . let ac e . z, = k3ade II3. Then

i3X = jik3a" = 13j3a= = z.

Therefore xz e fl0 and xz is a representative of 1lZ. Since j3(c- Aa- ) = d3" - nz = 0 we have

3c -ma = e r(H2)

Let 9 = -i3' e G. Then hX~ = ksra = - = gX. Therefore the equivalence class of HIo is uniquely determined42 by g1(ri), *., gp(r,), where gX(n) e G,, is the residue class containing 9g.

42 See ?16 of [12].




It follows from the definition of b(m) that

b(m)c>0c = X\(x) e F (H2),TX

where cx is the homology class, mod. ax, which contains ca. Therefore

9X(n) = -i3(rX)b(rX)cX* ,

where i3(a): F(H2)T, -> GTA is the homomorphism induced by i3 . Therefore the equivalence class of Ho is determined by b, which determines G, and by

b(,r), - -,rp),

which determine gl(ri), * * * , p) Since b(m) = ump(l) the sequence :4(K) is determined, up to a proper iso-

morphism, by the groups

H2, H3, 1H4(m), H4{ F(H2) } (m = 0, Ti, ,rp),

the element p(1) e H4{4r(H2)} and the family of homomorphisms urn. If K is a (finite) simplicial complex all these items can, theoretically, be calculated by finite constructions.43

Let n > 2, let K = Kn+2 be (n - 1)-connected and let us make the natural identifications

r.+ = li/2 = Hn(2)

so that 0 = 1 at the end of ?14 above. Then Theorem 19 has an analogue, namely Theorem 4 in [5], which states that bn+2(2) is the dual of the Steenrod homomorphism ([24])

Sqn_2: H (2) - n+2 (2).

Further light is thrown on the calculation of ln+2(K) in forthcoming papers by S. C. Chang and P. J. Hilton. Chang defines certain numerical invariants, called "secondary torsions", which can be calculated constructively if K is given as a simplicial complex. An analysis which is similar to, but rather simpler than the one above, shows that ;n+2(K) is determined, up to a "proper" isomorphism, by the Betti numbers and torsions of K, together with the secondary torsions defined by Chang.

CHAPTER V. THE SEQUENCE OF A GENERAL SPACE

19. The Complex K(X)

Let P be any (geometric) simplicial complex, which may be infinite but which has the weak topology. That is to say, every (closed) simplex in P has its natural topology and a sub-set of P is closed provided it meets each simplex in a closed sub-set of the latter. By a local ordering of the vertices of P we shall mean an ordering, o(n), of the vertices of each simplex, on e P, such that, if ano1 is a face of an , then o(an-) is the ordering induced by o(an). Let such an ordering

43 This does not mean that we have found a finite algorithm for deciding whether or no 2;4(K) == 24(K'). Some of the difficulties in this question, even when K, K' have no torsion, are indicated on p. 88 of [3].




be given. Let the simplexes of P be divided into equivalences classes by an equivalence relation, _, such that

a) a' a' implies r = s, (19.1) b) if Tn where a = .t .-- . = wo wn and the vertices

vi, wi are written in their correct order, then vio *** Vir Wio ... Wir

for each sub-set 0 < io < i2 < *.. ir _ n. Let h (n, n): an __+ r be the order preserving barycentric map (onto) for every pair of simplexes en, rn e P. Let pi, p2 be points in P. We write P2 =P2 if, and only if, there are equivalent simplexes, a' , a' e P, such that pi e a- and P2 = h(f, X')pl . Obviously pi =P2 is an equivalence relation. Let K be the space whose points are these equivalence classes of points in P and which has the identification topology determined by the map k:P K, where kp is the class containing p.

LEMMA 3. K is a CW-complex, whose cells are the sets k(o- n for each simplex, an, of p.

First assume that the following supplementary conditions are satisfied: a) v $ v' if v, v' are distinct vertices of the same simplex of P

(19.2) b) if a V .. * n d if vi-wji for each i = O , n, then O =n.

Let a be the cardinal number of the aggregate of classes kv, for each vertex, v e P. Let kv -+ e(kv) be a (1 - 1) correspondence between the aggregate {kv} and a set of basis vectors, {e}, in a non-topologized vector space, A, of rank a (Cf. [25]). Then a simplicial complex, L, with { e } as the aggregate of its vertices, is defined as follows. Let eo , --, en be any finite sub-set of { e }. Then the rectilinear simplex, eo ... en C A, is a simplex of L if, and only if, there is a simplex v0 v. *n e P, such that es = e(kvi) (i = 0, * , n). We give L the weak topology.

Let an = v0 * n v*n be a given simplex of P. Then it follows from the definition of L that e(kvo) ... e(kvn) is a simplex in L. Therefore a simplicial map, 1:P -+L, is defined by lv = e(kv), for each vertex v e P. Since P has the weak topology it follows that 1 is continuous. Notice that, in consequence of (19.2a), the map I1 On is non-degenerate. Let p q, where p, q are points in P, and let

an = Vo ... Vn , 'n = Wo .Wn

be the simplexes of P, whose interiors contain p, q. Then an _ and it follows from (19.1b) that kvi = kwi, for each i = 0, * * *, n. Therefore lo = 1r n. Also q = h(7rn, crn)p and the map h rn, _n), likewise Ii an and I I enare barycentric. Therefore ip = lq. Therefore the map 1k-1:K -* L is single-valued. Since K has the identification topology determined by k the map 1k-1 is continuous."

Similarly it follows from (19.2) that kf-l:L -* K is single-valued. It is obviously continuous in each simplex of L, and hence throughout L, since L has

44 See ?5 of [6].



100 Y. H. C. WHITEHEAD

the weak topology clearly (Uk-')k17' = 1, (kl-')Ik-1 = 1. Therefore kl-' is a homeomorphism onto K. Therefore K is a simplicial complex, with the weak topology, whose cells are the interiors of the simplexes kJ7'(1oa) = ka', for each simplex, oa e P. This proves the lemma, subject to the conditions (19.2).

Now assume that (19.2a) is satisfied and let P' be the derived complex of P, in which each new vertex is placed at the centroid of its simplex. We define a local ordering in P' by placing the centroid of an' after the centroid of am if m < n. The equivalence relation between the simplexes of P induces a similar relation between those of P', in such a way that the equivalence classes of points are unaltered. Also it may be verified that the equivalence relation in P' satisfies both (19.2a) and (19.2b). Therefore kP' is a simplicial complex and K = kP is a "block complex", in which the blocks are the sub-complexes of kP', which cover the sets kn. It may be verified that K is a CW-complex, with the combinatorial structure described in the lemma.

Finally let P be general. Then the induced equivalence relation between the simplexes of the derived complex, P', obviously satisfies (19.2a). Therefore the induced equivalence relation between the simplexes of the second derived complex of P satisfies both (19.2a) and (19.2b). Again it is easily verified that K is a CW-complex, with the structure described in the lemma. This completes the proof.

We now proceed to the definition of K(X). Let v,0 be the origin and vi the point (th, t2, *.. ) in Hilbert space, RX, where ti = 1 and tj = O if j # i. Let vo, v1, - - - be ordered so that vm < vX+i and let An C RX be the rectilinear simplex vo .*. *n -

Let f: A' -* X be a given map and let (f, An) be the rectilinear n-simplex, whose points are the pairs (f, r), for every point r e An, and whose topology and affine geometry are such that the map r - (f, r) is a barycentric homeomorphism. If oa is any face of A' we shall denote the corresponding face of (f, An) by (f, 0i). We emphasize the fact that (f, r) $ (f', r) if f, f': An -- X are different maps, even if fr =f'r. Also (J, r) $ (g, r) if r e An-1 andg = f I An-l. Therefore no two of the simplexes (Jf, An) (g, Am) have a point in common.

Let P(X) be the union of all the (disjoint) simplicial complexes (If, An), for every n _ 0 and every map f: An -> X. We give P(X) the weak topology, which makes each (f, An ) both open and closed in P(X). The simplexes of P(X) are the simplexes (f, at), where at is any face of An. The ordering vo, v1, , - n, for each n > 0, and the maps r (f, r) determine a local ordering in P(X). Let (f, at) and (g, r') be faces of (f Am) and (g, /f). We define (J, o g) (g, r') if, and only if, i = j and45

f I = (g I r')h(r', a ) It is easily verified that this is an equivalence relation and it obviously satisfies (19.1). Therefore a CW-complex, K(X) = kP(X), is defined as in Lemma 3. Notice that K(X) is uniquely determined by X. Notice also that fr = f'r' if k(f, r) = k(f', r'), though the converse is not necessarily true.

45 h(ri, as) will always mean the order preserving, barycentric map, as -+ Ti, onto Ti.




Let S(X) be the abstract singular complex of X, as defined in [16]. Any n-cell, sn e S(X), has a unique representative map,46 f(sn): An -* X. It is obvious that the correspondence sn -* k {f(.fl), An } determines an isomorphic chain mapping of S(X) onto K(X), when the latter is treated as an abstract complex.

Let 4:X -* Y be a given map into a space Y. Then a map, KO:K(X) -*K(Y), is obviously defined by

(Ko)k(f, r) = k(cf, r), where k:P(Y) -* K(Y) is defined in the same way as k:P(X) -- K(X). More- over the correspondences X -* K(X) and 4 -* K4y determine a functorZ ? - , where ? and tk are the topological categories of all topological spaces and all CW-complexes.47

20. The maps K and Ak

It is obvious that a (single-valued and continuous) map, K: K(X) -* X, is defined by Ak(f, r) = fr. Let +:X -* Y be a given map. Then

okk(f, r) = ofr = K{(Kc)k(f, r)},

where K:K(Y) -* Y is defined in the same way as K:K(X) -* X. Therefore K is natural in the sense that

(20.1) 4K = KKO.

Let Q be any simplicial complex with the weak topology and with a local ordering of its vertices. Let 4: Q -- X be a given map. Then a map, X+: Q -K(X) is defined by

(20.2) Xoq = k{ (q5 I o-)h(o-, An), h(An, otn)q}

for each point q e Q, where an is any simplex of Q, which contains q. Obviously KeX = 0. Moreover, if Qo is a subcomplex of Q, with the local ordering induced by the one in Q, and if 4)o = I | Qo, then

(20.3) XA0 = X0 I Qo Let 0: Q -* L be an isomorphism of Q onto a simplicial complex, L, with the weak topology. Let L have the local ordering which makes 0 order preserving in each simplex of Q. Let A~: L -* X be a given map. Then it may be verified that

(20.4) X fe = "0: Q -* K(X).

46 Strictly speaking the unique representative of s'n is the pair (f s8n) On), where on is our fixed ordering of the vertices v., * *, v". Eilenberg has communicated to me a simplified definition of S(X), to be used in a forthcoming book with N. E. Steenrod, in which a cell of S(X) is simply a map f:An -_ X and its faces are the cells (f I ai)h(at, Ai), for each face, at, of An.

47 Notice that K0 maps each cell of K(X) homeomorphically onto a cell of K(Y). There- fore SE --+ T maps SE into the category of complexes in which the mappings are of this restricted sort.




Let Q* be any simplicial subdivision of Q and let a local ordering be defined in Q*, which is independent of the one in Q. Let O*: Q* -> X be a given map and let X* be defined in the same way as X,,, but in terms of Q* and the local ordering in Q*.

LEMMA4. If q 4 P* then AX c X *. Let L = Q X I, let Oo, 61 be the maps of Q into Q X 0, Q X 1, which are

given by Ooq = (q, 0), rq =(q, 1) and let Qo , Q* be the triangulations of Q X 0, Q X 1 which make Oo, 01 isomorphisms. Then L is a polyhedral complex, which consists of the simplexes in Qo, Q,* and the convex prisims an X I, for every simplex, an' e Q. Let v(o-') = (q, ), where q is the centroid of a-, and let L' be the triangulation of L, which is defined inductively by starring each a' X I from v(^') as centre, taking 'm X I before a' X I if m < n. We define a local ordering in L' by giving Qo , Qr the local orderings which make 0 , 0* order preserving and placing v(^') after all the vertices of L', which lie on the boundary of a' X I.

Let #:L' -> X be a map such that #(q, 0) = 0q, 41(q, 1) = O*q. Let XO:L' -+ K(X) be defined in the same way as X,,. Then it follows from (20.3) that XA determines a homotopy Xp,,Oo c- X*,10 where To = P Qo, 41, = 41 1 . Since 4 = y#000 , = #61 el the lemma follows from (20.4).

Let A, ,u':Q K(X) be given maps. THEOREM 21. If KC -- KtI', then ,i Al. We first show that, in the presence of Lemma 4, this is equivalent to

(20.5) SA -

For K/ = KXKP, . Therefore (20.5) follows from Theorem 21 with Ai' = A,,, . Con- versely, if KA /I KA's it follows from (20.5) and Lemma 4 that A C A,;, X a X,, -u' Therefore Theorem 21 is equivalent to (20.5). We shall prove (20.5).

Let K = K(X). Though K is not a simplicial complex we shall describe A: Q K as simplicial if, and only if, it can be defined as follows. Let a barycentric map

(-+ P(X)

onto a simplex of P(X), be defined for each simplex a' e Q in such a way that the (simplicial) map A:Q -* K is single-valued, where puq = kO(on)q if q e a'. Since ,I a' is continuous and Q has the weak topology it follows that Ai is continuous.

Let the simplex O(a')a' be d(a')-dimensional, let j = d(^') and let

/(a') = h(A', A(i )&&)Ot(c ):- A',

where 0'(or) > -O(or)a' is the map induced by O(^'). Let

O(W)r (g(l), ai) C P(X),

where a u C Ak, for some k > j, and let

fAa') = (g(0) I CF)h(a' X A') A' > X.

Let 'm be any face of a' and let I(a')a = c1 C A'. Since A: Q > K is single-




valued it may be verified that i = d(a') and that

a) f/(a') m = th(al ,$)A(a" (20.6) b) f() I= f(cm)h(At '),

where L: cl -* A' is the identical map. Also (f( ), A) (gq(a'), a0), whence

(20.7) A= k(f(n), 4(ar)q) (q e an)*

Conversely, given .(an): an Ai, f(n) :)A' -> X, satisfying (20.6), for each an e Q, a simplicial map A: Q K is defined48 by (20.7).

We shall describe the simplicial map A as non-degenerate if, and only if, d(^n) = n for each Srn e Q. Let this be so and let A (an), f(an) mean the same as in (20.6). Then we can order the vertices of each simplex arn e Q so that .L(cn) preserves order. It follows from (20.6a) that we thus define a local ordering in Q and from Lemma 4, with Q* = Q. 0* = 4 = KU, that we lose no generality in assuming this to be the one by means of which X,,, is defined. Then

/I(an) = h(An, crn), KA/I |an = f(an)a(n),

in consequence of (20.7). Therefore it follows from (20.2) and (20.7) that Au =

X. . Therefore the theorem will follow when we have proved that a given map, Q -+ K, is homotopic to a non-degenerate, simplicial map Q* -* K, where Q* is a simplicial sub-division of Q.

Let P" be the second derived complex of P(X). Then K" = kP" is a simplicial sub-division of K, as shown in the proof of Lemma 3. Let bt:P" -> P(X) be the canonical homotopy in which atv = (1 - t)v + tasv, where v is any vertex of P", atv is treated as a vector and Alv is the last vertex of the simplex of P(X), which contains v in its interior. Obviously tpt a3tp' if p = p'. Therefore44 a homotopy, pt:K" -> K, is defined by ptk = kat. Clearly pi: K" -? K is simplicial.

Let uo: Q > K be a given map. By Theorem 36 on p. 320 of [7] we have /o I il, where .l: Q K is simplicial with respect to K" and some simplicial sub-division, Q*, of Q. Then o c- pigi . The resultant of simplicial maps, Q -* L -* K, is obviously simplicial, where L is any simplicial complex with the weak topology. Therefore piul is simplicial and it follows that we lose no generality by assuming that the given map u Q -S K is simplicial.

Let A be simplicial and let A (an), f(an) mean the same as in (20.6). Let

A*(On) = h(An oan), f*(ofn) = f(on)/(on)h(on, An).

Let al = /*(c)cr where am is any face of an. Then

A*(an)Iain = h(An, n )ai = th(an, ct)

= th(, m)/*(m)

where t: a-* is the identical map. Let al = i(c)cm C A'.

" When ;& is thus defined it is to be understood that (u(an)an = Ai(j = d(on)) and that i = d(-) in (20.6).




Since h(a', )a`) = a' we have

f*(au) I '= f(a')A(ao)h(a, A') I am

= f~n~~a)I amha awt)

= {f (a ) i al } ,'h(u, al ),

where .L': a'n -> al is the map induced by .L(a'). It follows from (20.6a) that

= h(or , A i)(rn)

Hence, and from (20.6b), we have

f*(u') i = a f (am)h((A' u)h(a1 , Ai)g))h(Un a)

= f (a )yi(u)h(um, al )

= f(anm)A (am) h (am, /") h (A, i/)

= f*(an)h( Am al p).

Therefore the families of maps ,.*(a,), f*(aun) satisfy (20.6) and a non-degenerate, simplicial map, ,u*:Q -* K, is defined by

A*q= k(f*(0lt) n,*(an )q) (q e an).

Finally we prove that A Ai"*. Let uo, * , un be the vertices of an, written in their correct order. Let j = d(a') and let

Vo(a ), Vl(a,) an Aijrn+i

be the barycentric maps which are given by

vO(a )ua = I.L(a )Ua , Vl(cn)Ua = Vj+l+a (a = 0, , n).

Let Vg(on)q = (1 - t)Yo(a n)q + tvi(au)q, for each q E an, where 0 ? t ? 1 and vt(o )q is treated as a vector in A+'+'l Let p(&n) AQ+n+1 __ A' be the barycentric retraction which is given by

p(an) |A= 1, jl+a = A(an)Ua (a = 0, ... , n)

and let F(an) =f (an)p(an) Aj+n+l -+ X. Let

,(On): ,n__ P(X)

be the homotopy which is given by

(20.8) Og(an)q = (F(&n), Vg(an)q) (q e a"n).

I say that a homotopy, ,ti:Q - K, of no = A into A1 = A* is given by tq=

kOt (an)q. This will follow when we have proved that

(20.9) kOi(an) = 'i, I an (i 0, 1)

and, since Q has the weak topology and kOt(a") is continuous throughout a", for each an e Q, that At is single-valued. The fact that At is single-valued will follow




when we have proved that

(20.10) kOt(am) = kOt(a') a"

where am is any face of &n. Since (o(af)q = 4&')q (q e &') and F(a') IA = f(&') it follows from (20.7) and

(20.8) that kOo(c) = cry'. Let Al = *. . Then ,u(an)h(an, An) and h(An , &n) are the maps induced by p(&n) and v1(n). Therefore

F(crn) I 'An = f(cn)()4n)h(cn, An)

= f(n)g(n()han, An)h(An, An)

= f*(f )h(An, An')

V(an)q = h(An n

,)q (q e O'n)

= h(An , An)h(An, &n)q

= h(An, An),*(,n)q.

Hence it follows that k0i (&&n) = an cr and we have proved (20.9). Let

0-m = Uro .. Urn, 2 = V8 0 .

.Va. (r. < r,+; s < s+1)

V*0 VsVj+1+rO ...

Vj+r0 = i+m+l

Then P( n)0l+m+l = = a1 Let p': al+m+ a' be the map induced by p(an). Then it follows from (20.6a) that

p(am) h(/A+m+l, al+m+1)Vj+l+r.

= p(am)Vi+1+a = t(a )U'

= h(At, rai)t(an)Ua = h(At, ao)p'Vj+1+r.

Therefore h(A', a)p = p(am)h(At+m+l

n +m+)

and it follows from (20.6b) that

F(an) I ai+m~1 = {f( ,,)(lI) at+m+l

= {f(n) I ai I pi

= f(am)h(A' (A )p

= f(ca)p(ao)h(Ai+m+1 al+m+)

= F(am)h (At+m+l, r'+m+1)

Therefore (F(WWI, a +m+1) (F(rm), At+m+l) Also

h(l, At)v = v.0 = h(a1+m+l, At+m+l)vp ( i).




Therefore

vo(cr')urt = Mt(ac)u7. = h(al, At),u(am)Ur.

= h(as++l Ai+m+l)V(am)U

Vi(aY)ur = = h(a1+m+l, As+m+l)vi( m)Ur,

Therefore v,(o-)q = h(Ui+m+l, 'Ai+m+l)vt(am)q if q e a'. This proves (20.10) and hence the theorem.

Let 4 - 4':X -> Y, where Y is any space. Then it follows from (20.1) that

KKO) = OK Cq OK = KK)'.

Therefore K4O - K+', by Theorem 21. Therefore {K+} is a single-valued function of {lo}, where {A4} denotes the homotopy class of a map 4'. It may be verified that the correspondences K -> K(X), {l)} --{ K+} determine a functor of the homotopy category of all spaces into the homotopy category of CW-complexes.

21. The sequence X(X).

There is a unique map, f: ,A -> X, such that fA0 is a given point in X. There- fore K K0(X) is a (1-1) map onto X, whence KK(X) = X. Since K(X) is locally contractible, according to (M) in ?5 of CH I, each of its components is arcwise connected. Therefore each component of K(X) is mapped by K into a single arc-component of X. Let Ko(X), K1(X) be given components of K(X) and let ei e K? (X), xi = Keoi (i = 0, 1). If xo, xl, are in the same arc component of X there is a map, f: :A ,- X, such that fvi = xi. Then k(f, A') is a 1-cell in K(X),

0 whose extremities are e0o, e~. Therefore Kg(X) = K1(X) and K maps precisely one component of K(X) onto a given arc-component of X.

Let X be arcwise connected and let a 0-cell eo e K0(X) and the point xo = Ke0 be chosen as base points in K(X) and X.

THEOREM 22. Kn :rI{K(X)} ( irn(X) for every n = 1, 2, , where Kn is induced by K.

Let 4: (An+1, AO) (X, xo) be a map which represents a given element a e 7rn(X). Since OA0 = xo we have eo = k(O I AO, AO) and X<0A0 = eo. Therefore ,X &n+l -> K (X) represents an element ao e C-7 n{ K (X) }. Since 4 = K'X, we have a = Knao . Therefore Kn is onto.

Let Mt:An+l -> K(X) be a map which represents a given element ao e K-'(O) and let ,u' be the constant map An+_ e0. Since Knao = 0 we have K/I - KU/I.

Therefore z u- ,' by Theorem 21. Therefore ao = 0 and Theorem 22 is proved. It follows from (20.1) that the isomorphisms Kn are natural with respect to

the homomorphisms induced by maps 4: X -> Y and Ke, where Y is any arcwise connected space.

We recall from CH I that X is dominated by a CW-complex, L, if, and only if, there are maps +:X -> L, 4':L -> X, such that 4A4) - 1.

THEOREM 23. If X is dominated by a CW-complex then K:K(X) _X. This follows from Theorem 22 and Theorem 1 in CH I.




Let X be itself a CW-complex. Then it follows from Theorem 23 that K induces a proper isomorphism

T, g(X) I{K (X)} (q oo)

which is natural in consequence of (20.1). If X is an arbitrary, arcwise connected space we choose base points e0 e K0(X), Ke' e X and define 2q(X) as 2q{K(X)}.

The complex K(X) can also be used to extend the domain of definition of other invariants from CW-complexes to arbitrary spaces. For example the n-type of X may be defined as the n-type of K(X). The same applies to the injected groups discussed at the end of ?11.

APPENDIX A. ON SPACES DOMINATED BY COMPLEXES

We have seen, in Theorem 23, that any arcwise connected space, X, which is dominated by a CW-complex, is of the same homotopy type as some CW- complex49 K. Let X: X K and let K:K - X be a homotopy inverse of X. Let XX C Ko where Ko is a sub-complex of K, let X0: X -> Ko be the map induced by X and let Ko = K I Ko . Then KoXo = KX - 1. Therefore X is dominated by Ko . If X is compact, so is XX. Therefore XX C Ko, where Ko is a finite sub-complex of K, by (D) in ?5 of CH I. Therefore X is dominated by a finite CW-complex.

THEOREM 24. An arcwise connected space, X, which is dominated by a CW- complex with a countable aggregate of cells, is of the same homotopy type as some locally finite polyhedron.

By Theorem 23, K:K 3 X, where K = K(X). Let X:X -- K be a homotopy inverse of K. Let +:X -> L, i1:L -> X be such that i1i - 1, where L is a countable CW-complex. Then

X - X4)/? = X - K, where ,u = X4':L -> K. Let en be any cell of L. Since ,uen is compact it is contained in a finite sub-complex of K. Since L is countable it follows that ML and hence XOX, is contained in a countable sub-complex, Ko C K. Since X46 C X we have ;4:X a K and we may replace X by A. Thus we assume to begin with that XX C Ko .

Let pt K -> K be a homotopy of po = XK into P1 = 1. Then there is a countable sub-complex, K1 C K, such that ptKo C K1, for the same reason that ML C Ko. By repeating this argument we define a sequence of countable sub- complexes Ko, K1 ,X * * , such that ptKn C Kn+?. The union of the complexes Kn is a countable sub-complex, K*, such that XX C K* and ptK* C K*. There- fore X*K* - 1 (in K*) and K*X* = KX - 1, where X*: X K* is the map induced by X and K* = K I K*. Therefore X*:X K*. But K* P, where P is a locally finite polyhedron, by Theorem 13 in CH I. This proves Theorem 24.

It follows from Theorem 24, and the remarks which precede it, that any compact space which is dominated by a CW-complex, and in particular any ANR compactum, is of the same homotopy type as some locally finite poly-

49 This may be proved more directly by a construction of the sort used in [8].




hedron. We leave open the question whether or no it is of the same homotopy type as a polyhedron of finite dimensionality.

APPENDIX B. ON SEPARATION COCHAINS

Let X, X' and Y C X, Y' C X' be given topological spaces and let An= 7rn(X yo), Cn =r n(X, Y, Yo) (n 2 2)

= irn(XI, y'), C' = Tn(X', Y', YO),

where yo, yo are base points in Y, Y'. Let j:An -+C.,X j' A'n_ CI

be the injections and let fx f ? An An XA h, ho: Cn C'

be the homomorphisms induced by given maps 10 00o M Y) YO) --.(XI) Y', YO)-

Let 'k i Y = j Y: Then 0, 4? determine a separation homomorphism, A = (+ ? ) Cn An X

which is defined in the same way as Eilenberg's separation co-chain, except that attention must be paid to the base points. The purpose of this section is to prove that

(Bi) a) fh-h0=j'A b) lf-0 Aj.

We recall the definition of A. Let E , En be "Northern" and "Southern" hemispheres on an n-sphere, Sn, and let Sn-1 be the "equatorial" (n - 1)-sphere. Let In C Rn be the n-cube, which is given by 0 _ t, ... *, tn _ 1, where t1, -**, tn are Cartesian coordinates for Rn. Let Oi: Es - In be fixed homomorphisms (onto), such that 01 I Sn-1 = 02 | S'-1. Let En be oriented by means of the map 0 1 (i = 1, 2) and let S' take its orientation from E n . Thus, taking orientation into account, (B2) = Sn Sn S = E -E

We shall use maps of In and of the (oriented) n-elements Eln , Eln to represent elements of homotopy groups, both absolute and relative. We shall also use maps of Sn to represent elements of absolute homotopy groups. Let

2= (2 0 . *0) ( P. It will be convenient to take po and 0i 1po as base points in In and sn.

Let X: (In, in, po) -> (X, Y, yo) be a map representing a given element c e Cn Then Ac e A" is the element represented by X (, ,0) :Sn > X', where

X(4) OP)q = 0A01q if q En1

^0X02q if q E2n.




Obviously Ac is unaltered by a homotopy of the form

xt: (InY In, Po) >- (X, Y. Wo.

Therefore it does not depend on the choice of the representative map X. Any pair of elements c, c' e Cn may be represented by maps, X, :' P- Y.

such that '0t) ...*, tn) = Yo if t1 < 2

X ... ***, tn) = Yo if t1 > I,

and c + c' by X*: In Y, where X*(t, ** tn) 'X (t, ** tn) if ti >-

= X(t1, ,tn) if2t ? 2

Then X(+(, 40)E3 = Y, X'(4, )0)E4' = y0, where E3n , En C ,S are "Western" and "Eastern" hemispheres, and

X*(4) ,4)) = 'X (4) 00) in En

= V(t?> ?>) in E8n

Hence it follows that A(c + c') = Ac + Ac'. Therefore A is a homomorphism. Let bo e 7rn (Sn) and bi e 7rn(S', Sn-1) be the elements which are represented

by the identical maps Sn -* S' and En S n (i = 1, 2). Then it follows from

(B2) that bi - = j*bo,

where j*: 7rn(S') __ 7.(S n, Se'-) is the injection. On carrying this relation into X' by means of the homomorphism

Xr (sn) Sn-1) 7X (Xl Y%)

which is induced by X(4), O), we have (Bla). Let X: (In" P) -> (X, yo) be a map which represents a given element a e An.

Then X(0, 4O)Sn-l = yo and fa, f0a are represented by the maps

X(4X ,) IE0 X , (,)40) I En

Since Sn = En - E the map X(4, 40) represents fa -f0a. But X also represents ja and X(0, OP) represents Aja. This proves (Blb).

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MAGDALEN COLLEGE, OXFORD.



a certain exact sequence

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