simple homotopy types

Simple Homotopy TypesAuthor(s): J. H. C. WhiteheadSource: American Journal of Mathematics, Vol. 72, No. 1 (Jan., 1950), pp. 1-57Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2372133 .

Accessed: 09/12/2014 00:05

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

This content downloaded from 192.231.202.205 on Tue, 9 Dec 2014 00:05:19 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=jhup

http://www.jstor.org/stable/2372133?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


SIMPLE HOMOTOPY TYPES.*

By J. HI. C. WHITEHEAD.

1. Introduction. This is a sequel to two papers1 entitled "Combi- natorial Homotopy," Parts (I) and (II). It deals with what I have previously called the "nucleus," but which will now be called the simple homotopy type of a complex. It is closely related to parts of [1] and [3] but the treatment is so different that we shall start again from the beginning.

Let {K} be the class of all (cell) complexes,2 as defined in CH (I), which are of the same homotopy type as a given complex K. Let K' = K (i. e. K' ? {K}) and let +: K = K' be the class of maps which are homotopic to a given homotopy equivalence, c: K K. If 4': K' K", we define p'p by

= p'p: K = K".

It is easily verified that the classes p, with this multiplication, form a groupoid,3 G, whose unit elements are the classes I: K' - K', for every K' e {K}, where 1 :K'-- K' is the identical map. Our plan is to analyse this groupoid in algebraic terms.

First consider the group, GK C G, which consists of the classes +: K - K. We define an additive Abelian group, T, which depends only on 7r1(K). The group T admits GK as a group of operators and we shall define a crossed homomorphism -r: GKE-> T. We call T (g) the torsion of a given element g e GK. If p:K K=K', where K'#ZK, we define a class of elements T(4)) C T, which we call the torsion of p. We describe p as a simple (homotopy) equivalence if, and only if, T(4) = 0. We say that K and K' are of the same simple homotopy type, and shall write K = K' (E,), if, and only if, there is a simple equivalence O:KK K'. It will follow from the

1

* Received January 18, 1949. BuUlletin of the American Mathematical Society, vol. 55 (1949), pp. 213-45 and

453-96. These papers will be referred to as CH (I) and CH (II). 2 Until the final section we assume that any given complex is finite and connected.

We also assume that the points in our complexes are taken from some aggregate, O, which is given in advance. The power of a shall exceed that of the continuum, so that it is not exhausted by any one (finite) complex, and the points in Hilbert space shall be included in a.

3 See [6], p. 132.



2 J. H. C. WHITEHEAD.

definition of (p) that K -K' (>) is an equivalence relation. We then prove that K - K' (4) if, and only if, K can be transformed into K' by a "formal deformation," which is defined in much the same way as in [1]. Thus the elementary transformations, or "moves," do not appear in the definition of simple equivalence but in a theorem which is analogous to Tietze's theorem 4 on discrete groups. Similarly it is proved that two complexes are of the same n-type if, and only if, they can be interchanged by elementary transformations of the sort used in [1] to define the " n-group."

It was proved in [3] that the Reidemeister-Franz torsion,5 when defined, is an invariant of the simple homotopy type. Using this fact, examples were given of complexes, which are of the same homotopy type but not of the same simple homotopy type. However, if T 0, then K = K' (X) if K -= K. It will be obvious that this is so if iri(K) = 1. It follows from Theorems 14, 15 in [11] that T = 0 if 7r,(K) is of order 2, 3, 4 or cyclic infinite.

It is an open question whether or not the simple homotopy type is a topological invariant. However we shall prove that it is a combinatorial invariant in the following sense. If K' is a sub-division of K, then the identical map K -> K' is a simple equivalence.6 Any differentiable manifold has a "preferred" class of triangulations,7 any two of which are combi- natorially equivalent in the sense of Newman. Also any analytic variety has a preferred class of triangulations,7 any two of which have a common sub- division. Therefore the simple homotopy type has an invariant status in differential and alegbraic geometry and in the study of analytic varieties.

2. The group'T. Let R be a ring with a unit element 1. Eventually R will be the group ring8 of vri(K) but here we only assume that, if A is a free R-module of (finite) rank n, then any free R-module, which is isomorphic9 to A, also has rank n. This condition is equivalent to the

4See [7], p. 46. 5 See [8], [9] and p. 1209 of [3]. In Section 12 below it is shown, in the case of

Lens space, how this is related to our torsion. See also [10]. 6 This may turn out to be a wider definition, even for simplicial complexes, than the

one based on Newman's " moves," or on recti-linear sub-divisions (see [12], [13] ). For example, we do not enquire whether or not the vertex scheme of a given " curvilinear" triangulation of an n-simplex is a formal n-element, as defined by Newman.

7See [2] and [14]. 8 By the group ring of a group, r, we shall always mean the integral group ring,

in which the additive group is the ordinary free Abelian group, which is freely generated by the elements of r.

9 A module will always mean a free R-module and, unless the contrary is stated, a homomorphism will always mean an operator homomorphism.



SIMPLE HOMOTOPY TYPES. 3

conditioii that every regular R-matrix (i. e. one with elements in R and a 2-sided inverse) is square. Hence it is satisfied if there is a homomorphism, other than R -> 0, of R into a division ring, D. For such a homomorphism carries a regular R-matrix into a regular D-matrix, which is necessarily square. If R is the group ring of a group, Ir, then r -> 1 defines a homomorphism of R into the rational field. Therefore the condition of rank invariance is satisfied.

Let M be the module, of infinite rank, whose elements are the infinite sequences (r1, r2, * * ) (ri ? R), in which all but a finite number of r1, r2, * are zero. The elements in R will operate on M from the left.10 Thus an operator, reR, transforms m into rm, where

m = (r1, r2, . * ), rm = (rri, rr2, Let mn P M be the basis element which is given by r= 1, rj = 0 if j #& i. Let Mn C M be the module generated51[ by (in1, M , mn) and Mn the one generated by (Mfn+1 mn+2< * ), where n ? 0 and M- 0. Then M is the direct sum M = Mn + Mn and a given element in M is in Mn for some value of n. We shall describe an endomorphism, f: M -> M, as admissible if, and only if, fmi = mi for all sufficiently large values of i. If f, g: M llM are admissible endomorphisms 12 so, obviously, is fg: M -> M and if f: M MJI is an admissible automorphism so is f-1. Therefore the admissible auto- morphims form a group, a.

Let f: M -> M be an (admissible) endomorphism and let fmj = mi if j > p. Let ni be such that fmi ? Mn? (i-, * *, p) and let n Max(ni, p). Then fMin Mn for i=1, * * *, n and fmi = mj if j > n. Therefore fMn C Mn, fm n=M if Mn ?M,. We shall write f= (f)n: M -> M, and fn will denote the endomorphism, fn: Mn -> Mn, which is induced by f. That is to say, fnm = fM if i e Mn. Notice that (f) n (f)q if q > n. There- fore, if f/i: M -> M is any finite set of endomorphisms, we may take fi = (fi) n,

for any value of n which is sufficiently large to be the same for each i. Notice also that any endomorphism f': Mn -> Mn can be extended to a unique endomorphism, (f) n: M -> M, such that fn = f'. Obviously fn is an automorphism if, and only if, f ? a.

Let f= (f) n be given by

(2. 1) fmi Y f1jmj (fj ?_ R). j=1

10 This has the disadvantage indicated by (2. 3) below. But the convention n -* mr would be inconvenieht in the geometrical application.

11 I. e. generated with the help of the operators in R. 12 Unless the contrary is stated it is to be assumed that any given endomorphism of

M is admissible.




Theni the matrix f- [fij] is of the form

where fn is the matrix of fn : Mn -> Mn and 1. is the infinite unit matrix. Let g: M -- M be given by

g-E gtjmj. j

Since fr rf, where r e R is any operator, we have

(2. 3) fgmi - E gigfm E giifikMk j j,k

Therefore fg: M -> M corresponds to the matrix gf. Let g:M -M be given by

(2.4) gmi= mi%+rmj, gmk = Mk (j, 7 / i;reR).

Then g has an inverse, which is given by (2. 4), with r replaced by - r. It is therefore an (admissible) automorphism. Let >1 c a be the group generated by all such automorphisms, for all values of i, j, r.

Let A and B be the modules generated by disjoint sub-sets, mi, *,p and M11,, . , Mi, of the basis elements n1, m2, . , . Let h: A-- B be an arbitrary homomorphism and let g: M -> M be given by

(2. 5) g(a + b) =- a + (ha + b), gmz = mz,

where a e A, b e B and 1 Ap or j]. Then g is the resultant of the homomorphisms

m -> mp + hpa?mj, a fk -> Mk (k =7 ip),

where hmrp= hp1,mj +* + hpqnMj. These are of the form (2. 4), whence

g9 C1.

THEOREM 1. 1 is an invariant sub-group of a and a/:1 is Abelian.

Let f, f' C G. We shall write f = f' if, and only if, f - gf'g', where g, g' C >i. This is obviously an equivalence relation. Assume that ff' = f'f for every pair f, f' C a. Let g e :41, and let f' gfr. Then

fgf-1 _ gfrf = g.

Therefore fgf-1 e ?1, whence , is invariant in a. Also C/:1 is Abelian since ff' - f'f for every pair f, f' C (X.

We proceed to prove that ff' f'f. Let A =M-MP, let B be the module




generated by mp+, , m2p and let g be given by (2. 5). Then g (g)2p and

g2p [ l1 h]

where h = [hpa] and 1p is the unit matrix of order p. Let f (f)2p a and let

L [21 .22

where fil, f22 are square matrices of order p. Let f'2p and f'x.l (A, u = 1, 2) be similarly defined in terms Of f P (f')2p. We shall write f2p f'2r if, and only if, f r. Then

(2. 6) f2 f2g2 [fIl filh + f12]

Similarly a right hand multiple of the second column may be added to the first. Also j2p = g2pf2P, and g2Pf2P is obtained from f2p by a similar operation on the rows.

Let f, f' c a be given and let p be so large that

f (f)p_ (f)2p f= (f')V (f')2p.

Let r = PJ r' - P. Then r, r' are regular matrices. Therefore, beginning with (2. 6), with h r-1 anad f replaced by f'f, we have

f 2pf2p [rOr 0 ] [r?rlp] [-r lp] r 0 t r r rl= r rl_rr O0 L-r OJ L-r OJ-L r'J-L r'

Similarly

[r r ] [r 0] 2pf 2

Therefore ff'=f'f and the theorem is proved.

Since a/y1 is Abelian it follows that ac C Y, where ac is the commutator sub-group of a. Therefore we have the corollary:

COROLLARY. If Y C a is any sub-group, which contains >, then , is invariant and d/8, is Abelian.

The totality of automorphisms (f) n Ca, for a fixed value of n, is obviously a sub-group, (a)n C C. It follows from Theorem 1 that




(Y11)n = , n (a)n is an invariant sub-group 13 of (a) n and that (a) n/ (>z ) n is Abelian. Let an be the group of (operator) automorphisms, fn: Mn - > Mn, and let p: (a) n-> an be given by p(f)n=fn. Then p is obviously an isomorphism.14 It follows from the invariance of (.,) n in (a) n that =, 1 ( :,) n is invariant in Cy anad that y n is independent of the particular isomorphism (b: (a) n an. Also aGn/yn is Abelian.

Let A be a sub-group of the multiplicative group of regular elements in R (that is, elements with two-sided inverses), which contains both ? 1. Let g: M -> M be given by

(2. 7) gm =Xm- + rmj, gm,k=m& (i Mk7=/=),

where A c A., r ? R. Then g e aand g-1 is given by (2. 7) with A, r replaced by A-', -A-1r. Let >A be the sub-group of a, which is generated by all automorphisms of the form (2. 7), for every choice of i, j, A and r. Clearly - C . Therefore Et is invariant and T = a/YA is Abelian. We shall keep A fixed and shall write X, T for YAA, TA. The elements of . will be called simple automorphisms. We shall write T additively and T (f) c T will denote the co-set containing a given f ? a.

Our "torsion" will be defined in terms of T. An element of torsion will correspond to an isomorphism of one module, of finite rank, onto another. In order to classify such isomorphisms in term of T we need a standard class, which have "zero torsion." We therefore proceed to define a class of "basic modules" in M, which are related by a standard class of automorphisms, called permutations.

By a basic module, A C M, we shall mean the one generated by mi, . . *, m,p, for any (distinct) values of ii, * *, i,. We shall call (mq1, * , m,p) the basis of A. We allow p = 0, in which case the set (mq1, . . ., m,p) is empty and A = MO. Let p ? 0 and let MA be the module generated by the remaining basic elements, m P + mp of M. Then M is the direct sum M = A + MA. Let B be a basic module and let (mj,. * , mjq) be its basis. We shall only allow ourselves to form the direct sum A + B B + A, if A nB =0. In this case A + B will be the basic module, whose basis is

(m , . . ** r. - rn.~ *j * t.

13 The example II, on p. 1233 of [3] shows that (2;) l may be a larger group thani the one which is generated by transformations of the form (2. 4), with i, j ' n. I see no reason to suppose that the latter is necessarily aln invariant sub-group of (a ) n.

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




not the set of all pairs (a, b), with ag A, b ? B. Let C be a given basic module. Then C = A + B will always mean that A, B are basic modules, with disjoint bases, of which C is the direct sum.

Let i1, * , i*, be any permutation of 1, * . , n, for any n ? 1. Let P: M -> M be the automorphism, which is given by

Pmj =mj, Pm mk ( 1, , n; c > n).

We shall call P a permutation. It follows from (2. 7), with X,r ? 1, that the transformations

(mi, mj) -> (- m + mj, mj) *(- m-, mn) -> (mj, mi)

determine simple automorphisms. Therefore P ? >. Let A, B be basic modules of the same rank and let n be so large that the bases of A, B are both contained in Mn. Then there is obviously a permutation, P = (P) n,

such that PA B. The totality of permutations is obviously a sub-group of C7.

Let c: A At, where A, A' are basic modules. Since A and A' have the same rank, accoording to our condition on R, there is a permutation, P, such that PA' = A. Let fi: M -> M be given by

(2.8) f(a+m) = Paa+m (aDA, m ?SIA),

and let T(a) = T(f). Let P' be any other permutation such that P'A' A and let f' be defined by (2. 8), with P replaced by P'. Since P'P-1A - A the permutation PTJ-1 permutes the basis elemenlts of A among themselves. Therefore P": M -A I, given by

P"(a + m) -P`P-1a + m (m ? MA),

is a permutation. Since PaA = A, it follows from (2. 8) that f' = P"f. Therefore

7r(f') rT(P') + rT(f) T(f).

Therefore T (a) does not depend on the choice of P. We shall describe a as a simple isomorphism, and shall write a: A A' (Y), if, and only if, T((a) = 0. It follows from (2. 8) that T(a) = 0 if a = 1: A A. In particular T(a) = 0 if A = A' = Mo.

Let a, P, f, mean the same as in (2. 8), let a': A' A" and let P' be a permutation such that P'A" = A'. Then T(ac) = T(f), T(ac') =TQ(f"),

where f' and f" are given by (2. 8) with , P replaced by ', P' and by 'a, PP'. Clearly PK1MA A MA'. Therefore



8 J. H. a. WHITEHEAD.

Pf'P-f (a + rn) Pf'P-' (Paa + m4) =Pf'((aa + P-1m)

TP(P'ca'aa + P'1m) =PP'a'aa + m = f"(a + in). Therefore

T(cX'X) =T(PJPf) =T(f) +T(f) =(a') +TX(a)

Since 7-(1) =0 it follows that 1(a-') -T (a,). Let ca: A / A' be the isomorphism induced by a permutation, P': M -> M

such that P'A = A'. Then f, given by (2. 8), is a permutation. Therefore (a) =0.

Let A, B and A', B' be two pairs of basic modules such that A n B =A n B 0. Let y: A +B->A" +B be a homomorphism such that

yB C B'. Then y(a + b) =caa + (ha + /b), where aa e A', ha, /b C BR. It is easily verified that a: A -> A', h: A -> B, /: B -> B' are homomorphisms.

THEOREM 2. If either:

(i) y is an isomorphism 14 and either a is an isomorphism into or ,B is onto, or if

(ii) a, / are isomorphisms, then x,3,y are all isomorphisms and

T(Y) T (a) + 4T(/).

Let y: A + B A' + B'. If /8b 0, then yb =/b 0, whence b 0. Therefore /8 is an isomorphism into. Let a' ? A' be given. Then aa4+(ha+,-8b) y(a4+b) a'forsomeacA,b8B. Sinceha + 8b8B' we have aa= a'. Therefore a is onto. Let c: A A' and let b'gB' be given. Then aa + (ha+ fb) =b' for some acA, b B. Since aa cA' we have aa = 0. Therefore a = 0, ha 0 and Ab b'. Therefore /8 is onto. Let ,B:B B' and let aa 0. Then y(a -,81ha) aa+ (ha -ha) O. Therefore a - /ha = 0. Since ,8-'ha ? B it follows that a = 0. Therefore a: A A'. Thus a,,/, y are isomorphisms if (i) is satisfied.

Let a, /8 be isomorphisms. Then y = y *, where y*: A + B -* A' + B', 8: A + B -A + B are given by y* (a + b) aa + 8b, 8 (a + b) =a + (/-'ha + b). Obviously y*, 8 are isomorphisms and so therefore is y. Moreover g:M ->M is of the form (2. 5), where

g(a+b+m) 8(a+b)+m (msCMA+B).

Let P be a permutation such that PA' = A, PB"' B. Let f = fa be defined by (2. 8) and let fp, fy, fy* be similarly defined in terms of /3, y,y and the same permutation P. Then fab = b, foa = a and

fy(a+b) Paa+P(ha+ 8b)= + f+fl(f3'ha+b)= faf6g(a+b).




Since g ? Y it follows that

r(Y) rT(fy) = T(fa) + (I(f6) + T(g) rr(a) + T(f3)

and the proof is complete.

COROLLARY. If any two of a, /, y are simple isomorphisms, so is the third. Let 0: R R be an automorphism of R and let s8: M -> M be the trans-

formation which is given by

(2. 9) so (r1, r2, * (Or,, Or2,

Obviously so-1 = s-' and sos@ = so, where b: RI R. Also so(rm) =- (Or)som, where r F R., m ? M. Hence it follows that, if f: M -> M is an (operator) endomorphism, then (sofso) rm = (6(br) (sofso) m. Therefore fOr - rf9 where fJ = sofso-1. Since somi = mi it follows that fO ? a if f ? a. Let g:M->M be given by (2.7). Since som =mi we have g9m - (OA)m, + mji, g9Mk - Mk. Therefore g0 is also of the form (2. 7) if OX ? A. Clearly (g1g2)9 = g19g20 and it follows that f/ ? . if f ? Y, provided OA C A.

We shall describe 0: R IR as a A-automorphism if, and only if, OA = A. The totality of A-automorphisms is obviously a group E. Since fJ ? Y if f ? X and 06 c it follows that T admits O as a group of operators, according to the rule

(2. 10) 0T(f) T(f ).

Let x c R be any regular element, not necessarily an element of A, and let 0xr = xrx-1. I say that

(2.11) 06T T

for each T ? T. For let f ? a be given by (2. 1). Then

=O-m s0,jf, jmj = XY (xf ijx-) Mj.

Iet f= (f)n and let g9 (g,)n:M--M be given by

9$ (r1 *, rn, r+i . ) =* (r1x,* *, rnx, rn+i** ..

Then gxmi = xmi if i ? n. Since fxm = xfmn

n n

fgxmi = xfmin = xfijmj = E (xfjx-1)xxmj = g.Tf9xmi (i n). j=1 j=1

Therefore fOx$ gx lfgx and T(f9$) 7-T(gx) +- T(f) + T(gx) =T( f which proves (2. 11).




Let f ? aand let f and fr& mean the samne as in (2. 2). Let g be given by (2. 7) and let g be its matrix. Then gf is obtained from f by the following operations

(2. 12) a) multiplying a row from the left by an element A e A, b) adding a left multiple of one row lo another, the multiplier

being an arbitrary element reR.

Therefore f e Y if, and only if, f -> 1oo by a finite sequence of such transformations. Let f -> 1G by such a sequence, ,l, . . . , P, and let f = (f) n.

Then there is a c ? 0 such that no row of f, after the (n + k) -th is involved in any of (Oi, . . ., (p. Therefore osr, , arp transform 13 fn+I, into 1n+k,

where

- [P 0] Let R be the group ring of a group r and let A consist of the elements

+ y, where y , P. If r is Abelian, the determinant, I fn I, of fn can be calculated in the ordinary way. Obviously I fn I is unaltered by (2. 12b) or by an " expansion," fn ..> fn+k, and a transformation of the form (2. 12a) changes I fn I into + y Iff . Therefore +- f I , jer if f e.. Let r be cyclic of order 5 and let y#1. Then (1_ -y y4) ( _ y2 y3) . Therefore f: M -> M, given by f(r1 r2, r3, * *) = {rl(1-y y4), r2, r3,

is in a, but not in .. Therefore T # 0. On the other hand it follows from the theory of integral, unimodular matrices, in case P = 1, and from Theorems 14, 15 in [11], that T = 0 if r is of order 1, 2, 3, 4 or is cyclic infinite.

We continue, until Section 9, without the assumption that R is a group ring.

3. Chain systems. By a chain system, C = {C.}, we shall mean a family of basic modules, Cn C M, together with a boundary operator, a {an}, which is a family of (operator) homomorphisms, n: Cn ---C,, such that OnOn+1 = 0. For the sake of completeness we define a000o C-, = 0. Each Ch, being a basic module, is of finite rank. We do not require CO to be of rank 1, as we did in section 8 of CHl(II). For example, we allow C0 0. We assume that Cn = 0 for all sufficiently large values of n. If C,, 0 when n > N? 0, but CNx#0, we write N= dim C. We write C =0, and dim C =- , if Cn =0 for every n ? 0. We insist that cp n Cq=O if p=X=q and C shall be the set-theoretic union of the groups C,, C,, * < - . Thus c C C means that c C Cn and c + c' is only defined if c, c' C C,,, for some n ? 0. Also a is a map, 0: C--> C, of the set C into itself.




Until Section 9 we shall only consider chain mappings,15 f: C _> C', of C into a chain system, C' = {C'"}, such that each f: Cn -> C'n is an operator homomorphism. That is to say, in the terminology of CH (II), f is associated with the identical isomorphism, R -> R. Thus 16 Of = fO, fr = rf, where r ?1 R is an operator. Also fi g: C -> C' will mean that

(3. 1) gn -fn = dn+lqn+l + ?qndn (n > 0),

where \ = {rpn} is a chain deformation operator, and f: C -. C' will mean that there is a chain mapping, f': C' -> C, such that f'f 1, ff' 1 in the sense of (3. 1). We shall call f: C -> C' a simple isomorphism, and shall write fi: C C' (8), if, and only if, it is a chain mapping such that fn Cn Cn (Y), for each n ? 0.

Let B, C be given chain systems and let Bn = B'*n + B"n, Cn = C'n + C"n, where, according to our convention, B'n, B"n, C'n, C"n are basic modules. Let fi: B -> C be a chain mapping such that

f n (b' + b") = f'nb' + (gnb' + f"nb") (b' ? B'n, b" ? B'n),

for each n > 0 where fnb' eO C,' gab', f" b" C C In.

LEMMA 1. If any two Of {fn}, {f'n}, {f"n} are families of simple isomorphisms, so is the third.

This follows immediately from the corollary to Theorem 2.

Let Cn C' n + C"' and let OC'" C C', l for each n > 0. Let C' = {C'n} and let 0': C' -> C' be defined by O'c' = Oc'. Then O'a'c' = aac' = 0. Under these, and only these conditions, we shall describe C', with the boundary operator 0', as a sub-system of C. If also OC"n C C", (n ? 0), so that 0(C' + C") = O'C' + 'C"V (C' C C'n, C" ? C"), then C" = {C"n}, with boundary operator s", is also a sub-system. In this case we shall call C the direct sum, C C U' + C" = C" + C', of C' and C". Let C', C" be given, disjoint,17 chain systems. Then the direct sum, c' + C", will be the system which consists of the groups C'n + C"n, with 0 (c' + c"') = O'c' + a"c". Similarly we define the direct sum of any finite set of disjoint chain systems.

15 At this stage we do not impose any restriction such as f0m =- Xmj on f0, where ml is a basis element of C,. For example, C -- 0 is a chain mapping.

16 We shall often use 0 to denote the boundary operator in each of two or more systems, C, C', C",. . ., which occur in the same context. On other occasions we shall use 0, 0', 0", *. . to denote the boundary operators in C, C, C", . . .

17 We describe two or more basic modules, or chain systems, as disjoint if, and only if, 0 e M is their only common element.



12 J. H. 0. WHITEHEAD.

Let C' be a sub-system of C and let C', = C'n + C"n. Let in: Cn -CtI and a".: C-In -n> c'l be defined by

(3. 2) jn(C' + C ) C", all nc" = jn-PJnct.

Then 0",jn j=-10, since 0C' C C' and jin 'Cn = 0. Therefore

afl nOfn+ljn+lJ Anjnf+l l in-10n0n+1 = 0,

whence 0"tnd"n+i- 0. Therefore C"' = {C"n}, with 0" {a"} as boundary operator, is a chain system. We call it the residue system, mod C', and write C" =- C - C'. Notice, however, that an element in C" is an elemnent in the basic module C',, for some n> O, not a residue class of elements in C. Notice also that j = {jn} is a chain mapping, j: C -> C"; also that c - jc ? C', whence

(3. 3) ac," - a"C" = Wc" - jc" C C'.

Let B', C' be sub-systems of chain -systems B, C and let B" = B - B', C" = C - C'. Let f: B -> C be a chain mapping such that fB' C C'. Then f': B' -> C', given by f'b' = fb', is obviously a chain mapping. Let

=In jnfn: B"n C",.

Then f"j = jf, where j: B -> B" is defined in the same way as j: C -> C". Since a"j =j we have

f"'ji = ]fa = jaf = T'j

where 0 operates on B, C and ol" on B", C". Therefore f" is a chain mapping. We shall call f': B'-> >C', f": B" -> C" the chain mappings induced by f. It follows from Lemma 1 that, if any two of f, f', f" are simple isomorphisms, so is the third.

Let A be a common sub-system of B and C. Then we shall describe a chain mapping, f: B C, as rel. A if, and only if, fa = a for each a , A. We shall say that f g: B ->C, rel. A, if, and only if, g- f = 7+ ?0, where -: B -- C is a deformation operator such that 'qA = 0.

Let Zn (C) =0nj1 (0) and let

Hni(C) = Zn(C) - n+Cn+ (n ? 0).

A chain mapping, f: B -> C, obviously induces a family of homomorphisms

f*e: Ht, (B) -> H bn (C) i

Let C' be a sub-system of C, let i: C'- C be the identical map, which is




obviously a chain mapping, and let j: C -> C" mean the same as before. Let z" ? Z,n (C"). Then it follows from (3. 3) that Oz" ? Z,_1 (U'). There- fore 0 induces a family of homomorphisms d*: H. (C') -> H,- (C'), where H, (C') 0. It is known 18 that the sequence of homomorphisms,

(3.4) * * H.n(C') H.ln(C) - H. (Ct) *- - H, (C'), d* j* d* d*

is exact, meaning that the kernel of each homomorphism is the image group of its predecessor. We prove that dH. (C0") i*-1 (0). Let z e 1IL (X) (X = C, C' or C") be the residue class containing a given element z e Z" (X). Let z" ? Zn (C"). Then az" = iOz" e, Z,_1 (C), and

i*d*V ' i*Oz" = iOz"'= Oz =_ O.

Therefore d*H.n(C") C i*'-(0). Conversely, let i*z = 0, where z' Z (Ce ) This means that iz' _ C or that z' = ac = 0 (c' + z") (c' e C',, z" cn) Therefore, writing z' - Oc' z',, we have

=z' '1 =Oaz" dX1

whence d H.n (C') = i.- (0). It follows from similar arguments that iEnH(C') j*-'(0) and that j*Hln(C) =d*-'(O).

4. Deformation retracts. Let C' C C be a sub-system and let i: C -- C be the identical chain mapping. A chain mapping, k: C -> C', will be called a retraction if, and only if, ki = 1. We shall call C' a deformation retract (D. R.) of C if, and only if, there is a retraction, 7c: C -> C', such that Re- 1, rel. C'. Let ikc 1, rel. C', and let k': C ->C' be any other retraction. Then ik' - i7'ik = ik 1, rel. Ct.

THEOREM 3. A sub-system C' C C is a D. R. of C if, and only if, Hn(C -C') O for every n>0.

Let C' be a D. R. of C and let k: C -> C' be a retraction. Then

(4.1) ik Ory + ra

where q: C C is a deformation operator such that -qC' = 0. Let C" = C - C' and let z" e Zn (C"). Then Oz" ? C', whence 'qOz" 0. Clearly ji = 0, jz" z", where j: C -> C" is given by (3. 2). Therefore

z j (1 - ic) z" j (0-q + rd) e" = jz"' = jrz".

Therefore Hn(C") =0 (n >0).

18 See Theorem 3. 3 in [ 15] .




Conversely, let H, (C") 0 for every n ? 0. Assume that there are homomorphisms,

kCr: Cr > C`r' ,qr+1: Cr >O Cr+1 (r 1 n -

such that krir = 1, Orkr = kr-lfr and

(4. 2) r irkr - 1 = Or+lpjr+l + lqrjr9

these conditions being vacuous if n = 0. Let m"1, , m"', (m"x = X m ) be the basis of C"n and let

(4. 3) (1 + a.n) m"X = C'X + C"X (C'X C C`n' C"X =Cn) C

It follows from (4. 2) n-1 that

n (1 + --nd-n) ( + -n1-n) On = (in lkn-c1 -n-1n-f1)0n in-17en-1a n

Therefore On (C'X + C"X) = in-l kn 10nM1Xm kn= -10m"x. Therefore "nC"X 0-. Since Hn(0") 0 we have C"X =- "n+1a"x, for some a"x P-C"n+l. Let a'X C - n+la"x CC'n. Then it follows from (4. 3) that

(4. 4) (1 + 77nOn) MX cx? + ax? + n+ia`x.

Let kn: Cn -- C'n and mqn+l: Cn -> Cn+l be the operator homomorphisms defined by knC' = c', qn+lC' = 0 and

nM"X= c'x + a'x, qn+1M"x a"X.

Then (4. 2)n follows from (4. 4). Also

OnknC' a nC' kn-ldnCt

OnknM>"X= On (C'X + a'x) = On (c'X + C"X) kn=10nMn1X

Therefore, starting with kc1 - = 0, the theorem follows by induction on n.

COROLLARY 1. C O0 if, and only if, Hn(C)- 0 for every n? 0.

COROLLARY. 2. C' is a D. R. of C if, and only if, C - C' 0.

LEMMA 2. If C' is a D. R. of C then C C' + C" (.), rel. C', where C" C - C'.

Let C* C'H+ C". Then C*n Cn and 0*: C* -C* is given by

0* (c' + c") 0'c' +- "c". Let i, 7k, q mean the same as in (4. 1) and let

f: C* > C be given by

f (c' + c") = (c' -kc") + c"= c' + (an + 0) c".




Then Ofc' = dc' = fO*c' and

Of C 0 (a + no) c" = onoc",

fee,c (O + no)O"c" = (% + A- ) (Oc" + C',) (c,1 ? 0').

Since 0C' C C' and nC' = 0 it follows that fd*c" - Ondc" = Of c". Therefore f is a chain mapping. It follows from Lemma 1 that f: C* C (.) and the lemma is proved.

5. Simple equivalence. We shall describe a chain system, B, as elementary if, and only if, Bn = 0 when n ,7 r - 1, r, for some r > 1, and Or: Br Bri, (E). This being so, it is obvious that Hn (B) = 0 for every n ? 0. Therefore B - 0, by Theorem 3, Corollary 1. We shall describe B as collapsible if, and only if, it is the direct sum of a finite set of elementary systems. Clearly B 0 if B is collapsible. It is obvious that B' is collapsible if B' 1B (:,), where B is collapsible; also that, if B, B' are disjoint and collapsible, then B + B' is collapsible; also that the direct sim of a set of r-dimensional elementary systems is itself elementary.

Let Bn = An + Zn, let 0'n: An, Z1n- (Y.) and let On: Bn -> Bn-l be given by Ona 0'na, OnZn 0, (n= 1, 2, ). Then B = {Bn}, with a = {an}

as boundary operator, is the direct sum of the elementary systems (. * 0, Ann Zn ,l 0, * * ). Therefore B is collapsible and any collapsible system is obviously of this form.

We shall say that C, C' are in the same simple equivalence class, and shall write C -C' (E) if, and only if, there are collapsible systems, B, B' such that

(5.1) f: B +C --1B' +C' (E.).

This being so, it follows from Theorem 3, Corollary 2, that C, C' are D. R.', of B + C, B'+ C'. Let

i:C->B+C, i':C'->B'+C'

c: B + C-> C ':B' + C'-> C'

be the identity maps and any retractions. Let

(5.2) g -c'fi:C ->C'

and let g' -f-lil: C'--> C. Then

(5 3) g'g - kf-1i'lk'fi -lff 1.




Similarly gg' 1. Therefore g: C C'. We shall describe a chain mapping, g: C -> C', as a simple equivalence and shall write g: C C' (X), if, and only if, it is related by (5. 2) to some simple isomorphism of the type (5. 1). It follows from (5. 3) that, if g: C C' (E) and if g": C'"-- C is such that gg"* 1, then g" is a simple equivalence. Obviously g: C - C' (>) if g:C C' (,).

The relation C = C' (E) is obviously reflexive and symmetric. We proceed to prove that it is transitive; also that, if g: C = C' (>) and g': Ci C" (E), then g'g: C C" (E). Let C, C' be related by (5. 1), let f*B* + C' B' + C" (E), where B*, B" are collapsible, and assume that

(5.4) B' nB* Bl n (B + C)== B n (B"+ C") =O.

Then B + Be + C B' + B* + C' B' + B" + C" (.), whence C=-C (). If (5. 4) are not satisfied we apply a permutation, Pn: B'n -> A',, to each module B's, thus transforming B' into a new system, A', such that P:B' A' (Y), where P= {P.}, and A' n c - O. Let

h:B'+ C' A'+ C'(E)

be given by h(b' + c') = Pb' + c'. Then

(5. 5) hf :B + C A' +C' (Y.).

Similarly we can replace B* by A* P*B*. We can choose the basic modules A'., A*. in such a way that (5. 4) are satisfied when B', B* are replaced by A', A*. Therefore C -C" (E), and it follows that C- C' (:) is an equivalence relation.

Let g: C-> C' be given by (5. 2) and let h mean the same as in (5. 5). Then g (lc'h-) (hf)ii: C -* C' and lch-1: A' + C' -> C' is obviously a retraction. Also kAh-1 can be extended to a retraction, A'+ A* + C' C 0', by mapping A* on zero. Therefore, if g: C C' (E) and g': C' C" (E) we lose no generality in assuming that g satisfies (5. 2) and that

f': B' + C' B"+ C" (E), g - kl"f'i': C' ->C"

where k": B" + C" -> C" is a retraction. This being so,

f'f: B + C B" + C" (E) and

g'g - k"f'i'lc'fi k"f'fi: C -> C".

Therefore g'g: C C" (Y).




A non-zero element, which is common to two chain systems, will be called an accidental intersection, unless it is in a common sub-systenm, which is explicitly mentioned in the context. Accidental intersections between any finite set of systems, C,* * *, can always be eliminated, as in the paragraph containing (5. 5), by replacing C, * * *, by a set of chain systems, PC, * * * , where Pn: Cn -> (PC)n, is a suitable set of permutations. When the context requires it, we shall always assume that this has already been done.

Let C C' C) C - C'* (E). Then B + C- B' + C'(E), B* + C* B'* + C'* (E), where B, B' etc. are collapsible. Therefore, in the absence

of accidental intersections, it follows from Lemma 1, in Section 3, that

B + B* + C + C * - B' + B'* + C' + C'* (E), whence

(5. 6) C + C* =- C' + C'* (Y.)

Let A be a common sub-system of C and C'. We shall write C - C" ( ), rel. A, if, and only if,

(5. 7) f : B+ C B' + C' (Y.), rel. A,

where B, B' are collapsible.

THEOREM 4. a) If C C' (.,), rel. A, then C-A C'-A' (Y).

b) If C A 0 (E), then C =A (E), rel. A.

Let C, C' be related by (5. 7). Since f induces the identity, A -> A (E), it follows from Lemma 1 that

fi': (B + C) -A - (B' + C') -A (.),

where f' is the chain mapping induced by f. Obviously

(B+C)- A B+ (C -A), (B'+ C')- A=- B'+ (C' A),

which proves (a).

Let C" 0 (E), where C" C -A. Then B + C" B' (E), where B, B' are collapsible. Since C" 0 it follows from Theorem 3, Corollary 2, that A is a D. R. of C. Therefore C A + C" (E), rel. A, by Lemma 2. Therefore

B + C B + C" +-A B' + A (,), rel. A.

Therefore C A (Y), rel. A, and the theorem is proved. By a (p, q) -system, C, we shall mean a chain system such that Cn O 0

if n < p or if n > q (p q).

2




LEMMA 19 3. If C = C' (>), where C, C' are (p, q)-systems, there are collapsible (p, q)-systems, B, B', such that B + C B' + C' (E).

Let fi: A + C A' + C' (>), where A, A' are collapsible. Let n dimA > q. Then n dim(A +C) =dim(A'+ C'). It follows from the definition of a collapsible system that

(5.8) A B? + . . . +Bm

where each Bi is an elementary system. Let E be the direct sum of all the n-dimensional summands, Bt, and let D be the direct sum of the others. Let D', E' C A' be similarly defined. Then En - (A + C)n, E', - (A' + U')n. and. nE: E.nl (E), a' : E'n E'n-1 (E) since E, E' are elementary systems. Also fn: En E'. (.). Therefore

fEn1 - fOEn = a'fEn - a'E'n E'"

and fn1 l-- O`fnO)n_1 En1 l-1 E.'n 1 )

Therefore fi: A + C A' + C' (E) induces a simple isomorphism E E' (E). Since A + C -E D + C, A'+ C'- E' D' + C', it follows from Lemma 1 that

(5. 9) D + C 1-_ D' + C'(T )

Now let Ar, 0 for some r < p and let s be the least value of r with this property. Since C's = 0 and A + C A' + C' (>) it follows that s has the same property in A'. Let E now denote the direct sum of the (s + 1)- dimensional summands in (5. 8) and let D be the direct sum of the others. Let D', E' C A' be similarly defined. Then D - D'8 0, by the minimal property of s. Therefore

(5. 10) (D + C)8= (D' + C')8 0.

Since E, E' are elementary systems we have

(5. 11) 8~: E8+1 -1- Esg ( On': E'8+1 1Ets(E

Let c e C,+1, d e D8+1 and let f(d + c) e' + d' + c'. Since a'(d + c') - 0, in consequence of (5. 10), we have Ye' -0'(e' + d' + c') = a'f (d + c)

fO(d + c) - 0. Therefore it follows from (5. 11) that e' = 0. Therefore f (D + C) C D' + C'. Let h: E > E' be the chain mapping induced by f. Then it follows from (5. 10) that hs - f,: E, E's (.,) and also, since he - fe ? (D' + C') S+1 if e ? E,+,, that

19 Cf. Theorem 1 on p. 1202 of [3].



SIMPL.E HOMOTOPY TYPES. 19

'hs,je, = O'f,+le = fJe h,8e. Therefore

h8.+1 hE : E8+1 1 E ( E ),8

where OB- = 1 E8+1, B ' j E'8+1. Therefore h: E 1 E' (X) and (5. 9) again follows from Lemma 1. Lemma 3 now follows by induction on m in (5. 8).

We are now approximately half way through the algebraic preliminaries. The simple homotopy equivalences will be defined as those which induce simple chain equivalences, in a sense explained in Section 10 below. But we have still to relate chain equivalences to the group T, which is defined in Section 2 above. The first step in this is to associate an element, r(C) ? T, with each system, C, such that C -0. We shall do this by transforming C into an (m, m + 1) -system, Cmn, in which 0m+1 : Cmm+i Cmm, and defining r (C) - (- 1) mr (am+i). In Section 8 below we define a chain system called the "mapping cylinder," C*, of a given chain equivalence f: c c0. This contains C as a sub-system and C* -0 C 0. We define Ir (fi) r (C* - C). We shall also need to consider the effect of a A-automorphism, 0: R B, operating on T, because the chain mapping induced by a homotopy equivalence, p: K - K, is "(associated" with an isomorphism 7ri(K) 7ri(K'), namely the one induced by p.

6. Null-equivalent systems. Let C 0. Then 7c 1: C - C, where kC = 0. Therefore there is a chain deformation operator, v: C C0, such that

(6.1) + I.

Let S = vqO. That is to say, 8 = {8,n}, where ain = qrnnn: Cn- - C.. There- fore 8 is also a chain deformation operator. It follows from (6. 1) that O=q (1 - ) = 0. Therefore 08 8+ = =-- --+0 =- vq=-- 1. Also

=X7 (1 -0) -> =(1 - O) = a. Therefore 88 = = 0. Thus

(6.2) as+80 1, 88=0.

Let P1, P2: M -> M be permutations and let B be the elementary system in which

Bo = O, B = P0Co, Bn 0 (i = 1, 2; n > 2)

and aP2CO =- PcO (co e Co). Let C'=- B + C. Then C'. = Cn if n m 1 or 2 and




5 C 1 P1CO + CN, C12 = P2Co Co aO', (Pico + C,1) = lclc, O'2 (P2Co + c2) = PlCo + 02C2,

where ci c CL. Let C* be the system which consists of the same groups, C*n= 'Cnn with 63_- if n > 2 and

(6. 3) O*l(Pico + C1) Co, O*2(P2Co + C2) 81cO + 02C2.

Let fi: C'-> C* be given by fn 1 if n :1 1 and f1(Plco + cl) =PlAc + (81co + cl). Then

O*ifi(Pico + c1) al= l fo0'1(Pico + C1)

O*2f2(P2co + C2) = 81CO + 02C2 = P101N2C2 + 81CO + 02C2

fl (Plco + 02C2) = f10'2(P2cO + c2)

and O*nfn - fn-1'n l - On 0n O if n > 2. Therefore f is a chain mapping. Let gi, hi: C', -> C,' be given by

g1(Pico + C1) =Pi(Co + 01c1) + c1

hi(Pico + c1) -Pico + (81co + c1).

Since 018 01= 8 + 8OO = 1 we have

g1h1 (Pico + c1) gl{- Pico + (8co + c1) }

P(- CO + co + Oicl) + (Pco + C)= fi(Plco + C0)

Therefore f, = g1h1. It follows from Theorem 2 in Section 2 that gl, hi: C', C', (Y) and hence that fi: C C' (Y) .

It follows from (6. 3) that C* = B' + C', where B'n 0 if n > 1,

B'o = Co, B'1 = P4Co, (* I B'1) = Pi-: B'1 B'o (Y,) and

(6. 4) clo ? 0, Cl=Cl, C12 % P2CO + C2, C C cn (n > 2)

with 01: C'> C' given by 3ln 63On if n > 2 and

(6. 5) al2 (P2cO + c2) - 81CO + 02c2.

Let m ? 1 and assume that there is a system, Cm, such that C = Cm (,) and

Cm0 - Cm 1 0, CmnW+p+l Cm+p+1 (P > 0).

Then it follows from the above argument, with Cmm+n playing the part of Cn, that Cm = Cm+' (Y), where Cm+' satisfies the same conditions as Cm, with m




replaced by m + 1. It follows by induction on m that there is such a system for each value of m. Moreover Cm C- 0. Therefore equations analogous to (6. 2) are satisfied in Cm.

Let N =dim C and let m 2 N- 1. Then Cm.=0 -unless n m= m or mn + 1. Therefore (6. 2) reduces to

.U 8m+m+1 1 :Cmm+i -> Cm,in+i.

Therefore Om+i: Cmm+l Cmm and 8m+l = 0-1m+,. We define the torsion, r(C), of C as

(6. 7) T(C) =( ) )MT(Om+l) -( M+1T(8m+j)

In (6. 4) and (6. 5) let C, C1 be replaced by Cm, Cm+', with Cmn = 0 if n > m + 1. Then (6. 4), (6. 5) become

C m+l1+ Cmm+i, CmM+2 = Pctn

where P is a permutation, and

=M+2 8m+iP-1: Cm++lm+2 -> Cm+lM+i.

Since r(P) =0 it follows from (6. 7) that r(C) ( +=7r(8m+l) - ( 1) m+1T (m+2). Therefore T (C) does not depend on the choice of

m ? N - 1. However T (C) appears to depend on the particular choice of 8 in (6. 2) and on the construction for Cm. The following theorem shows that it does not.

THEOREM 5. T(C) depends only on C. Also T(C) = T(C'), if and only if, C C' (E), given that C C'- O.

Let C-C' (E), where C = 0, and let Cm, C'm" be any given (n, m + 1) -

systems such that Cm-C (Y.), C' == C' (Y.). Let r (C) be defined by (6. 7), where Cm is now this given system, and let r(C') be similarly defined in terms of C'm. In particular we may have C = C'. Therefore, when we have proved that T(C) =T(C'), it will follow that T(C) depends only on C and also that r(C) - r(C') if C=C' (>)-

By Lemma 3 f: B + Cm B' + C'm (E), where B, B' are collapsible, and hence elementary (mn, m + 1) -systems. It follows from Theorem 3, Corollary 2, that B + Cm_ B' + C'm 0. Therefore it follows from relations analogous to (6. 6) that

0: (B + Cm) m+1 - (B + Ctn)m




Moreover a = fm10'f.m+ since fO = O'f. Since fm. fim+ are simple isomorphisms it follows that T (a) =T (') . Also

0 O(b + c) =Ob + Om+lc (b c Bim+i, c c Cmm+i) V '(b' + c') = Ob' + 0'1+jc',

where OB = 0 Bm+i, OB' 0' Blnz+,. By the definition of an elementary system, T (aB) =T ( =B) 0 and it follows from Theorem 2, in Section 2, that T (am+j) =rT (O) rT (0') T (' T m+im). Therefore T (C) =-T (C').

Conversely let r(C) =r(C') and let CGm=C (E), C'm=C' (E,), where Cm, C'm are (in, m + 1) -systems. Let Cmm be of rank p and C'mm of rank p'. If p -7 p', say p < p', we replace Cm by B + Cm, where B is an elementary (in, m + 1) -system, such that Bm is of rank p'- p. Therefore we assume that p = p'. Moreover, after applying suitable permutations to Cmm, Cmm+. we assume that Cmm= CYmmp Cmm+l = C'mm+i. Then g 3',m+n101m+1: Cmm - Cmm

and (1)mT(g) T=(C') -'r(C) O=0. Therefore g: Cmm. Cmm (4). Let f: Cm -> C'm be given by fm = g, fm+l 1. Then O'm+lfm+l =m+l fmam+i.

Therefore f: Cm - Ctm (Y.) and the theorem is proved.

Obviously r(0) 0. Therefore we have the corollary:

COROLLARY. C 0 (E) if, and only if, r(C) 0. Let C' be a sub-system of C, and let C" = C C'.

THEOREM 6. If any two of C, C', C" are chain equivalent to 0, so is the third and Tr(C) T=r(C') +Tr(C").

Let X, Y, Z denote C, C', C" in any order and let X - Y--0. Then Hn (X) = O H,,( Y) = 0 for every n ? 0, according to Theorem 3, Corollary 1. Therefore it follows from the exactness of the sequence (3. 4) that H, (Z) 0, for every n ? 0, and hence that Z - 0.

Let Ct C'-C"-O. Then C C' + C" (E), according to Theorem 3, Corollary 2, and Lemma 2, and it follows from Theorem 5 that r(C) = r((C' + C"). For a sufficiently large value of m we have C' - A' (,), C" = A" (4), where A', A" are (in, m + 1) -systems. Therefore it follows from (5. 6) that C' + C" A' + A" (4) and from Theorem 5 that

Tr (C) = r (A' + A"). Let 0': A'm+i > Am, s": A"m+i > A"mp be the boundary homomorphisms, which are isomorphisms since-A' c A" 0. Then it follows from Theorems 2 and 5 that

T(A' + A"') = (- 1)m{T(0') + T (8")} T (A') +T (A") = T((') + T(0C")

and the theorem is proved.

For purposes of calculation we exhibit the structure of the system, C(m,




which is defined by reiterating the construction C -> C', leading to (6. 4). To begin with we do not require m + 1 > dim C. Let m = 2k or 2k + 1 (kc ? 0) and let Do, DJ C M be the basic modnles

(6. 8) 5 Do-Co0+C,+- * +Cr (6.8) D1=C, + C3 +. * + CNCk+l,

where r=m if m=-2k, r-=m+1 if m=2k+1. Let D=Do+D1 = Co + C, + - - + C.m+1 and let 0: D -* D be the homomorphism which is determined by a: C -> C. Let 8': D -> D be the homomorphism which is given by

8'c=8c if csC, (s< m+A )

=0 if c eC.m+,.

Then 8'S' = 0. Also it follows from (6. 2) that

(6.9) (0('?8'0)c=c if ceC, (s<m+1)

(69 c if c C C,m+,.

Also ODi C Dj, 8'Dj C Dj (i#-j;i, j 0,1). Let

(6.10) A= 0+8 (D,Do,DI)->(D,D1,Do)

and let Ai: Di Dj be the homomorphism which is induced by A. Then

(6.11) &iAjd =A*Ad AAd (d c Dj).

Since O=- 8'S' = 0 it follows from (6. 9) that

(6.12) AAc =- (a + 8') (0?+ 8') C

(c if c CO (s < m + 1); =80c if c C C.+,.

Also 8'OCm+2 C 8'Ci, = 0, whence AOCm+2 = 0. Let i 0, j 1 if m =2k and let i 1, j= 0 if m =2k + 1. Then

Cm+1 C Dj and it follows from (6. 11) and (6.12) that

(6. 13) AA

AAjc c if C cC, (s < m + 1); =80c if Cc C.+,.

Let Cm, with bou'ndary operator Om, be the chain system, which is given by 20 C =0 O if n < m,

Cmm =T Di= + Cm-2 + Cm

(6.14) . CmM+= Dj = ** + Cm-, + Cm+,

L Omm+ A1,

20Cf. (5) on p. 205 of [10].



2 4 J. H. C. WHITEHEAD.

and Cmnn Cn, Omn a On if n > m + 1. Since AOCm+2 0 it follows that =man 0. Let 8'n be the deformation operator, which is given by

( 6. 15 ) 8mm+l A 6

8m+2C 0 if cC CC (s < m +1); =m+2C if c ECm+,

and mn a8n if n > m + 2. Then 8"'n+18mn O if n > m + 1 and

8m M+28m (* + CM) = 8m+28M+lCM =n

where cm E Cm. Therefore 8m8mn =0. Also it follows from (6. 13) that

am m+8mm+= 1, amm+28rnm+2 + 8mOm M+= 1,

whence Om8m + 8mOm

Let Cm+1 be the system, with boundary operator On,+', which is obtained from Cm by the construction, C -> C1, leading to (6. 4), with Cmm+n playing the part of Cn and with P2Co replaced by Co and P2 by 1 in (6. 4), (6. 5). Then

C rn+1 Cmm+ * + Cm-1 + Cm+i

C n+1 2 = mm ? CMmm2 * + CM + CM+2

and (6. 5) becomes O m+lqn2 (cm + Cm+2) = Omm+1Cm + O'm+2Cm+2 = Cm + OM+2CM+2 A* (Cm + CM+2),

where Cm ? Cmm, C?n+2 C Cm+2 and A* is defined by (6. 10), with m replaced by m + 1. Therefore Cm+' is defined in the same way as Cm and we define 8m+1 by (6. 15), with m replaced by m + 1. Starting with CO - C,. it follows by induction on m that the construction C -> C1, reiterated m times, leads to Cm. Therefore C = Cm (E). We now take m ? dim C-1, thus giving an explicit definition of Om+j, am+, in (6. 7).

Let R' C R be a sub-ring, which is the image of R in a homomorphism, : R -> R', such that or' r' if r' e R'. Let 1 cR' and let A' = A. It is

easily verified that A' is a (multiplicative) group . Let M' C M be the sub- module, which consists of all the elements (r'j, r'2, * * ), where r', e R', and let R' be such that every admissible automorphism, M'-> M', is A-simple. That is to say, every matrix of the form (2. 2), with elements in R', can be transformed into the unit matrix by a finite sequence of the transformations (2. 12), with XcA, r zR.

Let +':M-->M' be given by +V(r1,r2, ) (4r1,,r2, ). Then m, m, and /(rm) = (or)>Im. Let f: M 1M1 be an admissible automorphism, which is given by fm, = Yjf,jmj, (f,j e R). Then qfmi = Yj(pfij)rnj and it follows that fij e R' if (and only if) if = ff. Therefore f e E if f= ff.




Let C be a chain system, with boundary operator 0, and let i: M -> M mean the same as before. Then iCr C Cr, since +imn = mi, and aq, /i are two families of homomorphisms Ot, t&: C. -* Cn-.

LEMMA 4. If C O and if O8 = +O, then C=O (E). Let C = 0 and let Ot = f0. Let n: C -> C be a deformation operator,

which satisfies (6. 1), and let $: C -* C be the deformation operator determined by $mn = mim, $r = r$. Since O/ = +O and t/mi mi we have (O4 + tO) mi t(On7 + rq0)mi

- mi. Therefore O4 + $= 1. Also J' =

since 4t = . Therefore 74qmi $7nm =-t/$mj, whence ~t = t. Moreover 8 04 is a deformation operator such that 84' = fr, which satisfies (6. 2). It follows from (6. 5) and induction on m, or from the explicit formulae (6. 14), that Om+l, in (6. 6), may be constructed so as to commute with J. Therefore 0m+, is a simple isomorphism. Therefore T(C) = 0 and the lemma follows from the corollary to Theorem 5.

Let R be the group ring of a group 1, let A consist of the elements ? y (y ? r) and let R' consist of the integral multiples of 1 C P. Let 4: RR - RI be given by 4r = 1. Then -we have the corollary:

COROLLARY. Let C -0, let (mn1, * mP) be the basis of Cn Mn my ) and let

Omni jdn,jMn-ij (n 1= 2, )

where dn,j are integers. Then C O (E).

7. Conjugate systems. Let C be a given chain system with boundary operator 0. Let 0: R R be a given automorphism and let S : M -* iIl mean the same as in (2. 9). We shall also use so to denote the semi-linear transformation s': Cr > Cr, which is given by 21 sIc = SOC for each c F Cr (r O,1, ). Let

f30 SoOS-1: Cn ->Cn-1 (n 2

Then 0%00 so00so1 - 0. Obviously O0r = r00 for each operator r c R. There- fore 00 is a boundary operator. Let CO be the chain system, which consists of the modules Cn with the boundary operator D0. We shall describe CO as conjugate to C.

Let f: C -- C' be a chain mapping, let fn =- sof ,s ,l : C,- C'n, and let fo = {fnI}. Obviously fOr - rf0 and f0$ =-- sOf=s) soOfs,-' = 00f0. There-

21 s C, = Cr since s mi = mi andCr is a basic module in 11.




fore f0: CO C'@ is a chain mapping. On transforming the relevant equations by so we see that, if fi: C C', then

(7. 1) f0: CO e C'.

Let C! 0 and let 0, S satisfy (6. 2). Then 00 and 8 s-8s obviously satisfy (6. 2). It follows from (6. 5) and induction on m, or from the explicit formulae (6. 14), that the construction for Cm, with 0, 8 replaced by 0, 80, leads to the conjugate system (Cm) 0. Therefore

(7. 2) T(CO) =( 1)nr(0m+i).

Let OA = A. Then it follows from (7. 2) that

(7. 3) -r(CO) 0 OT(C),

where 0:T->T is defined by (2.10).

8. Mapping cylinders. Let C, C' be disjoint chain systems, with boundary operators 0, O', and let f: C > C' be a chain mapping. Let acnGl be the image of Cn-l in a simple isomorphism a: Cn- a Cn-, which is induced by a permutation Pn,l M -> M. Then caCPn l is a basic module. Let C@* be the direct sum C*n - C'n + Cn + (cxC.-. Let 0: C*b C*n1 be defined by

(8.1) a Q a%'a) PC =a k _ 'c' (c C Cn, c, C'n) ( * ) 1 b) Paoc (f 1 os) C (c C Cn-,) .

We shall write (8. ib) as Pa =- f - 1 - ca, using 1, f as abbreviations for i, i1,f, where i: Cn -> C*n, i: C'. C*, are the identities. Obviously 0*0*(C'n + CO) 0 and 0*ao tt'f E - PA E (f 10- (f - 8)a = 0. Therefore C* - {C*"}, with P as boundary operator, is a chain system. We shall call it the mapping cylinder of f. Clearly C, C' are sub- systems of C.

LEMMA 5. C* -C' is collapsible. Let C" = C* - C'. Then C"'n = Cn ? c+CG- and

(8. 2) a" (c2 + ac1) = ac2 - (1 + ac) c, = (ac2 - c,) - Ac,

where C2 ? Cn, C1 ? C.-,. Let 00: C" -> 0" be given by

00c 0. o 8o(c c.

Then 0OO? 0 and it follows that {C"n}, with 00 as boundary operator, is a chain system C0, which is obviously collapsible. Let g: C" -> CO be given by




g (c2 + aci) =c2 + a (Oc2-c,). Then g9"(c2 + aci) =- g (c2- cl - c) c - c1 + ac(- Oc + &ci) = OWg(c2 + acc). Therefore it follows from

Lemma 1, Section 3, that g: C" CO (E), and Lemma 5 is proved. It follows from Lemma 5 and Theorem 4 that C* C' (E), rel. C'.

Therefore C' is a D. R. of C* and any retraction k': C* -> C' is a simple equivalence. Let k' be given by k'c = f c, k'c' = c', k'ac = 0 (c ? C, c' ? C'). Then it follows from (8. 1) that k'O* - O'k'. Therefore k' is a chain mapping, which is a retraction, since k'c' = c'. Also k'c =- fc fic. There- fore we have the corollary.

COROLLARY. k': =* C' (X), rel. C', and f = k'i. Let C(f) = C* - C. Then C.n(f) = CGn + C,,,-,, and it follows from

(8. 1) that the boundary operator, Of: C(f) - C (f), is given by

(8. 3) Ofc' = 'c', O tof = f - a9.

LEMMA 6. C(f) -0 if, and only if, f: C0 C'.

Let C(f) 0. Then C is a D. R. of C*, according to Theorem 3, Corollary 2. Therefore i: C C* and, by the corollary to Lemma 5, k': C* = C'. Therefore f: C = C'.

Conversely, let f: C0 C'.' Then

f - =c'*i*: Hn (C) - Hn (C') (n = 0, 1, *

and it follows that i*: HM (C) H Hn(C*). Therefore it follows from the exactness of the sequence

Hn (C) > Hn (C*) > Hn {C (f }Hn1 l(C) ->Xn1 l(C*)

that HnU{C(f } 0 for every 22 n > 0. Therefore C(f) - 0 by Theorem 3, Corollary 1.

LEMMA 7. If f g: C ->C', then C(f) ~.C(g) (s). Let g - f -- O- + 7q,, where -: C -> C' is a deformation operator, and

let 83Cn and 83: CnO 183Cn be the analogues, in C(g), of asCn and a. Let h: C(f) ->C(g) be given by h(c' + ac) = (c' --c) + /Gc, which we write as hc' c', ha - 'q. Then Oh-hO in C' and since hf f we have 23

Oha =(/3 - = g -G - (g - f - ) f - (, - =) hf - ha = h&a.

Therefore it follows from Lemma 1 that h: C (f) C 0(g) (X) and the lemma is proved.

22 Cf. Section 3 in [5]. 23 We now use a to denote the boundary operator in all our systems.




Let f: C0 C'. Then it follows from Lemma 6 that C(f) 0. We define T (f) r{C (f) } and call T (f ) the torsion of f. It follows from Lemma 7 that T (f) depends only on the chain-homotopy class containing f and we shall also call it the torsion of this class.

Let f: C -> C', f': C' - C0" be any chain mappings.

LEMMA 8. There is a chain system1 D, containing C(f) as a sub-system, such that24 D -C(f) = C0(f') and D C0(f'f) (>).

Let 'Ct',, C C (f') and a"C"0 C 0(f'f) be the analogues of aCO. Let C'* be the mapping cylinder of f' and let D be the direct sum D - C'* + C(f) with the " united sub-system" C'. That is to say Dn 0"-C`,- + C,n + a"C'0 i +4aC'0, and 0:1D ->D is determined by 6:0C'*-0C'* and 0: c(f) ->C(f), which coincide in C'. Thus Oa' f' - 1 - a'0, Oa = f - A. Moreover C(f) is a sub-system of D and obviously D1- 0(f)- C (f').

Let D' be the direct sum D' =C'* + C(f'f), with the united sub- system C". Then D'n OC"n + C'n + a'C'n-1 + aCn l and Oa" = f'f - "O.

Let g: D ->1D' be given by

g(c'* + occ) ((c'* - ofc) + a'c (c C C, C0 * C C'*),

which we write as gcf* - c'*, ga - a" - af. Then Og gO in C'*. Since gfc = fc we have

aga = Oa" - Oa'f = f'f - c"0 - (f' - 1-a'a)f = f - "0 + a'af

f (a" - 'f)a =- g(f-cab) = gba.

Therefore g:D D 1' (X). Clearly D'- C(f'f) - 0'- C" and it follows from Lemma 5 anid Theorem 4(b) that C(f'f) _D'z1 (D). This completes the proof.

Let f: C C', f': C' C".

THEOREM 7. T(f'f) =r(f') +?r(f). Let D mean the same as in Lemma 8. Then it follows from Lemma 8

and Theorems 5, 6 that r{C(f'f)} r(D) r{C(f')} + T{C()}, and the theorem is proved.

LEMMA 9. If f: C 0 C' (E) then r(f) 0.

Let f: 0C, C' (E) and let C0 be the mapping cylinder of f. Let g: C(f) ->0C -C' be given by g(c' + ac) = f-1c' - ac. Then Ogc' = gOc' and it follows from (8. 2) that

24 I. e. D- C(f) C (f') if the permutations a': C', -a'C', C C(f') are suitably chosen.



SI'MPIE HOMOTOPY TYPES. 29

L+ flf + a g (f ) g0

Therefore g: C(f) C* -C' (E) and the lemma follows from Lemnma 5 and the corollary to Theorem 5.

Let fi: C - C' and let us discard the (implicit) condition thatC n co0 O. Let h: A 0 C (E), where A is a chain system such that A n C' 0. Then fh: A = C' and we define r(f) by r(f) = r(fh). Let (h', A) be any other pair such that h':A 0C (Y), A' nC' 0 . Let h":A" C (E), where A" is disjoint from A, A', C'. Then fh: A = C', h-1h": A"l A (E,), fh": A" 0C. Therefore it follows from Theorem 7 and Lemma 9 that r (fh") = r (fh) + -(h-1h") =r(fh). Similarly r(fh") =ir(fh'). Therefore r(f) is independent of the choice of h, A.

Let f g: C= C', where C n0 c0. Then fh gh: A=C', and in consequence of Lemma 7 we have:

THEOREM 8. If f g: C=-C' then r (f) r(g). Let f: C=C', f':C'_C". Let h: A C (Y), h':A' C' (Y), where

A. A' are disjoint from C', C" and from each other. Then f'h': A' 0C", h'-Ifh: A = A', f'fh: A - C", and h'-': C' A' (E), fh: A = C'. Therefore it follows from Theorem 7, and Lemma 9 that

r (f'f) T (f'fh) r(f'h' * h'-'fh) =r (fh') + - (h'-1fh)

=r(f'h') -r(h') + r(fh) =r(f') + r(f).

Therefore Theorem 7 is valid, even when C, 0', C" are not disjoint from each other.

Similarly Lemma 9 is valid, even if C n c0 =A O

THEOREM 9. Given g: C =C', then 7-(g) =0 if, and only if, g: C -0 C' (E).

It follows from Theorem 7 and Lemma 9 that we may assumeCn 0 ] O This being so, let r (g) = 0 and let C* be the mapping cylinder, of g. Since r{C (g) }, T (g) =0 it follows from Theorem 5 that C (g) _ O (E). There- fore it follows from Theorem 4(b) that C* =- C (,), rel. C, whence i: C -C* (X). Therefore it follows from Corollary to Lemma 5 that g: C C' (E).

Conversely, let g: C-C' (E). Then f: B + C -0 B' + C' (E), g -'fi, where B, B' are collapsible systems, i: C -- B + C is the identity and

D': B' + C' -- C' is a retraction. Assume that r(i) = -r(i') = 0, where

i': C'-- B' + C' is the identity. Then r (') = 0, since r (') + r(i') = - (Fi'') = 7-(1) = 0. Also 7 (f) 0, according to Lemma 9, and it follows from Theorems 7, 8 that r (g) = 0.



30 J. Hn. C. WHITEHEAD.

It remains to prove that r (i) = r (i') = 0. Let h: A IC (), where A n (B + C) =- 0. Since ihA = C it follows from (8. 3) that C (ih) =B + C(h), and from Theorems 5, 6 and Lemma 6 that r{C(ih)} =-r(B) +- T{C(h) } = 0. Therefore r (i) = r (ih) = 0 and similarly r(i') = 0. This completes the proof.

Let A, A' be sub-systems of C, C' and let h: C -> C' be a chain mapping such that hACA'. Let B C -A, B' =C' A' and let f:A->A', g: B -> B', be the chain mappings induced by h.

THEOREM 10. If any two of f, g, h are chain equivalences so is the third, and r(h) =r(f) + r(g).

Assuming that C n c0 =0 we have

Cn (h) = Cn l + +C., = A'n - B'o + aA -, + aBn t Cn (f) + Cn ( g)

Let D C(h)-C(f). Then Dn Cn(g) and I say that D C(g). For let Ax denote the boundary operator in X, where X stands for any of the systems C, 0', . Since A' C C(f) we have 25

aDb' = ac(h)b' =cab'- aB b' ac(g) b, mod. C(f),

where b'V B'. Since aA C C(f) we have ac1 =ac2, mod. C(f), if c1 -C2

mod. A, where c1, c2 C C. Therefore

aDclb = c(h)b hb - gcb =gb - caBb = Oc(g)b, mod. C(f),

where b e B. Therefore JDd = aO(g)d, mod. C(f), for any d ? D. But D n C(f) = 0. Therefore aD aC(g), whence D =- C(g). The theorem now follows from Lemma 6 and Theorem 6.

In consequence of Theorem 6 we have:

COROLLARY. If any two of f, g, h are simple equivalences, so is the third.

Using the same notation as in Theorem 10, let A' =A = c n0c and

let h: C-> C' be rel. A. Then we define the mapping cylinder, C*, of h in the same way as when A = 0, except that C*n C'n + B-n+ AB. o l = B'n + Cn + cxBn l where a: Bn cBn (E) is induced by a permutation, Pn:M->M5 and 0* is given by (8. 1), with ccB and a aB in (8. ib). Obviously C* - C0 C (g). Therefore it follows from Theorem 10, with f =, and from Lemma 6, that h: C C' if, and only if, C* C 0. It

25 If Y is a sub-system of X, then x =x', mod Y (x, x' C X) means that x- x' C Y.

Clearly O,x 3 ,x, mod Y, where Z = X - Y.




follows from the corollary to Theorem 10 and Theorem 9 that h is a simple equivalence if, and only if,

(8. 4) C _ C O (

Let f: C = C' and let 0: R ft be any A-automorphism. Then fO: CO =- C'@, according to (7. 1). Since a, in (8. 1), is the isomorphismn induced by a permutation it follows that as0- soa. Therefore it follows froin (8. 3) that C(f) =- C(f)0 and from (7. 3) that

(8.5) '(fO) --07(f)*

Let f: C =- C'. In order to calculate r (f) we need to- know a deformation operator, : C0(f) ->C(f), such that 04 + - I 1. Let f and f': C' -C be related by f'f - 1 = J-j + q, ff' - 1 - 0-' + -'a, where -: C -> C, -q': C' -> C' are deformation operators. Let

(8. 6) C f: C'.

Then

Ott + +f= af a-q,f ? fiO -q'fo fQ(an + v0) - (n' + ?q'a)f

ff(f'f - 1) - (ff' -1)f = 0.

Hence it follows by a straightforward calculation that 04 +? = 1, where :0(f) C(f) is given by

(8.7) C/ C af'- /4

9. The groupoid $. Let R be the integral group ring of a group r. We need to consider chain mappings which do not necessarily commute with the operators r ? R. By a chain mapping, f: C C', associated with an automorphism, 0: R fR, we shall mean a family of homomorphisms, fn: un -* n5 0, such that fO = Of and fr - (Or) f. We now insist that CO + 0 and that, if mi is a basis element of C0, then f0,mj shall be a basis element of C'O. This ensures that f is associated with only one automorphism 0 (unlike C 0, for example). If f': C' - C" is associated with 0': R R

then f'f: C C-> " is obviously associated with O'O. Let x e R be any regular element. Then Ax = x9 and x (rc) = (xrx1) xc. Therefore x: C -> C, given by c -> xc, is a chain mapping associated with the inner automorphism Ox. We shall confine ourselves to chain mappings associated with those automorphisms of R, which are determined by automorphisms of F, and we shall




use the same symbol to denote 0: r P r and the corresponding automorphism of R.

We define chain homotopy and chain equivalence as in CH(II), with r playing the part of j,. Thus f g: C -> C' means that

(9.1) yg f 7 + 05

where y e r and q: C -> C' is a chain deformation operator associated with the same automorphism, 0, as f.. As in CH (II) it follows that g is associated with 0y-10. We shall write f g if, and only if, f, g are related by (9. 1), with y = 1. As in the ordinary theory of homotopy or chain homotopy, f g implies fh gh, h'f h'g, where h, h' are any chain mappings of the form h:C ->C, h':C' ->C".

We say that f: C -> C' is a chain equivalence and write f: C = C', if, and only if there is a chain mapping, g: C'-- C, such that gf 1, fg 1. Let y, y' C r be such that ygf -_ 1, y'fg = 1. Then f'f _ 1, where f' = yg. On transforming y'fg _ 1 by y' we have fgy'- 1. Therefore ff' - ff'fgy' _fgy'_ 1.

Let f: C = C', where f is associated with 1: R IR. That is to sav, fr = rf. Let f': C' -> C be such that f'f 1, ff' _ 1. Since f'f is associated with only one 0: R1 R and since fr = rf it follows that f'r = rf'. Therefore f: C - C', in the sense of Lemma 6.

Let fi: C - C' be associated with 0: R R and let 0&, 02: R R be given. Using the same notation as in Section 7 we have

0'Os0lfs02-l = s01'fsO2-1 = soJfSO2 = sOJfs2-1002.

Therefore

(9. 2) seL SO2-1: C02 .> C'01

is a chain mapping, which is obviously associated with 01002-1. Let f': C'-> C be such that f'f 1 , ff' 1 I. On transforming f'f _ 1 by so we have sof'fso- : CO -> CO. Also fso-1 s4f' 1 I: C'-- C'. Therefore it follows from (9. 2), with 01 = 1, 02 = 0, that fs-": CO= C'. Moreover (fs0-1)r =r(fso-1). We define r (f) by

(9. 3) r (f) = (fso-')

and call it the torsion of f. Let 01, 02: R R be arbitrary. Then s0lfs02;' is associated with 01002-1 and it follows from (9. 3) and (8. 5) that

(9. 4) T((s6lfsO2-1) = 7(solfsO21 * S02s5-01-1') = T(sol . fsj-l . so=-i ) 01r (f).




Let f': C' = C" and let f' be associated with 0': R - R. Then f'f: C _ C" is associated with O'O, and it follows from Theorem 7 and (9. 4) that

T(f'f) = T(f'fso'S0',1) T(f'S0_1 * sOfsO1 Sot-1)

=T(f') + T(So * fso 1 sol') =T(f') + 0'T(f).

Let f - g: C_ C' and let f, g be related by (9. 1). Then yg,f and q are all associated with the same automorphism, 0, and

ygso-l - =fsl O'sSO-1 + so-'sooso-1 =a,@ + ?ao where =so-1. Therefore fs01 -_ ygso-1 and it follows from Theorem 8 that T() T (yg). Also it follows from (9. 5) and (2. 11) that r(-yg)

T=(y) +Or(g) =-T(y) +T(g), where y:C' -C' is the chain mapping c' -> yc', which is obviously a chain equivalence. Since yso -lmi =-ymj, where mi is any basis element of C'", it follows from (2. 7) that ysa-': C'@e ~ C' (>), whence v-(y) 0. Therefore f -_ g implies

(9.6) T(f) T (g).

Let (M be the totality of chain homotopy classes of equivalences between all the chain systems, which are equivalent to a given one. Let f: C C', f': C' -- C" be such equivalences and f, f' the corresponding chain homotopy classes. We define f'f f'f. It may be verified that, when multiplication is thus defined, ( is a groupoid. Let f be associated with 0. Then we define f: T -- T by fT = OT. Let f g. Then g is associated with O0-10, for some -ye r, and it follows from (2. 11) that gT =Oy1OT = OrT fT. Therefore a single-valued map f: T -* T is defined by fT = fT. Obviously 1 T = T if 1 is any identical map, C-> C. Since f'f, if it exists, is associated with 0'0, where f, f' are associated with 0, 0', it follows that f'(fT) (f'f) T. There- fore we say that T admits ( as a groupoid of operators.

It follows from (9. 6) that a single-valued map T: - T is defined by T(f) = T(f) and from (9. 5) that

(9. 7) T(gf) T(= ) ? 7r(f) .

Therefore T is what, by a natural extension of the language of group theory, we call a crossed homomorphism of ( into T. We call T(f) the torsion of f. Given C it is easy to construct a chain system C'-=C and an equivalence, f: C C', such that T(f) is a given element 'T , T. For example, let d: A B be an isomorphism such that T(d) =Ton, where A, B are basic modules, which are disjoint from C and from each other Let rt > dim C and let C' be the system which consists of C, with its own boundary operator, and

3




CmM =.B, C0m+i = A, with O'm+i = d. Then it follows from Theorem 10 that r(i) = To, where i is the identical map C - C0'.

10. Homotopy types of complexes. Let K be a given complex and let a 0-cell e? c KO be taken as base-point for 7ri(K). Let Ki be the universal covering complex of K, in which the points are classes of paths joining e? to points in K. Let 26 C(Kf) be defined in the same way as C(K) in Section 12 of CH (II) and let (Cn,, * * *n Pc.) be a natural basis for Cn(Z) = Hn(fn, J4-1). Let R, r, M mean the same as before, with y: 7ri(K) r. Let R1(K) be the group ring of 7ri(K). Let Cn(K) C M be a basic module of rank pn (n O, 1, * * ), such that C0(K) n Cj(K) = 0 if i 3 j. Let (mni, * , Mn,p') be the basis of Cn(K) and let kne: C.(K) 0n(K) be defined by

(10. 1) kn1(r,Cni + ? * * + rpcnp) = (yri)mni + * + (yrp.)mnp ,

where ri c R (K). Let

(10. 2) An = kn-11nkn1 Cn0(K) -C l(K),

where W' is the boundary operator in C (EiT). Obviously =o 0 and Or =r (rcR). Therefore C(K) = {C0(K)}, with a = {On} as boundary operator, is a chain system and kc =- {cn}: C(K) 0 C(K) is an isomorphism associated with y.

The arbitrariness in the definition of C(K) consists of

a) the choice of the base point e?,

(10.3) b) the choice of y: 7ri(K)-r,

c) the choices of the bases {c"n} and of the basic modules C., (K).

Let another 0-cell e10 c KO be taken as base point and let ki and C(K1) be the corresponding universal covering complex and chain system. Let

a':7r, (K, e1?)7r,(K, e?), +: Ki1K-1f

be the isomorphisms 27 determined by a path (I, 0, 1) -* (K, e, e,0). Let h: C(Lf) C 0(K) be the isomorphism induced by 4. Then h is obviouslv associated with X, and the same system, C(K), is defined by

26 Here we reserve the symbol C (K) for a system in which C,, (K) is a basic module in M, as defined by (10. 1), (10. 2) below. We no longer insist that the elementary chain c0 e Co (K), associated with a e-cell which covers e?, shall be associated with the base point in K.

27 We use the symbol to denote the relation of isomorphism between complexes as well as between groups and chain systems.



SIMPLE HIOMOTOPY TYPES. 35

a s: 7r, (K, e1?) r, kch: C (:Z1) C C(K),I

as by a and 7k. The effect of choosing a different path, (I, 0, 1) -* (K,eO, e10), is to replace a, h by 06x, xh, where x e wi(K, e?). Since

x (r,Cn1 + ? * * + rp.Cn P) = (Oxrl) XCni + ? * * ( Oxrv.) xCnV.

it follows that the resulting alterations, -ya -* yGx and kch -> k7xh, are included in (b) and (c).

Let -y be replaced by y': 71r(K) r and let 0 6 y'y-1r: r ~r. Then it follows from (10. 1) and (10. 2) that kn is replaced by s0k1n and 0 by 00 s00so0'. Therefore C (K) is replaced by the conjugate system CO (K).

Any other natural basis for Cn(k) is of the form (? x,Cnl,. * * *,4+PnP.), where xi c 7ri (K). Any other basic module of rank pn is of the form P'Cn, where P: M -> M is a permutation. Therefore a change in (c) leads to a new system C'(K) --C(K) (C).

Therefore C (K) is determined up to a transformation, C (K) -* C'(K), which is the resultant of a semi-linear transformation, C (K) -> CO (K), followed by a simple isomorphism CO (K) C'(K) (E)).

Let K'_K and let k': C((K) OC(K') be defined in the same way as kc: C(k) C(K), in terms of an isomorphism y': 7ri(K') r. Let g: C(K)- CO(1) and a: 7ri(K) 7r1(K') be the chain equivalence and the isomorphism induced by a homotopy equivalence 28 0: K - K'. Let

(10. 4) f= k'gkl': C(K) -*C(K'), a ya-y-1 : r r.

Then it may be verified that f is a chain equivalence associated with 0. We describe it as the chain equivalence induced by 4 and we define r(4) =-r(f). Let g*: C((k) = C(K') be the chain equivalence induced by a homotopic map 4)* 4) and let f* k'- lg*7lr. Then g* g and it follows that f* f. Therefore (o*) = 7()). Hence, and by the two preceding paragraphs, T7(4)

depends only on the homotopy class, +, of maps K -* K', which contains 4, on y, y' and on the choice of base points 29 in K, K'. We define r(Q)-T(0) Let -y and -y' be replaced by 71: 7r- (K) r P and y'i: 7r, (K') r I. Let 0 = yly'l, 0' = -y'l'y. Then 7k, kc' are replaced by s0lc, so,lc' and f by

so0fso-'. Therefore it follows from (9. 4) that r(+) is replaced by 6'r(+). Let us write 7 =,r7 where T, T C T, if, and only if, T' =B T, where

28 All our maps and homotopies of complexes will be cellular and it is always to 'be understood that a given map, K -E K', carries e? into e'?, where e? e K'? are the base points.

29 Actually T (0) does not depend on the choice of base points since 0 T = r for any ,yer, TeT.




6: r r. Obviously r =' is an equivalence relation and we shall describe the corresponding equivalence classes as 0-classes. It follows from the preceding paragraph that the 0-class, T(+), which contains r(i), is uniquely determined by the homotopy class +. We call it the torsion of +, or of any map q cq.

We shall describe q: K K' as a simple (homotopy) equivalence, and shall write p:K K'(>), if, and only if, T(p) =0. We shall say that K,K' are of the same simple homotopy type, and shall write K = K' (Y,), if, and only if, there is a simple homotopy equivalence 4: K IK' (X).

Let q': K'= K" and let C(K") Z C(K") be defined in the same way as C(K) and C(K'), in terms of an isomorphism y": 7r(K") Fr. Let 4/ be associated with x': ri(K') w1(K") and let 0'- =' rty'1:P 1. Then it follows from (10. 4) and (9. 5) that

(10. 5) 'rQ(0) =rp) [+ 0r(f).

Therefore, if p, 4/ are simple equivalences, so is +'4. Obviously r(s) 0 if i/i 1: K -> K. Therefore, taking K" K and p'4 1, it follows from (10. 5) that a homotopy inverse of a simple homotopy equivalence is itself a simple homotopy equivalence. Therefore K =K' (X) is an equivalence relation.

Let GK be the aggregate of homotopy classes, ,, * , of homotopy equivalences, p, v,* , of K into itself. Let Q+ - . Then GK, with this multiplication, is obviously a group. Let e0 e K0 and Ic: C(K) C(K) be fixed and let (MK be the sub-group of the groupoid (, which consists of the chain homotopy classes of chain equivalences C(K) -C (K). Let fq f, where f is given by (10. 4), with K' I K, Y' Y y, X' = k, and let To ? ( be the class which contains fo. Then + -> 70 is obviously a homomorphism of GK into OK. Let TK: GK->T be the map which is given by TK(+) =r(+). It is the resultant of + -- 70, followed by the crossed homomorphism J -> r (J) . Therefore TK is a crossed homomorphism, in which GK operates on T according to the rule +r fr.

Let us take r = 7r(K, po), where po - K, and let y: 7ri (K, e0) 1r be the isomorphism determined by a path in K, which joins po to e?. Then the degree of arbitrariness in y is that it may be replaced by 6y, where 0: r r is an inner automorphism. In this case fp is replaced by fq/ and TK by T,: GK-> T. But 6- ~-, according to (2. 11), whence 6EK i-EK. Therefore

7- is uniquely determined by K when r = 7r, (K, po). For the reasons given in discussing (10. 3), TK is independent of the choice of e?.

Let KO be a connected sub-complex of K, which contains e? and is such




that i: 7r1(Ko) t 7r, (K)) where i is the injection homomorphism. Then Ko p-1Ko may be taken as the universal covering complex of Ko, where p: -> K is the covering map. Let C.(I?o) c C,,(E?) be the sub-module consisting of the n-chains carried by Kfo and let 7cn mean the same as in (10. 1). A natural basis for Cfl(1?o) is part of a natural basis for C0(1i) and it follows that C(Ko), with

(10.6) Cn(Ko) =1c7CnC(A7o),

is a sub-system of C(K). Let U = K - Ko and let us denote the residue system C(K)- C(Ko) by C(U) ==C(K)- C(Ko). Let rTa p-=1U and let Cn(tU) C Cn(k) be the sub-module consisting of the n-chains carried by U. Then obviously Cn ( U) = 1Cn (CT).

When dealing with such a pair of complexes K and Ko C K we shall always assume that C(Ko) is imbedded in C(K) in the way described above.

Let Ko C K, Lo C L be s-Lb-complexes of given complexes K, L. Let (:(K, Ko) -> (L, LO) be a map such that (jK - Ko is an isomorphism onto L - Lo and let 00: Ko -> Lo be the map which is induced by (.

THEOREM 30 1. If (P is a simple equivalence, so is (.

Let h: C(K) ->C (L) and f : C(Ko) ->C(Lo) be the chain mappings which are induced by ( and (P. Then it is obvious that hC(Ko) C C(LO) and that f is the chain mapping induced by h. Since (P K - Ko is an isomorphism onto L - Lo it is also obvious that g: C(K - Ko) C(L - LO) (Y), where g is the chain mapping induced by h. Therefore the Theorem follows from Theorem 10.

As an application of Theorem 11 let 00: Ko Lo, where Lo consists of a single 0-cell, let L be formed from K by shrinking Ko into the point Lo and let (P:K -> L be the "identification map." Since 7r1(LO) 1 it follows that (o is a simple equivalence and so therefore is (. In particular we can take Ko C K1 to be a tree containing KO. Then LO consists of the single 0-cell Lo.

11. Combinatorial invariance. In this section we prove:

THEOREM 12. If K' is a sub-division of K the identical map, i: K -> K, is a simple equivalence.

30 Cf.. Theorem 12 in Section 8 of (I). Obviously bo: Ko _ Lo(z) if Ko0= e?, Lo,= e'?, where e? e KO, e'? e LO are the base points. In this case the theorem states that




Let P be a given complex and Q C P a sub-complex, which is a D. R. of P.

LEMMA 10. If every circuit in P Q is contractible to a point in P, then the identical map, Q ->P, is a simple equivalence.

We first prove the theorem, assuming the truth of the lemma. Let 4': K' L, where L is a new complex, which does not meet K. Let

= <b'i: K -> L. Then i = {'-l and p'-1:L = K' (X) according to Theorem 11. Therefore it is sufficient to prove that 0: K L (X). Let P be the mapping cylinder of tf. We regard P as K X I, with (x, 0) = x (x e K) and K X 1 sub-divided to form L. Let e0 be a principal cell (i. e. one which is an open sub-set) of K and let K1 K - eO. Proceeding by induction we define a sequence of sub-complexes

K Ko, K1,j . Kn K-1,

such that KxA+1 = x- ex, where ex is a principal cell of Kx. Let Px= K U (Kx X I). Then Pl+ is a D. R.31 of Px and PxP+1 is the point-set ex X (0, 1>, where (0, 1> is the half open interval 0 < t? 1. Therefore 7r (Px - Px+1) ) 1 and it follows from . Lemma 10 that ix: =Px,j Px (X), where ix is the identical map. Therefore

i=~ iXo . . .

in-,: P,, K ==P (X)..

Similarly k: L P (-), where k is the identical map. Let i: P -> ,

be given by + (x, t) - x. Then b: P -L (E), since ji is a homotopy inverse of k. Therefore p =j: K = L (X) and Theorem 12 is proved.

It remains to prove Lemma 10. Since Q is a D. R. of P it is easily proved that the chain system C(Q) is a D. R. of C(P) and that a retraction : P -> Q induces a retraction k: C(P) -C0(Q). We have to prove that ic

is a simple equivalence and this will follow from Theorem 4, Section 5, when we have proved that C(U) O0 (X), where U=- P -Q.

Let Ui, * , Ur be the components of U, which are finite in number since U is the union of a finite number of (connected) cells. Let P be the universal covering complex of P and let p: P -P be the covering map. Let Ux be any component of p-1Ux and let U* Ui U J * U m. Since P and P are locally connected, Ux and 1t7 are open sets. Let e, e q be the n-cells in U. It follows from the condition on the circuits in U, which is satisfied a fortiori by the circuits in Ux, that p I O7x is a homeomorphism onto Ux. Therefore p I U* is a homeomorphism onto U. Therefore

31 See Theorem 1. 4(ii) in [16].




U* contains precisely one, en,, of the cells in P, which cover ent. Let Cne c Cn(P) be the element which is represented by a characteristic map for jn,. Then (Cn_, * *, cnqn) is a basis for Cn (U), which is part of a natural basis for Cn(P). Moreover Cn(U*), which consists of the n-chains carried by U*, is the ordinary free Abelian group, which is freely generated by Cn1. n c,,, without the help of the operators in 7rl(P).

Since each component of UC is open it follows that no cell in 17 U meets the closure of jn. Therefore

qn-1

( 11. 1 [) =aCn E dijnCjn-1 + c',n-, J=1

where dijn are integers and c'in I Cn(Q) (Q p-1Q). Let m nC where kn1 means the same as in (10. 1). Then it follows from (11. 1) that

,qn-.t

O:GC(U) -> C(U) is given by OM in T nMjn-1 Therefore C(U) ! O (:), j=1

by the corollary to Lemma 4, in Section 6. This proves Lemma 10.

12. Lens spaces. By way of an example let A, B be the chain systems determined by lens spaces of types (in, p), (in, q), where m is the order of their fundamental groups and q = k2p (n). That is to say, A, B play the part of C(K) in Section 10 and 7r1(P), 7ri(Q) in Section 15 of CH1(II) are both replaced by P. The generators $, - in CH (II) are replaced by a generator y e r and we denote the integer r by h, to avoid confusion with r ? R. Other- wise the notations will be the same as in CI (II). Thus a: A -- A and a: B -> B are given by

a , (-y - 1)ao, Oa2 crm(y)a, aa3 (yI 1)a2

(12. 1) ).Ob, (y- 1)bo0 ab2 =qm(y)bi, ba = (yq - 1)b2,

where t(y8) ==1 + y8 + * * * + y(t-1)8

The Reidemeister-Franz torsion in A and B is rA and rB, where

(12. 2) rA (y-1) (yP- ), rB (y1-) (Q --1)

Let : r P r be given by y-y y and let u: A->B, v: B-A be the chain mappings, associated with 0 and with 0-1, which are given by uac, b=

vbn an if n 0 or 3 and

(12. 3) { ua1 = 7k (y)b1, ua2 Uk(ykP)b2

vbl 1= (y) a,, vb2 o (-ylq) a2,

where kl 1 + mh. As shown in CH1(II), vu-1=a + qa, uv-1



40 T R. C. WHITEHEAD.

-= a + -qO, where yan = ?7bn 0 if n -7 I and -ral = ha2, 'qb, hb2. Notice that 7m 0 and so so-q, since an, b. are generators, Min, rnhjn of M.

It follows from (12. 1) that

(12. 4) 00a (-yk - 1)aao, ea2 m(y)a,, OOa3 (ykp _1)a2,

and from (12. 3) that

(12. 5) sovb, = o, (yk) a,, sovb2 u(-yq)a2,

since 1 - 1(m). We proceed to calculate T(U) T{CO(uSO1)}, where C(f) means the same as in Section 8. Let C = C (us0-1) and let a", aa= , where a means the same as in (8. ?2). Then (ba, a',-,) is a basis for Cn (a' = b4 0). Since uso-a'n ua, it follows from (8. 2), (12. 3) and (12.4) that 0:C -OC is given by

b01 (y 1) bo, Oa'o bo 0b2 U,n(y)bj, Oa',=oa k(y)b_- (yk 1)a'o ab3 (yq-1) b2, aa'2 =ok(yk) b2-oamr(y)a'l

aa'3 b3 - (ykp l1) a'2

It is easily verified that fA 0, where u is given by (8. 6), and similarly that sov71 = -qsov. Since q-q 0 a straightforward caluculation shows that 88 0, where S is given by (8. 7), with S = . It follows from (8. 7) and (12. 5) that Sbo a'o and

Sb - hb2+ ? I (y7l) a',, *8a'O = 0

Sb2 (/q) a'2, &a/' = ha'2

Sb3 a'3, Sa'2 0.

Let Do =-- Co ? 02 + C4, D1 = C, + C3, as in (6. 8) with m = dim C-1 3. Then Do, D, have (bo, a',, b2, a'3), (a'o, b,, a'2, b3) as bases and

A0 = + 8: Do -> D1 is given by

Aobo = a'o

oa' 1 - (7_/)ka'o + ? k(y)bI + ha'2

Aob2 =m(y) bi + a1 (yq) a'2 Aa' (yP -1)a'2 + b3.

Let fo: Do Do, fl: D, D, be the simple automorphisms which are given by

foa= a', + (yk 1)bo, foc = c (c = bop b2,a'3) f1b3 = b3 + (ykp - 1)a'2, f1c c (c = a'0, bl, a'2).




Then A'O = fApfo: DO -- D, is given by z'ObO a'O, A'Oa'3 b3 and

A'oa'1 = kO('ky)b, + ha'2

zVob2 crm (y/) bi + a, (yl) a'2.

Let i, j be any integers and let p = (i, j). If i < j we have, writing as Us (ey) (ao= 0),

ji - Oyj-i j-i.

Therefore it follows by induction on i + j that the matrix [of, Jj] can be reduced to [up, 0] by a -sequence of transformations of the form

[i, Oj] -> [, j - ySd or [0f yj - 0j]]

followed, if necessary, by [O, up] -- [up,O]. Since (k, m) = 1 it follows that

[mk(Y) h ] [O t] (r,t C R)

by such elementary transformations of the rows. Since these transformations alter the determinant by a factor + 1, at most, we have t + d, where

d ak (,I),u (7a) m() I say that

(12.6) drB OrA,

where rA, rB are given by (12. 2). For let X be the homomorphism of R into the complex field, which is given by Xr = w, where On = 1. Then

X(drB) X(OrA) 0 if o 1, and if w #1 we have

x (drB) N(k 1 <)q )/( @ )] -1 1) (a - (<sk1) (@Iq ) ==X(OrA).

Therefore X (drB rA) 0 and (12. 6) follows from the orthogonality relations between the group characters. Therefore r (u) is essentially the same as the inverse of the element 7r in Lemma 5 on p. 1209 of [3].

13. Formal deformations. As in CH(II) let In be the n-cube in Hilbert space, which is given by 0 ? t, . . ., tn < 1, ti 0 if i > n, with IO= (0, 0, ). Let

Eon-1 In -(In- -aIn-1) (n? 1).

Let en be a principal cell of a complex K and let en-' be a principal




cell of K - en (n ? 1). We shall describe the transformation K -* K1 K K- e- en-' as an elementary contraction if, and only if, en has a

characteristic map, f Jn > e, such that fEor-l C K1 and f In- is a characteristic map for en-1. The inverse, K1 -* K, of an elementary contraction, K -- K1, will be called an elementary expansion. An elementary expansion may also be defined as follows. Let En be an n-element, which is disjoinlt from a given complex, K, and let En-' be a hemisphere 32 of aEn. Let en E- _ aE, e- = aEn- En-' and let f: (En-1, aEn-1) -* (Kn-, Kln-2) be an arbitrary map. Let K1 K U en-1 U en be the complex formed by identifying each point p e En-1 with fp e Kn-1. Then K -* K1 is obviously an elementary expansion.

Either an elementary expansion or an elementary contraction will be called an elementary deformation. The resultant, Ko0-> Kr, of a filite sequence of elementary deformations,

(13.1) K-- Kj+j (i =On o r - 1),

will be called a formal deformation. We also include the identical transformation, K --> K, of any complex, among the formal deformations. We shall denote a formal deformation by D: Ko -* Kr, and Kr = DKO will mean that Kr is obtained from Ko by a formal deformation D. If each Ks -*K+1 is an elementary expansion (contraction) then Ko * Kr will be called an expansion (contraction) and we shall say that Ko expands (contracts) into Kr. If D: Ko + Kr is the resultant of the sequence (13. 1), then the resultant of the sequence Ki+i -) Kt is the formal deformation, D-1: Kr -* Ko, inverse to D.

Let D: Ko -> K1 be an elementary contraction. Obviously Eon-1 is a D. R. of In. Therefore " K1 is a D. R. of KO and [D] will denote the homotopy class of maps, Ko -* K1, which contains a retraction. If D: Ko - > K1 is an elementary expansion then [D] will denote the homotopy class of maps, Ko0 - K1, which contains the identity. Let D = Drw- Do: Ko -* Kr, where

Di stands for (13. 1). We define [D] by [D] = [Dr-1] [Do]. If 1:K X->K is the identical formal deformation then [1] will denote the

homotopy class of the identical map K -K. It is obvious that, if D: K -> K' and D': K' -*> K" are formal deformations, then K" = D'DK and

(13. 2) [D'D] = [D] [D].

Also it is easily verified that [D--] [D] = [1].

32 I. e. the image of Eon-1 in some homeomorphism oIn - OEn.

33 See Lemma 2 in Section 4 of [4].




Let L be a sub-complex of K, which may be empty. By a formal deformation, D:K-* K', rel. L, we shall mean the resultant of a sequence of elementary deformations, none of which removes a cell of L. By K'I DK, rel. L. we shall mean that K' is the image of K in such a formal deformation, D. If K' = DK, rel. L, then [D] obviously contains at least one map 4c::K -K', rel. L. where rel. L means that p0y=y if yeL. We restrict [D] to maps, c: K -> K', rel. L, such that p -0, rel. L.

We shall describe a map c: K -> K', rel. L, as a restricted equivalence, rel. L, if, and only if, K'= DK, rel. L, and c e [D]. We shall write K = K' (E). rel. L, if, and only if, there is a simple equivalence

p: K IK' (E), which is rel. L.

THEOREM 13. K' = DK, rel. L, if, and only if, K K' (K ) rel. L, in which case the restricted equivalences, K -> K', rel. L, are the same as the simple equivalences, K -> K', rel. L.

In order to prove this we shall need some lemmas, which are proved in the following section.

14. Lemmas on formal deformations. Let K, K' be complexes, with a common sub-complex, L, which may be empty, and let (K - L) n K'= 0. Let 4: K -+ K' be a map rel. L. By the mapping cylinder of p we mean the complex, P, which is formed from 34 K X I by identifying (x, 0) with x, (x, 1) with ox and y X I with y, for each point x e K and y e L. Let

(14.1) Or: 7rr(K) Z rr(L) (r 1, . m),

where or is the homomorphism induced by p. Then the argument 35 used in the case L = 0 shows that

(14. 2) ir 7rr (K) ~7rr (P) (r , . .,m,),

where ir is the injection. Therefore it follows from the exactness of the sequence

7rr (K) ->7rr (P) 7rr (P, K) -7 r-1 (K) ->7rr-, (P) n

that 'rr(P,K) =0 for r 1 * ,m , where 7r1(P,K) 0 means that i1 is onto 7r,(P).

34 After replacing K X I by a homeomorph, if necessary, we assume that it has no point in common with K or K'.

S5 See Section 3 of [5].




LEMMIA 11. P contracts into K'.

Let K = Ko U en, where en is a principal cell in K - L. Then P Po U enU e+', where en+1 = en X (0, 1) and PO is the mapping cylinder of 4) Ko: Ko -* K'. Let f: In > jn be a characteristic map for en. Then g: In+1 > jn+1, given by g(t5, , . tn, t) = {fW(ti . . . tn), t}, is obviously a characteristic map for en+' and gEon C PO and gx = fx if x C In. Therefore P contracts into PO. Therefore the lemma follows by induction on the number of cells in K - L.

Let &: P -+ K' be given by

IKI / 1, 5 (X5 t) X (x cK).

Since q K' = 1 and since any two retractions P -> K' are homotopic to each other, we have ql e [D], where D: P -> K' is any contraction. Also

= ti, where i: K -> P is the identical map. If K = D,P, rel. K, then

i e [D1-']. Hence, and from (13. 2), we have the corollary:

COROLLARY. If K = D1P, rel. K, then 4 e [DD1-1].

Let K, K' and L be as in Lemma 11, except that K - L and K'- L may now have points in common. Let c: (K, L) (K', L), rel. L.

LEMMA 12. c is a restricted equivalence, rel. L.

First let K n K' = L and let P be the mapping cylinder of c. Then P may also be regarded as the mapping cylinder of p-1 and the Lemma follows from Lemma 11 and its corollary.

If K'- L meets K we replace the points in K' - L by new ones, thus forming a complex K", such that 0': K" K', rel. L, and K n K" = n K" = L. By what we have already proved, f' and 4)`10: K I K" are restricted equivalences, rel. L. Therefore it follows from (13. 2) that tp is a restricted equivalence, rel. L, and the lemma is proved.

Let Ko, K1 be complexes with a common sub-complex, K, and let K =K U e, (i= 0 n1; 1). Let f,:In-.*e,n be a characteristic map for ein in K,.

LEMMA 13. If fo I Ohn f, I aln in K, then K1 = DKO, rel. K.

First assume that eon n eln = 0 and unite Ko, K, in the complex K* =Ko U K1. Let gt: 0In ->K be a homotopy of go tfo 0In into

= f, I 0Tn and let f: oIn+1 -+ K* be given by




f ( tin ** tnn, i) = ft ( tin ** tn ) {i = O, 1; ( ti, * tn ) ? I n}

f (Si,** Sn, t) gt (Si,** Sn) {t e I; (Si, * S.) ? al"} .

We attach a new cell, en+1 . to K* by means of the map 36 f, thus forming a complex, L = K* U en+i, in which e1+1 has a characteristic map, h: In+,- .> jn+l

such that h I OIn+ I f. Since h(x,0) =fox (X In) and37 hEon C L-eon it follows that L -- K1 is an elementary contraction. Similarly L -> Ko is an elementary contraction. Therefore K0 -* L -> K1 is a formal deformation, rel. K.

If eon n e1n .7 0 we attach a new cell, e'on, to K, by means of the map

fo I 01n, taking care that e',n ne en 0 (i O 0, 1). Then Ko -> K U e'0 8- K, is a formal deformation, rel. K, and the lemma is proved.

Let P be a given complex, let PO C P be a sub-complex and let kn(P - PO) denote the number of n-cells in P -PO. Let K be a sub-complex of PO and let Do: PO -> Qo be a formal deformation, rel. K. We shall describe a formal deformation, D: P -- Q. rel. K, as an extension of Do if, and only if, QO is a sub-complex of Q and

kn (Q -Qo) -- n (P -Po) (n 0, 1 * )...

LEMMA 14. Do: Po -> Qo has an extension D: P -> Q. Let D: P -Q be an extension of Do and let D': Q-> Q', rel. K, be an

extension of a formal deformation D'o: Qo -> Q'o. Then D'D P -> Q' is obviously an extension of D'oDo: PO -* Q'O. Let P1 C P be a sub-complex, which contains PO. Let D1: P1 -> Q,, rel. K, be an extension of Do and let D: P -- Q be an extension of D1. Then D is obviously an extension of D,. Therefore the Lemma will follow by a double induction on the number of elementary deformations in Do and on the number of cells in P- PO when we have proved it in case Do is an elementary deformation and P -P0 is a single cell.

Let P = Po U en and let Do be an elementary expansion Do: Po -Qo = P,, U ep-' U ep. If en -has a point in common with ePm" U eP we apply a preliminary formal deformation, P -* P', rel. PO, as in Lemma 13, so as to replace en by a cell which is disjoint from eP-' U eP. Then P' and Qo may be united in a complex

Q = P' U eP-' U eP

36 That is to say, we attach an (n + 1) -element, En+1 (en+1 = En+i- aEn+1), to K* by means of the map fh': OEn+l > K*, where h': OEn+1 -- OI1+1 is a homomorphism.

37We recall that Eon+1= 1n+l- (In A-I).




and P -* P' -> Q is an extension of D,.

Let Do be an elementary contraction,

Do: PO -Q = PO - eP -e

and let f: In - ll be a characteristic map for en. Since QO is a D. R. of PO there is a map, f': 0h -*> QO, which is homotopic, in PF, to f I 0ln. We attach a new cell, en , to PO by means of the map f', thus forming a complex P' = Po U e'l. Then P' = D'P, rel. PO, by Lemma 13. Since ae'n C QO it follows that P'-? Q = F' - eP - eP-1 is an elementary contraction and P -* P' -* Q is an extension of DO. This proves the lemma.

Let K be a (connected) sub-complex of P such that 7r. (PF K) = 0 (n=1 ,r.

LEDMMA 15. There is a formal deformation D: P -> Q, rel. K, such that 7kn(Q- K) =0 if n?rand n (Q -K) = kc(P- K) if n>r+2.

Let 0 < p ? r and assume that, if p > 0, then knc(P -K) 0 for n = 0, , p - 1. For the sake of clarity we consider the case p= 0 separately. Let p =0 and let e10y *, ek? be the 0-cells in P - K. Since P is connected there is a map

g: (Eol, I, Eo?) .. (P', ei0, KO).

Let E12y * E*,k2 be a set of 2-elements, which are disjoint from P and from each other. Let hi: 0E 2 __> I2 be a homeomorphism and let

ei2 =E 2 _ - E,2 e.l hr-' (I' h1-) _- (OI2 E01)

We attach E 2 to P by means of the map gihi: hi-'E01 -p* F, thus forming a complex P* P u L I (eil U et2). Then P -> PF is an expansioni. The

complex K* =Kt Ul~J (ei? U eil-) contracts iiito K. By T emma 14 there

is an extension, P* -> MO, rel. K, of the contraction K* -> K. Then

ko (Mo-K) -o (P* K*) 0

1nc(Mo- K) = kn(P*- K*') =7kn(P -K) (n > 2).

Thus we have eliminated the 0-cells from P - K at the expense of introducing

A-,(P K) new 2-cells. Now let P > 0, let e1P, . . . , ekP be the p-cells in P -K and let

fr IP -4 je be a characteristic map for eiP. Since kp-, (P - K) = 0, whence




PP-1 = KP-1, we have f%OIP C K. Since 7rp (P, K) 0 and since IP, EOP are two hemispheres of OEoP+_ - aIP+l, the map fi can be extended to a map,

(14. 3) gi: (EoP+l, EoP) .-> (PP+1, KP).

It now follows, in exactly the same way as when p = 0, that there is a complex Mp = DrP, rel. K, such that k1, (Mp - K) = 0 if n < p and knl(Mp-K) =kn (P-K) if n > p + 2. Therefore the lemma follows by induction on p.

15. Proof of Theorem 13. Let D: K -X K, = K U en-l U eU, (n ? 1) be an elementary expansion. Then i ? [D], where i: K -> K1 is the identity. Also K is a D. R. of K, and K1 - K is simply connected. Therefore it follows from Lemma 10 that i: K -> K1 is a simple equivalence. Since k: K1 -> K is a homotopy inverse of i if k c [D-1], it follows that k is also a simple equivalence. Therefore 4: K -> DK is a simple equivalence, rel. L, if 4 c [D], where D is any elementary deformation, rel L. Therefore it follows from an inductive argument that 4: K -= K' (Y.), rel. L, if K' = DK, rel. L, and 4 s [D].

Conversely, let 4: K =- K' (.), rel. L. Then it follows from Theorem 11 and Lemma 12 that we may, without loss of generality, replace K' by K", where K" K', rel. L. Therefore we assume that K n K' = L and also that e? = e'? P L, where eo ? KO, e'? c K'0 are the base points, thus excluding the case L 0. Let P, with base point e?, be the mapping cylinder of 4. Since 4: K=K' the relations (14. 1), (14.2) hold for every n? 1. Also K' is a D. R. of P. Therefore i'1: 7r1 (K') m,r (P), where i', is the injection. Therefore C(K), C(K') are sub-systems of C(P), according to the convention (10. 6). Let j: C(L) -> C(P) be the chain mapping induced by the identical map L -4 P. Then A = jC(L) = C(K) n C(K'). Let h: C(K) C-> (K) be the chain mapping induced by 4. Then h is obviously rel. A, since 4 is rel. L. It may be verified 38 that C(P) is the mapping cylinder of h, as defined in the paragraph containing (8. 4). Since, by hypothesis, h is a simple equivalence we have

(15-1) C(P-K) O 0 )

according to (8. 4).

38 See Section 14 of CH (II), with ye'f = 0 if c;<" corresponds to a cell in L c K (not the L in CH (II)).




It follows from (14. 1) that 7rr(P, K) = 0 for every r ? 1. Let

(15. 2) q Max{dim(P-K), 3} = Max{dim(K-L) + 1, dim(K'-L), 3}.

Then it follows from Lemma 15 that there is a complex Q - D1P, rel. K, such that

(15. 3) Q K U eig-1 U . . . U esg-l U eig U . . . U etq.

By the first part of the Theorem, with K, K', L replaced by P, Q, K, we have C(Q) =C (P) (X), rel. C(K). Therefore it follows from Theorem 4(a) and (15.1) that C" C(Q -K) =C(P -K)-0 (), whence 8"q: C"q C"q C (X), by the corollary to Theorem 5. Therefore s =- t. Let

a"M,q = d jMjq-, (dje R), j=1

where (Mlr,* , rm8) is the basis of C"r (r = q 1, q). Since r(8"q) =- 0 the matrix d [d,j] can be annihilated by an expansion

rdo

followed by a sequence of elementary transformations of the form (2. 12), followed by a contraction 1s+k --> 1, where 10 is the empty matrix. Since q - 1 ? 2 it follows from arguments on pp. 289, 290 of [1], with minor alterations,39 that the transformation d -> lo can be "copied geo- metrically" by a formal deformation Q -> K, rel. K. Therefore K = D2P, rel. K, and the Theorem follows from the corollary to Lemma 11.

Let us describe r as the order of an elementary expansion K -> K1 =K U eri U er, and also of its inverse, K1 -> K. Let p: KN K' (X), rel. L and let KP, K'P C L, for some p 2-1. Then the following addendum to Theorem 13 is implicit in the proofs of Lemmas 11-15 and of Theorem 13.

ADDENDUM. K'= DK, rel. L, and q e [D], where D is the resultant of elementary deformations, whose orders lie between p + 2 and q + l inclusive, where q is given by (15. 2).

This addendum has the following application. It follows from Theorems 11 and 13 that, by means of a formal deformation, we can reduce a given complex to one which has a given point, e?, as its only 0-cell. It is some-

39 Let r(a,, . . ., a,) mean the same as in Theorem 19 of [1], with K* = Qq-1 and = s. Then mj1- - aj determinies an isomorphism CG",_ -r (a, a , .).




times convenient to restrict ourselves to a class of complexes, all of which have the same point, e?, as their only 0-cell. Let K0, K, be two such complexes and let Kr = PKO. Then it follows from Theorem 13 and its addendum, with p-- 0, that [D] = [D'], where D': Ko ->. K, is the resultant of elementary deformation, Ki --> Ki+1, whose orders exceed 1. Therefore Kj? --e? for each j=O,* *,r.

16. n-types. By a cluster of n-spheres, attached to a space X, at a point xo ? X, we shall mean a set of n-spheres, {S,n}, such that X n S,n = XO and Sin - xo does not meet Sjn if i $/ j. If X is a complex and Xo ? Xn-1

then X U {Sil} is the ' complex X U {ein}, where etn = S -n - xo. At this stage we assume that, X being a finite complex, the number of n-spheres in a cluster attached to X is finite.

Let Kn, Ln be complexes of at most n dimensions (n > 1) and let >: Kn -* Ln be an (n - 1) -homotopy equivalence, as defined in Section 2 of CI (I).

THEOREM 14. There is a simple equivalence,

V':Kn U {Slin} - Ln U {S2jn} (s),

such that A/x = ox if x ? Kn-l, where {S1l"}, {S2jn} are clusters of n-spheres attached to Kn-1, Ln-.

Assuming that Kn n Ln 0, let P be the mapping cylinder of q. Then Pn is the union of Kn, Ln and the cells er X (0, 1), where er 8 Kn-1. We attach a cluster of n-spheres,

{S2Pn} = S21n U * U S2k0,

to a 0-cell e'0 e LO, where k is to be determined later. Using Lemma 13, we transfer these over Pn to a 0-cell of Kn, so that they become a cluster,

{Spn}, attached to Kn-1. Since : K= n-Ln it follows that (14. 1) and (14. 2) are satisfied with m = n - 1. Therefore

(16.1) a) 7rr(Pn,Kn) =0 if r=-1,* * *n -

( ) b) the injection, (K~) -> 7rn-(Pi ), is an isomorphism (onto).

These conditions are obviously satisfied by K* and P*, where

K* Kn U {Sp}, p* = pn U {SP}.

4




Therefore it follows from (16. la) and Lemma 15 that there is a formal deformation

D1: P -> Q- K* U e,n-1 U . . . U ean-1 U eln U U ebm, rel. K*.

Now let k= a. On considering the effect of a simple elementary deformation, rel. K*, it follows inductively that (16. 1) are also satisfied by K* and Q. Let

gp: (Eon, In-1) -* (Q, Pn-1) (p 1 * a)

mean the same as in (14. 3). Since gp 1 ain _ gp I OEon is homotopic in Q to a constant map, it follows from Lemma 13 that there is a formal deformation Q -> Q', rel. Kn, which replaces each Spn - e? by a cell, eb,pf,

with a characteristic map g'p: In -> pn such that g'p I aIn = gp I aIn. There- fore it follows, as in the proof of Lemma 15, that there is a formal deformation

D2: Q' Q =- Kn U elln U * U eb,n rel. Kn.

Let h: In eJ ,n be a characteristic map for e'tn. Then it follows from (16. lb) that ht i aIn is homotopic in Kn to a constant map. Therefore it follows from Lemma 13 that there is a formal deformation

D3: Q" -E Kn U Slin U U Slbn, rel. K",

where {Sj?n} is a cluster of n-spheres attached to Kn-1. On reversing these constructions we have

pn U S21n U * * * U S2,n = D (KA U Slln U * . . U Slbn), rel. Kn.

Let pOn C pn be the mapping cylinder of 4 j Kn-1. Then pn is the union of Pon and the n-cells in Kn. Let f: In en> gn be a characteristic map for an n-cell en? Kn. Then f I aIn is obviously homotopic in Pol to p (f l aIn) :ln - Ln-1. Since the latter can be extended to 4f: Im - Ln it follows that f I ain is homotopic in POn to a constant map. Therefore

P*o pon U S21n U . . . U S2 m=P'(pn U 21n U USS,kn), rel. pon,

where I ? k and S2 + , * ,S2 In are n-spheres, attached to e'0 e Ln, which correspond to the n-cells in Kn. It follows from Lemma 11 that

L*=L U S21 U . . .U S2n=-- D*Po*,

where Dl* is a contraction. Let i/*: Po* - L* be given by L* 1 - 1,




f* (x, t) = ox (x c Kn-1). Then * c [D*] and the conditions of the theorem are satisfied by a map 6p e [D*D'D]. This completes the proof.

It follows from this theorem that any two complexes of the same n-type can be interchanged by a finite sequence of elementary deformations and transformations of the form

(16.2) KL= K er, L ->K L U er (r>n),

where er is a principal cell of K. For the transformations K ->K" Kn U En1 -> Kn U Sn are the resultants of such sequences, where

Enl'-e0 U en U en+1, Sn -M1n =- eo U e", and Kn n Enl-e0sK0. Con- versely a formal deformation preserves the homotopy type, and hence the n-type of a complex. Also Kn = Ln, whence K, L are of the same n-type, if they are related by (16. 2). Thus the n-type may be defined in terms of formal deformations and elementary transformations of the form (16. 2).

It follows from Lemma 2 in Section 9 of CH1(I) and from Lemma 13 and Theorem 12 above that, if K is any complex, there is a simplicial complex K* = DK. Moreover, Sections 14-16 may be interpreted as referring to formal deformations of the kind considered in [3]. Therefore the class of simplicial complexes, which, when treated as cell-complexes, are of the same simple homotopy type, or n-type, as a given one, K, is the same as the "nucleus," or "n-group," of K, as defined in [1].

17. Homotopy systems.40 We proceed to the simple equivalence theory of homotopy systems. In this section we confine ourselves to systems, p, such that dim p < oo and each group pn has a finite basis.

We modify the definition of a homotopy system, p, by associating a class of preferred bases with each pn. Let (a,, * * *, ap) be a preferred basis for pn. Then (a'1, * , a'p) shall be a preferred basis for pn if, and only if, a'-i aj1, in case n-1, or a'- = xaj, if n > 1, where xi epi and

jl, * *

jp is a permutation of 1I , p. We shall only admit that f: p p' if f carries a preferred basis for each pn into a preferred basis for p'n. The preferred bases for p(K), where K is a complex, shall be the natural bases, as defined in Section 5 of CH (II). If KO is a single 0-cell, then the natural bases for pi(K) are uniquely defined. In general they depend on the choice of a tree T C K1, which contains KO. In this case p (K) is the homotopy

40 The main purpose of this section is to prove Theorem 17, which was anlounced in Section 7 of CH(II).




system of the pair, (K, T). However we shall continue to write it as p (K). A complex, K, will be called a (geometrical) realization of a given system, p, if, and only if, p p (K), subject to our condition concerning preferred bases. The process of realizing p by K will consist of defining a particular f: p (K). Having (implicitly) done this, we shall use p and a c p to denote p (K) and fa. By a basis for pn we shall always mean a preferred basis.

Let C and h:p-*C be defined as in Section 8 of C11(II). Then C shall be a chain system of the kind introduced in Section 2 above, B being the group ring of p,/dp2. We insist that, if (a, , a.) is a (preferred) basis for pn and if (rn'1, * * , m' ) is the basis of UC, then ha, = ? rn,'jM,

where ft ? Tp and xT = 1 if n - 1. We are going to define a sub-system, p', of a homotopy system p. This

is not quite so simple as in the case of chain systems, for the following reason. Let p'i C p, be the sub-group generated by part of a basis for p,. Let

p'2 C p2 be the sub-group generated by p'l, operating on a set of elements, (a', * **, a'-), in a basis for p2. Let dp'2 C p', and let d': p'2 - p'i be the homomorphism induced by d: p2 -- p,. Then p'2 is not necessarily a free crossed (p'i, d')-module. For example, let p' have a single free generator, x, and p2 a pair of basis elements, a, b, such that da = x, db = 1. Let p'1= p' and let p'2 be generated by p1, operating on b. Since db =1 we have a + b = b + a, whence xb-b = a + b-a - b =0 . Therefore P'2 is not a free p1-module.

Let -' = p'/dfi'2 and let i*: p-' j, be the homorphism induced by the identical map i1: p'i -> p,.

LEMMA 16. Let i*-1(1) = 1. Then p'2 is a free crossed (p'i, d')-module, having (a'1, - - , a'7) as a basis.

Let P2 be the free crossed (p'i, d")-module, which is defined in terms of the symbolic generators (x', ci) and the map ac -* da', (i = 1, ,;

XI p"). Obviously d'p"2 - dp'2. Let a"'i P"2 be the basis element which corresponds to the generator (1, a+). It follows from Lemma 2 in Section 2 of 0H1(II) that an operator homomorphism, i2: P"2 -- P2, associated with i1: p'i -> pl, is defined by i2a"i = a'i. Obviously i2p"2 =- P'2 and the lemma will follow when we have proved that i2- (0) = 0.

Let C2 = hP2, 02 = h-"p"'2 be P2, P"2 made Abelian and let j: C0'2 C2 be the homomorphism induced by i2. Since a' i2ad' and jh" = hi2 it follows that (jh"a"1, * . . , jh"a'k) is part of a basis for C2. Since d"p"2 =- dp'. and i*-1(1) = 1 it follows that j-1(0) = 0. Let. a" e i21(0). Then




'IEdfla' = di2d" =- 1, h,"a" hita" = 0. Therefore d"a" = 1, h"a" = 0 and it follows from Lemma 1 in CH1(II) that a" = 0. Therefore i2- (0) =0

and the lemma is proved. Let p'i, P'2 satisfy the conditions of Lemma 16. Let p'p C pp (p 3,

4, * ) be the sub-group which is generated by -1 operating on part of a basis for pp, and let dp'p C p'p-1. Let d': p'p -> p'p-l be the homomorphism induced by d: pp -* pp-1. Then p' = {p'p}, with d' as boundary operator, is a homotopy system, which we describe as a sub-system of p, on the under- standing that a (preferred) basis for p'n (n ? 1) is part of a basis for pn-

Let p be a given homotopy system, let Z. (p) = d.-1 (0) and let 41

Gi(p) =p-,l Gn(p) Zn(p) - dn+ipn+l (n> 1).

A homomorphism, f: p -> p', obviously induces a family of homomorphisms f*: G.(p) -> Gn(p') (n =- 1, 2, * ). It may be verified in the same way as in ordinary homology theory that f*: Gn (p) Gn (p') if f: p -p'. The converse is proved below.

Let p' C p be a sub-system and let Zn (p, p') = dnlpIn.1 (n > 1). Let a e Z2(p, p), a' e p'2. Then a + a'-a = (da)a' ep'2, since da e p'I. There- fore P'2 is an invariant sub-group of Z2 (p. p') . So therefore is the direct sum p'2 + dp,, since dp, C Z2 (p), which is in the centre of P2. Let

Gn(p, p') = Zn(p, p) - (p'n + dpn+l) (n> 1). Let

(17.1) Gn (p') Gn (p) Gn (p, p') > Gn- l(p') G (p) * j* d* i* i*

be the homomorphisms, which are induced by i: p' --* p, the identical map Zn(p) -- Zn(p, p') and by d I Zn(p, p'). Then it may be verified, as in ordinary homology theory, that the sequence (17. 1) is exact.

Let f: p - p' be a homomorphism of p into a system p', with boundary operator d'. Let f*: pl pt'. We proceed to define a system, p*, which we shall call the mapping cylinder of f. We realize the systems p2 = (pI, P2)

and p'2 by complexes K = K2 and K' = K'2, such that- KO =- I' e-

=K- K'. By Theorem 4 in CH1(II), :p2-_p'2 can be realized by a map 0: K -> K'. Let P be the mapping cylinder of p, with e? X I shrunk into the point el. Then P= e?. We define Pt=pn(P)=pn(P2) (n =- 1, 2).

Since p induces f*: jp' and since K' is a D. R. of P, it follows that

41If p=p(K) then G1(p) ir, (K), G2(P) ir, (K),On (p) Hn1(K) if n>2.



54 J. H. C. WHITEHBAD.

i*: p-1 ~ p*1, i*: p1 p*1, where i*, '* are the injections. Therefore it follows from the proof of Lemma 16 that the injections

(17. 2) i p2 > p(P2), i:p 2.__> p(P2)

are isomorphisms (into).

Let 8: p2 (> p(F) be the deformation operator determined by the homotopy St: K-P, which is given by Stp= (pr t) (p e K). Then

(17. 3) d*n_n =f l-1-1 Snldn (Sd= 0),

where n = 2, 3 and p*1 is written additively. Let n 2 3 and let Snpn-i be a free p*1-module, which is the image of pn-1 in an operator homomorphism, An, whose kernel is the commutator sub-group of p.,n Thus Sn prn 1 Snpn-1 n > 3. We take 8382 C p3 (P) and S3 shall mean the same as before. Let pen be the direct sum pen =pZn + pn + Snpn-1 (n_ 3). We imbed pn, p'n in pen by means of i: pn -.- p* i': P'n > p*n, where i, i' mean the same as in (17.2) if n =1, 2, and ia= (0, a, 0), i'a' (a', 0, 0) if n > 2. We define d*b: p*n >p*n1 by d*na- dna, d*nal d'n,a' and by (17.3), with n?2. If {at"} and {a'jn} are bases for pn and p',t then the union of {at"}, {a'jn} and {8nakn-l} shall be a (preferred) basis for pen. It follows from an argument in Section 8 above that d*nd*n+l O=0 if n 2 3. Also d*d* =O in p(P). Therefore d*nd*n+l = 0 for every n > 1. Clearly d*n is an operator homomorphism and it follows that p* - {p*n}, with d* - {d* } as boundary operator, is a homotopy system. We call it the mapping cylinder of f: p -> p'.

Let i': p' > p* be the identical map and let k': p -> p' be given by

ka =fa, kIa' a', k'8aa 0 (a p,a'ep').

Then k'i' = 1 and it is easily verified that d'k' = k'd* and that i'k' - 1 d*8* + S*d*, where 3*a - aa, *p' = 8 0. Therefore Ic': p* p' and

7c'*: Gn (P*) ~ Gn (p) . Clearly f = k'i, where i: p > p is the identical map, whence f* k'*i*. Therefore, if each f* is an isomorphism (onto), so is i*. In this case it follows from the exactness of (17. 1) that

(17. 4) Gn(p*,p)=O (n?1),

where Gi (p*, p) = 0 means that p*1 i*pl.

Let r ? 2 and let (ao,, al" , apnn) be a preferred basis for p,n

(n = r - 1, r). If r = 2 let p', C pi be the sub-group generated by (a1l, * ,a,,1.) and if r > 2 let p'= pi. If n > 1 (n =r- 1, r) let




P'n C pn be the sub-group which is generated by p'i operating on (a * , ap.). In any case let

daor aor -- at0 dar e pr (=Pr)1

where a'o P'r-, and pi is written additively if r = 2. Then dp4r C PIr-1

and the conditions of Lemma 16 are satisfied 42 by p'1, P22 Therefore p= {p',}, with p'n = pn if n z7 r - 1 or r, is a sub-system of p. Let i: p' -* p be the identical map and let kn: pn p'n the operator homomorphism, which is given by kn ip' = 1 and kraor 0, r laor-l a'o. Then it is easily verified that kd = dhk, whence 7c: p -> p' is a homomorphism. We have i =- 1 and ik -1 = d$ + $d, where 6: p -> p is the deformation operator

given by

ep' O, eaor 0, eaor-l aor.

Therefore i:p' = p and : p =p'. Notice that, if f: p ->p' is any homomorphism such that fi - 1, then f - fik h k.

We shall describe a homomorphism f: p0 -> pl as an elementary eqwi- valence if, and only if, p0o, pl are related to each other in the same way as p, p' in the preceding paragraph, and f = i or f _ k, according as p? C P1 or pI C pO. We shall describe a homomorphism f: p >p as a simple equivalence, fi: p =- po (s), if, and only if, it is the resultant of a finite sequence of isomorphisms and elementary equivalences.

Let C, C' be chain systems associated with given homotopy systems p, pi. Let g: C -* C' be the chain mapping induced by a homomorphism f: p -> p'.

THEOREM 15. f: p p'(>) if, and only if, g: Cd-C' (X).

This follows from the lemmas in Section 14 and the proof of Theorem 13, restated in terms of homotopy ssytems.

Let p be a homotopy system and a a free p1-module, with a finite basis (bjy *, bq). Let pn0 =p-n+ C, Pp pp (p #Ln), for a given value of n > 2. Let d? pr 0-pr-10 be defined by d I p =d, d O=0. Then43 pO = {pro}, with d? as boundary operator, is a homotopy system. We say that p --> p? is the result of attaching a cluster of n-cycles to p. If {at} is a preferred basis for pn, then {aj, bj} shall be a preferred basis for pn? and the preferred bases for pp (p #? n) shall be the same in p0 as in p.

42 The homomorphism k,: p, -- p'i, defined below, induces i*-1: p, pi v 43 If n = 2 then Pn? is a free crossed module since d%= 1.




Let dimp, dimp'<n (n?2) and let f:p -p' be a homomorphism siuch that

(17. 5) f*: Gr(p) ? Gr(p') (r 1 . ,n 1).

Then we have the following generalization of Tietze's theorem.

THEOREM 16. There is a simple equivalence, f?: p0 = p'? (:4), such that fOa = fa if a c pr (r < n), where pO, p'? are formed by attaching clusters of n-cycles to p, p'.

Since f*: - - j'1 we can construct the mapping cylinder, p*, of f. Then the theorem follows from the proof of Theorem 14, translated into algebraic terms.

The following corollary may be deduced from Theorem 16, or proved directly with the help of (17. 4).

COROLLARY. If dim p, dim p' ? n - 1, then (17. 5) implies f: p = p'.

THEOREM 17. If f: p (K) =p', where K is a complex, then p' can be realized by a complex, K', and in such a way44 that f has a realization 4:K = K'.

This follows from Theorem 16 and an argument which is essentially the same as the proof of Theorem 9 on p. 1228 of [3].

18. Infinite complexes. Let K1 be a CW-complex, as defined in CH (I), which may be infinite. Let Ko C K1 be a sub-complex such that K1 = Ko U U (ean-1 U ean), where {ean-1, ean} is an indexed aggregate of

a cells such that ea,n1 U ean is an open subset of K1 and Ko -.- Ko U ean-1 U eaa is an elementary expansion, for each a. Then K0 -* K1 will be called a composite expansion and K1 -.- Ko a composite contraction. It follows from the argument used in the finite case, and (I), in Section 5 of ClH(I), that Ko is a D. R. of K1. By a formal deformation, D: K -* L, we shall mean the resultant of a finite sequence of composite expansions and contractions. We restrict ourselves to complexes of finite dimensionality. Then the proofs of the lemmas in Section 14 and of Theorem 14 apply to infinite complexes, after a few trivial alterations.

We also admit homotopy systems of finite dimensionality, in which the

'" In general p' can also be realized by a complex, K', in such a way that f has no realization K -e K'.




groups may have infinite bases. We define a simple equivalence, f: p p', where p, p' are two such systems, by analogy with a formal deformation D: K -X L. Then Theorems 16, 17 can be extended without difficulty to systems in which the bases may be infinite.

It remains to be seen whether or not the purely algebraic theory developed in Section 2-9 can be extended to systems of modules with infinite bases, in such a way as to yield a generalization of Theorem 13 to infinite complexes.

MAGDALEN COLLEGE, OXFORD.

REFERENCES.

1. J. H. C. Whitehead, Proceedings of the London Mathematical Society, vol 45 (1939), pp. 243-327.

2. , Annals of Mathematics, vol. 41 (1940), pp. 809-824.

3. , ibid., vol.. 42 (1941), pp. 1197-1239.

4. , Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 1125-32.

5. , ibid., vol. 54 (1948), pp. 1133-45. 6. N. Jacobson, The theory of rings, New York (1943).

7. K. Reidemeister, Einfiihruing in die kombinatorische Topologie, Brunswick (1932).

8., Journal filr die reine und angewandte Mathematik, vol. 173 (1935), pp. 164-173.

9. W. Franz, ibid., vol. 176 (1937), pp. 113-134. 10. G. deRham, Commentarii Malthematici Helvetici, vol. 12 (1940), pp. 191-211.

11. G. W. Higman, Proceedings of the London Mathematical Society, vol. 46 (1940), pp. 231-248.

12. M. H. A. Newman, K. Akademie van Wetenschappen, Amsterdam, vol. 29 (1926), pp. 611-626; 627-641.

13. J. W. Alexander, Annals of Mathematics, vol. 31 (1930), pp. 292-320.

14. S. Lefschetz and J.. H. C. Whitehead, Transactions of the American Mathematical Society, vol. 35 (1933), pp. 510-517.

15. J. L. Kelley and Everett Pitcher, Annals of Mathematics, vol. 48 (1947), pp. 682-709.

16. R. H. Fox, ibid., vol. 44 (1943), pp. 40-50.



simple homotopy types

Documents