Annals of Mathematics

On the Realizability of Homotopy GroupsAuthor(s): J. H. C. WhiteheadSource: Annals of Mathematics, Second Series, Vol. 50, No. 2 (Apr., 1949), pp. 261-263Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1969449 .

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AxNwEs OF MATHEMATICS Vol. 50, No. 2, April, 1949

ON THE REALIZABILITY OF HOMOTOPY GROUPS

By J. H. C. WHITEHEAD

(Received June 11, 1948) The object of this note is to prove a theorem, which provides an affirmative

answer to the following question, proposed by S. Eilenberg at the Princeton Bicentennial Conference. "Given a (finite or infinite) polyhedron, P, and given r > 1, does there exist a polyhedron, Pr , such that

ri(Pr) = 0 if i ? r

7ri(Pr) 7 ri(P) if i > r."

It should be stated at once that we allow dim PT = ??. The same question, with the restriction dim PT < CC Iis, presumably, much more difficult.

By a polyhedron we shall mean one of the kind defined on p. 316 of [1]. Such a polyhedron, P, is covered by a simplicial complex, K. The latter need not be star-finite but the topology of P is such that a sub-set, X C P, is closed if, and only if, X n a is closed for each (closed) simplex, o, of the complex K. We shall describe P as locally finite if, and only if, K is star-finite. The combinatorial processes used here are described in [1], [2] and, more clearly, I hope, in [3]. By a "complex" we shall mean a connected "membrane complex" of the kind introduced on p. 1211 of [2]. These complexes, which are treated in some detail in [3], may be infinite provided they cover polyhedra, in our sense of the word.

Let iri be an arbitrary (multiplicative) group and let r2 , 1r3, ... be a sequence of arbitrary (additive) Abelian groups, each of which admits ir1 as a group of operators. If the number of elements in each group i-xn is countable we shall describe the set irn } as countable. We prove:

THEOREM. There is a polyhedron, P, which is locally finite if Jr,, I} is countable, whose homotopy groups are related to {7rn} by isomorphisms (onto), f'n :7,(P) a-*r (n 1, 2, ),such that

(*) fk(xa) = (fix)fka (k > 1.)

for each x e 7r1(P), a e 7rk(P).

Let Ko be a single point. Let r ? 1 and assume that there is a complex, Kr-1 X such that fn :wn(Kri1) 7 Onfor n = 1, , r - 1, if r > 1, where fk is an operator isomorphism in the sense of (*) if k > 1. Let A C irr be any set of elements, which generate 7rr (e.g. A = irr). Corresponding to each a E A we attach an r-sphere, S', to Kr-, at a 0-cell, po E Kr,-, taking care that S' has no other point in common with Kr,- or with any of the other r-spheres Sr, . We thus form a complex

L = Kr-i U U Sa. aeA

Let a' I ir(L) be the element which is represented by a homeomnorphism, 261

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262 J. H. C. WHITEHEAD

St -t Sa, and let R be the group ring of 7r1(K,1), with integral coefficients.' According to Theorem 19 on p. 285 of [1], the group 7r,(L) is the direct sum

7r,(L) = 7r,(K,.4) + GI where G is a free R-module, which is freely generated by the set of elements I a' . Since G is free an operator homomorphism, h: 7r,(L) -- 7r, , is defined by ha' = a, hr,(Kr,) = 0, and h is onto since A generates ir, . Let B C h-'(0) be a set of elements which generate h-'(0) (e.g. B = h-'(O)). Corresponding to each b e B we attach to L an (r + 1)-cell, eb+lI whose boundary is a map which represents the element b, thus forming a complex

Kr, = L u Ueb+. beB

According to Theorem 18 on p. 281 of [1] the kernel of the injection homo- morphism, i: 7r,(L) ir,(K,), is h-'(O). Therefore an operator isomorphism (onto) is defined by

f,. = hi-': 7r,(K,) --> r,.

If s < r the group 7r.(Kl) is undisturbed by the addition of the r-spheres St and the (r + l)-cells es'. Therefore we may identify r.(K,.1) with ir(K,) and

fAn : rtn, (Kr) --+ T. (n = 1,* *1 r)

is an isomorphism onto, which is an operator isomorphism if n > 1. It follows by induction on r that Kr and fn(1 < n ? r) may be thus defined

for every value of r. Moreover Kr_1 C Kr and Kr - Kr_1 consists of the r-cells -Po and the (r + 1)-cells c'+'. Therefore

Kr - Kr-1= b Kn

if n < r, where X' denotes the p-section of a complex X. Let

K = U Kr re-1

and let a topology in K be defined by the condition that XC K is closed if, and only if, X n Kr is closed for each r = 0, 1, * . . . Then K is a complex. Since

n+1 = Kn+1 if r > n + 1 it follows that K"+' = Kne1. As in the case of a finite complex we may identify Vn(K) with Vn(K n+) = 7n(K n+). Then fn may be interpreted as an isomorphism fn nr,(K) n.

If each of the groups rn is countable, then all the groups which are involved in the above argument are countable and the number of cells in K is countable. Therefore it follows from a theorem in [3] that K is of the same homotopy type as a locally finite polyhedron, P, and the proof is complete.

' The wording is appropriate to the case r > 1. It is obvious what changes to make if re= 1.

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ON THE REALIZABILITY OF HOMOTOPY GROUPS 263

Eilenberg's question is now answered, since irn(P) is countable if P is any connected, locally finite polyhedron (cf. [2], p. 1237).

MAGDALEN COLLEGE OXFORD

REFERENCES

1. J. H. C. WHITEHEAD, Proc. London Math. Soc., 45 (1939), 243-327. 2. J. H. C. WHITEHEAD, Annals of Math., 42 (1941), 1197-1239. 3. J. H. C. WHITEHEAD, Combinatorial Homotopy, Bull. Amer. Math. Soc.

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