fiber spaces and the eilenberg homology groups

Fiber Spaces and the Eilenberg Homology GroupsAuthor(s): George W. WhiteheadSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 38, No. 5 (May 15, 1952), pp. 426-430Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/88596 .

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426 MATHEMATICS: G. W. WHITEHEAD PROC. N. A. S.

products) in the seminormal form, or simply from the standard diagrams themselves, one can construct a matrix A which transforms simultaneously all these elementary matrices according to (8). Of course new non-zero elements are introduced which, as yet, it has not seemed possible to keep track of in the general case, but the transformation has been applied to the seminormal form of [3, 2, 1] to yield a representation, which on re- duction modulo 2, 3 or 5 behaves as one would expect.2 It seems likely that this direct approach, taken in conjunction with our knowledge of the hook-structure of the diagrams and the ideas outlined above will

eventually lead to a complete solution of the problem. 1 Littlewood, D. E., Modular Representations of Symmetric Groups, Proc. Roy. Soc.

(London), A209, 333-352 (1951). 2 Robinson, G. de B., On the Modular Representations of the Symmetric Group (II),

PROC. NATL. ACAD. Sci., 38, 129-133 (1952). 3 Rutherford, D. E., Substitutional Analysis, Edinburgh, 1948, p. 41.

FIBER SPACES AND THE EILENBERG HOMOLOGY GROUPS

BY GEORGE W. WHITEHEAD

DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Communicated by S. Lefschetz, March 1, 1952

At the Princeton Bicentennial Conference in 1947 the following problem was proposed by Hurewicz:l if X is a O-connected space, does there exist, for each positive integer n, a fiber mapping p:T -- X such that (1) T is

n-connected; (2) p induces isomorphisms of r,(7T) onto rQ(X) for q > n. If n = 1 and X satisfies certain local smoothness conditions the universal

covering space provides a positive solution. If n = 2 and X is a simply connected finite polyhedron with no 2-dimensional torsion, a positive solution has been found by Hirsch.2 In both cases the fiber space in

question is actually a fiber bundle. In this note we show that the Hurewicz problem has a surprisingly

simple affirmative solution for an arbitrary O-connected space X if the notion of "fiber space" is used in a very general sense.3 The fiber spaces we obtain are very far from being fiber bundles and we do not attempt to answer the question whether a solution with fiber bundles is possible.

We use this result to study the Eilenberg homology groups of a space X. These groups, introduced by Eilenberg,4 are topological invariants of X intermediate between the homology and homotopy groups. They are defined as follows: choose a base point xo eX and, for each integer n, let Sn(X) be the subcomplex of the total singular complex S(X) consisting of all singular simplexes whose n-dimensional skeletons are mapped into


products) in the seminormal form, or simply from the standard diagrams themselves, one can construct a matrix A which transforms simultaneously all these elementary matrices according to (8). Of course new non-zero elements are introduced which, as yet, it has not seemed possible to keep track of in the general case, but the transformation has been applied to the seminormal form of [3, 2, 1] to yield a representation, which on re- duction modulo 2, 3 or 5 behaves as one would expect.2 It seems likely that this direct approach, taken in conjunction with our knowledge of the hook-structure of the diagrams and the ideas outlined above will

eventually lead to a complete solution of the problem. 1 Littlewood, D. E., Modular Representations of Symmetric Groups, Proc. Roy. Soc.

(London), A209, 333-352 (1951). 2 Robinson, G. de B., On the Modular Representations of the Symmetric Group (II),

PROC. NATL. ACAD. Sci., 38, 129-133 (1952). 3 Rutherford, D. E., Substitutional Analysis, Edinburgh, 1948, p. 41.

FIBER SPACES AND THE EILENBERG HOMOLOGY GROUPS

BY GEORGE W. WHITEHEAD

DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Communicated by S. Lefschetz, March 1, 1952

At the Princeton Bicentennial Conference in 1947 the following problem was proposed by Hurewicz:l if X is a O-connected space, does there exist, for each positive integer n, a fiber mapping p:T -- X such that (1) T is

n-connected; (2) p induces isomorphisms of r,(7T) onto rQ(X) for q > n. If n = 1 and X satisfies certain local smoothness conditions the universal

covering space provides a positive solution. If n = 2 and X is a simply connected finite polyhedron with no 2-dimensional torsion, a positive solution has been found by Hirsch.2 In both cases the fiber space in

question is actually a fiber bundle. In this note we show that the Hurewicz problem has a surprisingly

simple affirmative solution for an arbitrary O-connected space X if the notion of "fiber space" is used in a very general sense.3 The fiber spaces we obtain are very far from being fiber bundles and we do not attempt to answer the question whether a solution with fiber bundles is possible.

We use this result to study the Eilenberg homology groups of a space X. These groups, introduced by Eilenberg,4 are topological invariants of X intermediate between the homology and homotopy groups. They are defined as follows: choose a base point xo eX and, for each integer n, let Sn(X) be the subcomplex of the total singular complex S(X) consisting of all singular simplexes whose n-dimensional skeletons are mapped into



VOL. 38, 1952 MAITHEMA TICS: G. W. WHITEHEAD 427

xo. Then S(X) = S_-(X) : So(X) D .... The Eilenberg homology groups are the groups H,(S&(X)). Eilenberg proved that (1) Hn+I(Sn(X))

7rn+l(X) (abelianized if n = 0); (2) if ,,(X) = 0, then the inclusion

map of S_n-(X) into S,(X) is a chain-equivalence. It is also easy to verify that (3) if X has a universal covering space X', then Hq(Si(X)) - H (X') .

From (2) it follows that 7n(X) = 0 implies that H,(S,1(X), Sn(X)) = 0 for all q. This suggests the question: do the groups TH(Sn,_(X), Sn(X)) depend only on Tn(X)? The answer to this question is negative; for if X is real projective n-space, then it is easily seen that H3(So(X), Sl(X)) is cyclic of order 2 if n > 3 and infinite cyclic if n = 2, while i7r(X) is cyclic of order 2 if n > 2. However, the answer is partly in the affirmative- we shall see that H,(S,_,(X), Sn(X)) is a function of Tnr(X) provided that

q < 2n (Theorem 4). Let f:X - B be a mapping. We shall say that f is n-connective if and

only if X is n-connected and the homomorphism of ri(X) into Tri(B) induced byf is an isomorphism onto if i > n.

THEOREM 1. Let B be a O-connected space. Then for every n > 0 there exists an n-connective fiber mapping p :X - B.

Proof: By a general theorem of J. H. C. Whitehead5 there is a O-connected space B' containing B such that (1) B is closed in B'; (2) ri(B') =

0 for i > n; (3) for i , n, 7i(B) ivri(B') under the inclusion map. Let bo e B and let X' be the space of all paths in B' which start at bo. Let p':X'---B' be the map such that p'(F) -= F(l) for all FEX'. Let X = p'-I(B), p = p' X. Then p' and p are fiber maps3 with fiber F = p-l(bo) = the space of loops in B'. The homotopy sequences of the fiber maps p and p' are connected by the commutative diagram

i i(X) ------> v(B) b

rqW(F) i' J k a' 7,q_(F)

7r(XZ) - >. 7r (B')

Now X' is contractible and therefore ir,(X') = 0 for all q. Hence a' is an isomorphism onto. If q > n, then rq(F) = vrq1-(F) = 0 and therefore p is an isomorphism onto. If q < n, then k is an isomorphism onto and therefore b is an isomorphism onto. Hence ir,(X) = 0 for q ( n.

THEOREM 2. Let f:X- B be a map such that the homomorphism of 7rq(X) into irq(B) induced by f is an isomorphism onto for q > n. Then the chain-mapping of S,(X) into Sn(B) defined by f is a chain-equivalence.

Proof: Let Z be the mapping cylinder of f, F:Z - B the projection, i:X-*Z the inclusion map. Then i:,rq(X) ri t(Z) for q > n. By a




theorem of Blakers,6 i:Sn(X) -- S,(Z) is a chain-equivalence. But

F:Sq(Z) -- S,(B) is a chain-equivalence for all q. Hence F i = f:Sn(X) --

Sn(B) is a chain-equivalence. COROLLARY 1. If q > n, then f induces an isomorphism of Hr(S,(X),

S,+1(X)) onto Hr(S,(B), Sq+1(B)) for each r. Proof: By Theorem 2, f:IH(S,(X)) m H,(S,(B)) for all q > n. The

conclusion follows from the five-lemma. THEOREM 3. Let B be an n-connected space, p:X -B an (n + 1)-

connective fiber map, Z the mapping cylinder of p. Then Hq(Sn(B), Sn+1(B)) H(Z, X) for all q. Proof: Since Z is n-connected, Sn(Z) is chain-equivalent to S(Z).

Since X is (n + 1)-connected, Sn+,(X) is chain-equivalent to S(X) under the inclusion map. By the five-lemma, Hq(Sn(Z), Sn+l(X)) a Hq(S(Z), S(X)) = Hq(Z, X) under the inclusion map. Since p is (n + 1)-connec-

tive, Sn+,(X) is chain-equivalent to Sn+l(Z) under the inclusion map and therefore H,(Sn(Z), Sn+l(X)) is isomorphic with H,(S,(Z), Sn+l(Z)) under the inclusion map. Finally, Hq(Sn(Z), Sn+1(Z)) E H,(Sn(B), Sn+i(B)) under the projection P:Z -- B.

Let n be a positive integer, G a group (abelian if n > 1). Then Eilen-

berg and MacLane7 have defined an abstract complex K(G, n), whose

homology groups are denoted by H,(G, n). If B is a space they also define a chain-mapping t:Sn-_(B) -- K(rn(B), n). If ri(B) = 0 for i > n, then t is a chain-equivalence. If also B is (n - 1)-connected then S,-_(B) is

chain-equivalent to S(B) and therefore Hq(B) s Hq(rn(B), n). For any space B, t maps Sn(B) into a subcomplex kon of K(n,,(B), n)

which has the homology groups of a point. Hence there is a natural homo-

morphism t:Hq(S,_-(B), Sn(B)) -- Hq(inv(B), n) defined for all q > 0. THEOREM 4. If B is a space, then the homomorphism t:Hq(Sn_l(B),

Sn(B)) -- Hq(rn(B), n) is onto if q < 2n + 1 and an isomorphism if q ( 2n.

Proof: Let p:X -B be an (n - 1)-connective fiber map. Then

p:Hq(Sn-_(X), Se(X)) a Hq(Sn_1(B), S,(B)) by Corollary 1. Also

P:rn(X) :,n(B) and therefore p induces an isomorphism p:H,(rr,(X), n) Hq(n,(B), n). The diagram

H2(Sn1(^X), Sn(X)) p> Hq(Sn_I(B), Sn(B))

t t

Hq(7rn(X), n) - + HQ- (n (B), n)

is commutative. Therefore, replacing B by X if necessary, we may assume that B is (n - 1)-connected.

Suppose that B is (n - 1)-connected. As in the proof of Theorem 1, let B' D B be a space such that B is closed in B', 7r(B') = 0 for i > n,



VOL. 38, 1952 MA THEMA TICS: G. W. WHITEHEAD 429

and 7ri(B) 7 ,ri(B') under the inclusion map for i < n. Let X' be the space of paths in B' which begin at bo, p:X' -, B' the usual fiber map, X = p-'(B), Z' the mapping cylinder of p:X'-B', Z the mapping cylinder of p | XX - B. The diagram

J t Hq(Z, X) - Hq(Sn_-(X), Sn(X)) -- Hq(rn(X), n)

{i ii ji j ' t'

Hq(Z, X) H q(Sn-l(X'), Sn(Xt)) H q(rn(X'), n)

(the homomorphisms denoted by "i" and "j" are induced by the appro- priate inclusion maps) is commutative. Also j, j', i", and t' are isomorphisms onto. Hence it suffices to prove that i:Hq(Z, X) -* Hq(Z', X') is onto if q < 2n + 1 and an isomorphism if q < 2n.

By excision, Hq(Z, X) a Hq(Z u X', X') under the inclusion map. By exactness of the homology sequence of the triple (Z', Z u X', X') it is sufficient to prove that H,(Z', Z u X') =- 0 for q _ 2n + 1. By the relative Hurewicz theorem, it suffices in turn to prove that the pair (Z', Z u X') is (2n + l)-connected. This follows from: (1) the triad (Z u X'; Z, X') is 2n-connected; (2) rq,(Z', Z u X') a 7rq_-(Z u X'; Z, x').

Proof of (1): Since X is n-connected and Z is (n - 1)-connected, it follows that (Z, X) is (n - 1)-connected. Since X' is contractible, ri(X', X) s 7ri_l(X); since X is n-connected, (X', X) is (n + 1)-connected. By a theorem of Blakers and Massey8 the triad (Z u X'; Z, X') is 2n-connected.

Proof of (2): First note that the inclusion map induces isomorphisms onto: irq(X', X) a 7rq(Z', Z) for all q. For both groups are mapped isomorphically onto rq(B', B) by the fiber maps of X' and Z' into B'. Con- sider the diagram

7q-l(X , X)

i 7r( Z', Z u X') -- 7,i(Z u x', Z) _--(Z', Z)

k

7r,_(Z u X'; Z, X')

The vertical sequence is a segment of the exact homotopy sequence of the triad (Z u X'; Z,X'). The horizontal sequence is a segment of the exact homotopy sequence of the triple (Z', Z u X', Z). By the above



430 MA THEMA TICS: W. V. Q UINE PROC. N. A. S.

remark, joi is an isomorphism onto for all q. Hence i is an isomorphism into, j is onto and 7rq_(Z u X', Z) = Image iO Kernelj = Kernel k Kernel j, for all q. By exactness, a is an isomorphism onto Kernel j and k

maps Kernel j isomorphically onto 7r-_l(Z u X'; Z, X'). Hence ko?

maps 7r,(Z', Z u X') isomorphically onto r,-l(Z u X'; Z, X'), and the

proof is complete.' 1 Eilenberg, S., Ann. Math., 50, 247-260 (1949). 2 Compt. rend. acad. sci. Paris, 228, 1920-1922 (1949). 3 Serre, J.-P., Ann. Math., 54, 425-505 (1951). 4 Ibid., 45, 407-447 (1944). 5 Ibid., 50, 261-263 (1949). 6 Ibid., 49, 428-461 (1948). 7 Ibid., 46, 480-509 (1945). 8 Ibid., 55, 192-201 (1952). 9 Note added in proof: An independent discovery of the results of this paper has

been announced by H. Cartan and J.-P. Serre, Compt. rend. acad. sci. Paris, 234, 288- 298 and 393-395 (1952).

ON AN APPLICATION OF TARSKI'S THEORY OF TRUTHI

BY W. V. QUINE

HARVARD UNIVERSITY

Communicated by Hassler Whitney, March 15, 1952

Consider two interpreted systems of notation, L and L', in which state- ments can be formed. Let L' contain L together with what I have called the protosyntax of L;1 i.e., the elementary means of talking about the

expressions of L. For certain such systems L and L', we know from

Tarski's work2 (familiarity with which will not, however, be presupposed here) how to define truth for L in L'; i.e., how to translate 'y is a true

statement of L' (with 'y' as a variable) into a formula of L'.

But if L contains adequate notation for elementary number theory, and L' consists of L and its protosyntax and nothing more, then definability of truth for L in L' leads to paradox. For, Tarski has shown2 how to

derive paradox from definability of truth for L in L; and L' does not

essentially exceed L, since protosyntax can be reconstrued as elementary number theory by G6del's expedient of assigning numbers to expressions.3

Hence if in particular the system of my Mathematical Logic is con-

sistent, Tarski's method of defining truth for L in L' must somehow

break down when L is taken as the logical notation of Mathematical Logic and L' is taken as just L plus the protosyntax of L. It is the purpose of the present paper to follow out the details of this breakdown. An

outcome of the inquiry will be the discovery of a recursive definition which

430 MA THEMA TICS: W. V. Q UINE PROC. N. A. S.

remark, joi is an isomorphism onto for all q. Hence i is an isomorphism into, j is onto and 7rq_(Z u X', Z) = Image iO Kernelj = Kernel k Kernel j, for all q. By exactness, a is an isomorphism onto Kernel j and k

maps Kernel j isomorphically onto 7r-_l(Z u X'; Z, X'). Hence ko?

maps 7r,(Z', Z u X') isomorphically onto r,-l(Z u X'; Z, X'), and the

proof is complete.' 1 Eilenberg, S., Ann. Math., 50, 247-260 (1949). 2 Compt. rend. acad. sci. Paris, 228, 1920-1922 (1949). 3 Serre, J.-P., Ann. Math., 54, 425-505 (1951). 4 Ibid., 45, 407-447 (1944). 5 Ibid., 50, 261-263 (1949). 6 Ibid., 49, 428-461 (1948). 7 Ibid., 46, 480-509 (1945). 8 Ibid., 55, 192-201 (1952). 9 Note added in proof: An independent discovery of the results of this paper has

been announced by H. Cartan and J.-P. Serre, Compt. rend. acad. sci. Paris, 234, 288- 298 and 393-395 (1952).

ON AN APPLICATION OF TARSKI'S THEORY OF TRUTHI

BY W. V. QUINE

HARVARD UNIVERSITY

Communicated by Hassler Whitney, March 15, 1952

Consider two interpreted systems of notation, L and L', in which state- ments can be formed. Let L' contain L together with what I have called the protosyntax of L;1 i.e., the elementary means of talking about the

expressions of L. For certain such systems L and L', we know from

Tarski's work2 (familiarity with which will not, however, be presupposed here) how to define truth for L in L'; i.e., how to translate 'y is a true

statement of L' (with 'y' as a variable) into a formula of L'.

But if L contains adequate notation for elementary number theory, and L' consists of L and its protosyntax and nothing more, then definability of truth for L in L' leads to paradox. For, Tarski has shown2 how to

derive paradox from definability of truth for L in L; and L' does not

essentially exceed L, since protosyntax can be reconstrued as elementary number theory by G6del's expedient of assigning numbers to expressions.3

Hence if in particular the system of my Mathematical Logic is con-

sistent, Tarski's method of defining truth for L in L' must somehow

break down when L is taken as the logical notation of Mathematical Logic and L' is taken as just L plus the protosyntax of L. It is the purpose of the present paper to follow out the details of this breakdown. An

outcome of the inquiry will be the discovery of a recursive definition which



fiber spaces and the eilenberg homology groups

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