on the lusternik-schnirelmann category

Annals of Mathematics

On the Lusternik-Schnirelmann CategoryAuthor(s): Michael GinsburgSource: Annals of Mathematics, Second Series, Vol. 77, No. 3 (May, 1963), pp. 538-551Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970129 .

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AdINAL OF MATHUMATICS

Vol. 77, No. 3, May, 1963 Printed in Japan

ON THE LUSTERNIK-SCHNIRELMANN CATEGORY

BY MICHAEL GINSBURG*

(Received April 20, 1962)

In their study of the calculus of variations in the large, Lusternik and Schnirelmann were led to introduce the notion of "category" of a topological space X. Borsuk then began to investigate the relationship between the category of X and the more popular homotopy-type invariants of X. More recent results in this direction are due mainly to R. H. Fox, G. W. Whitehead, P. Hilton, and T. Ganea, but much remains unknown.

In this paper we will exhibit some relationships between the category of X and certain spectral sequences which arise from the bar construction of Eilenberg-MacLane [3] and Milnor's universal bundle construction [7]. In particular, if the category of X is k, then the differentials dr as well as many of the groups Er of these sequences are trivial for r>k. As applications of these results, we obtain an exact sequence relating the homology suspension [11] to the Pontrjagin product in &IX, the loop space of X. We also show that a bound on the category of X implies the vanishing of certain Massey (homology) products in 72X. In subsequent papers we will use these results to study the homology of fibre spaces.

Our method is to use a certain geometric analogue of the bar construction. This is due to G. W. Whitehead [10] and is described in ? 1 of this paper, along with other preliminary material. In ? 2, we consider the effect of the category of X on this construction; and in ? 3, we demonstrate its equivalence with the bar construction and Milnor's construction. In ? 4, we give a theorem about Massey homology products in &2X and, as an example, compute some of these products in &7CPn, where CPn is complex projective n space. The main results are listed, as theorems and corollaries, at the beginnings of ?? 2, 3, and 4.

Thanks are due to George Whitehead, who suggested many of the prob- lems we examine here, and who (see [10]) made the conjectures which we prove as Theorems 2.1, 3.1, and 3.2.

1. Definitions and preliminaries

All homology groups considered will be singular homology groups with integral coefficients. To simplify the appearance of many formulas, a

* This research was partially supported by the Air Force Office of Scientific Research contract number AF 49(638)-568.

538



LUSTERNIK-SCHNIRELMANN CATEGORY 539

homotopy F: X x I-- Y, where I is the unit interval, will usually be written as Ft: X- Y.

Let X be a space with base point x.. We will often need to use the excision axiom for a homology theory. To insure its validity in the places we will need it, we will always assume that x. is a non-degenerate base point in the sense of Puppe (see [8] and [11]). As usual, Xy will mean the space of all continuous maps of Y into X with the compact-open topology. The base point of Xy will always be the constant map of Y into x0.

Let A denote the infinite dimensional simplex with the weak topology. A is then a countable cw-complex. Label the vertices y*, YO, Yi, Define W = X; define Fi = [w e W I w(yi) =x] for i = 0, 1, 2, * * .; and finally define Wi = Un-=Fn for i = 0, 1, 2, ***. Since A is contractible, W is, up to homotopy type, just a large copy of X, while the Fi are all contractible. As Wi c Wi1, these subspaces form a sort of filtration of W.

This filtration gives rise to a spectral sequence ET with differentials dT, r i 1, which can be described as follows: let

i: Hp+q(Wp, WpVr)> Hp+q(Wp+^--l, WPl) jp q: Hp+q( Wp) > Hp+q( W)

be the homomorphisms induced by inclusion, and let

8: Ilp+q( Wp+r-lY Wp-1)) Hp+q-l(Wp l, Wp-r-])

be the boundary homomorphism of the triple (Wp+rl, Wp l, Wp-r-1). Then Eq = image i; E-q = image jpq/image jp-i q+1; and dp = aI Eq.

The following theorem, due to G.W. Whitehead [10], sums up the known facts about this spectral sequence. S2X denotes the loop space of X, while if (A, B) is a pair of spaces, B c A, then [A, B]P denotes the pth Cartesian power of the pair. The pth Cartesian power of A is denoted by AP.

THEOREM 1.1. If w1(X) = 0, then (1) the images of jP: H(Wp) - H(W) filter H(W), and Ed is the

graded group associated with H(W); (2) for each p, q, Eq Epq for large enough r; (3) El q = Hq([?2X, b]P), where b is the base point of f2X; (4) under the above isomorphism, dl is induced in a natural way

from the maps &2XP -2XP-1 given by

(Wr ju to Y Wp) mes Wp ultwili c Y wi pl)

where juxtaposition means path multiplication.



540 MICHAEL GINSBURG

Let xk be the base point of Xk and d the diagonal map of X into Xk. Define T(X) = [(x1, *, k)C X I xi = x for some i, 1 < i < k]. The base point of T(X) is also Xk. We will say cat(X) < k if and only if d is deformable, preserving base points, into T(X). This definition of category is due to G. W. Whitehead [10].

Thus cat(X) ? k if and only if there exist k homotopies of the identity map, f': X > X, 1 _ i _ k, preserving the base point, and such that for each x C X, there is some i for which fi(x) = xo. The sets Ai = (fi)-1(xo) are contractible over X and form a closed covering of X. If Xis separable, metric, an ANR, and x0 is a non-degenerate base point, then this definition of category is equivalent to the classical one, using either open or closed coverings. (Classically, cat(X) _ k if and only if there exist k open (closed) sets which cover X and are contractible over X.)

2. Relations between cat(X) and the spectral sequence Er

THEOREM 2.1. If cat(X) < k, then dr = 0 for r > k. In ? 4, we show that if X = CPn, complex projective n space, then

dn # 0. Since cat(CPn) _ n + 1, Theorem 2.1 is a best possible result. The following theorem is noted in [10].

THEOREM 2.2. If cat(X) ? k, then Eq = 0 for p ? k and all q.

COROLLARY 1. If w1(X) = 0 and cat(X) _ k, then Erq 0 for r, p > k and all q.

This corollary follows immediately from the above theorems and Theorem 1.1.

COROLLARY 2. If w1(X) = 0 and cat(X) < 2, there exists an exact sequence

. * *> Hq([f2X, b]P) o Hq([X, b]-') ,...

* * Hq([i2X b]2) Hq(72X, b) Hq+i(X, x0) O 0,

where a is the homology suspension, and the other maps are induced from path multiplication as described in Theorem 1.1.

Corollary 2 follows immediately from Corollary 1, Theorem 1.1, and [11, Lemma 2.3]. We will not give the details.

In the remainder of this section we will prove Theorems 2.1 and 2.2. Since Er = 0 for p < 0, dr: Er Er is trivial for p _ r, so to prove Theorem 2.1, we need only be concerned with proving dr = 0 if p > r > k > 1.

The proof of Theorem 2.1 splits into an algebraic and a geometric part as indicated by the following lemmas.




LEMMA 2.1. Suppose that for some integers p > r > 1 there exists a homomorphism C* HJ(Wp) > H,,(Wp-) for each s, such that the diagram

H.(Wp-r) H8(Wp)

i\ k/~

HI(Wp-1)

is commutative, where i and j are inclusion maps. Then d; 0. PROOF. Consider the diagram

Hp+q(Wp, Wp-r) g Hp+q-l(Wp-r) * Hp+q-i(Wp)

lg2 0* z L/({ Hp+q(Wp+r-il Wp--> Hp+q( JWpV-l)

1 g3

Hp+q-,(Wp-ly Wp-,-,)

where the horizontal lines are portions of the exact sequences of the pairs (Wp, Wp-r) and (Wp+r1, Wp-1) respectively; a is the boundary operator of the triple (Wp+r1, Wp-1, WP-r-1); and g2 and g3 are induced by inclusion. This diagram is commutative: the upper right triangle by hypothesis, the rest for standard reasons.

Since Er q = image g2, and d r = a I Epr q, dr will be trivial if &g2 = 0. But g92 g3=i**g, and i*g1 = 0 because they are consecutive homomorphisms

in an exact sequence. Thus Lemma 2.1 is demonstrated.

LEMMA 2.2. If p > r > k, there exists a map C: Wag W,1 such that

Wp_ r Wp jin

WP-1

is homotopy commutative, where i and j are inclusion maps. PROOF. Recall that A V B is the space obtained from the disjoint union

of A and B by identifying their base points. Let 0 be the base point of I, and define K=IvI V ... VI, a k-fold wedge. We label the vertices of K ZOO Y

.. Zk-l, Z*, where z* is the point at which all the intervals meet. Letf': X > X, i 0, 1, ..., ck-1, denote the deformations which exist because cat(X) < k. For each x, f'(x) C XI. Since fi(x)= x for all x and i, the wedge of these functions define a map a(x) = V -Ok-1 fi(x) of K into




X. One easily checks that y: X - XK is continuous and preserves base points. What is important about Y is that for each x, Y(x)(z*) = x and y(x)(zj)= x. for some i=0, 1, *--, kI-1. Intuitively, a(x) looks like a spider sitting on x with k legs lying on the paths x describes during the categorical deformations.

Now let N= A V K, where the points identified are y0 and z*. The vertices of N are labeled z0, *.., Zk-l, n0, na, ... where the ni are the images of the yi under the inclusion of A in N. We now use y to construct pa: W-e X'. For any w e W, define ,t(w): N - X by ,t(w) = w V y(w(y0)). Since y(w(y0))(z) =w(y0), ,t(w) is well defined. Again it is easy to see that ,t is continuous and preserves base points. Intuitively, ,t(w) is obtained by attaching k legs to w(A) at the point w(y0).

In what follows we will often be using the fact that if A is a cw-complex and B a subcomplex of A, then any map of B into N can be extended to A since N is contractible.

We can now define g'. Geometrically, w is altered to g'(w) so that '(w)(y%), i = O. * * *, k-1, are the end points of the k legs which can be

attached to w(A) at w(y0). Specifically, let 0: A N be any extension of the map defined on A0, the 0-skeleton of A, by

(z 0?< i ? k-i 8(y)n= {Zi k _ i.

For any w e W. ,t(w)0 C W. By the construction of ,t and 0, ,t(w)0 C W,

c W,, so the correspondence w , ,at(w)0 determines a function from W to W,-1. It is trivial to check that this function is continuous and preserves base points. We define T' to be its restriction to W,.

To complete the proof of Lemma 2.2, we define a homotopy H,: W, W,1 such that H. = 'i and H, = j. By definition, H,(w) = p(w)t, where 0a A > N is a homotopy, to be defined, such that 00 = 0 and 01 = inclusion. Obviously some homotopy with these boundary values exists; the difficulty lies in making sure Ht(WWPT) c W,1 for all t. To do this we must construct 0a in several stages. By drawing a picture and watching what happens to the vertices yi as t varies, the reader will be able to see what our formulas for 0a mean.

Consider the sets of integers A = (k, kI+1, * , p-1) and B = (1, * ,

p-r). For a finite set Y let #(Y) denote the number of elements in Y. Since p _ r > k > 1, #(A) = p-k _ p-r = #(B), so #(A-AnB) >

#(B-AnB). Thus there exists a set C c A-AnB such that $(C) = #(B-A n B). Let v: C B-A A B be any one-to-one correspondence. We now define a,:




(a) for 0 < t < 1/4: define

00 -0

0 (y~) = Z 0?< i ? Ik-i ni(Yi) = {tZI k < i and i V C

9114(Yi= fna(i) for i G C,

and extend t0 arbitrarily over the rest of A x [0, 1/4]. The salient features of this part of the homotopy are

(1) Ht(w) G Wk, for any w G W; this is because Ot(y%) = zi for 0 < i < kI-1;

(2) for each i e B there is a j, k _ j _ p-1, such that 0114(y) = ni, and if ie AnB, then i=j. These statements are easily checked.

(b) for 1/4 ? t ? 1/2: define

01/2(yi) = ni 0 < i < kI-1

9t(Yi) = 19114(Ii) k ? i.

Since 01/4(yI) = z0 and 01/2(yO) = n0, we can extend it(yo) linearly over the edge z0n0 of Kc N. With Ot(yo) thus defined, extend t0 arbitrarily over the rest of A x [1/4, 1/2]. One checks that Ht(w) e W,1 for any w e Wp-, for

(1) if w(yi) = x0 for some 1 < i < p-r, then there exists a yj, k < j < p-1, such that 0114(Yi) = ni. Hence

It(W)(Y=) = A(w ) = PM(w)0114(Y) = p,(w)(ni) = w(y) = xo;

(2) if w(y0) = x0, then

Ht(w)(yo) = p(w)0t(yo) e a(w)(K) = Y(w(yo))(K) = y(xo)(K) = xo.

(c) for 1/2 ? t ? 3/4: define

03/4(Ys) = ni for i e C

Ot(yi) = ni for i i C

and extend t0 arbitrarily to A x [1/2, 3/4]. One easily sees that during this part of the deformation H,( Wp-r) C Wp-r, for Ot(yi) = &j for all i = 0, *.., p-r. Furthermore, 03/4 1 A, = inclusion.

(d) for 3/4? t < 1: define

Ot(yi) = ni for all i 01 = inclusion




and extend St arbitrarily to A x [3/4, 1]. This finishes the proof of Lemma 2.2.

Theorem 2.1 follows immediately from Lemmas 2.1 and 2.2. PROOF OF THEOREM 2.2. The map wk+1: W > X'+' defined by wk+1(w)=

(w(y*), w(y0), ***, w(ykl-)) is a Hurewicz fibre map; i.e. the covering homotopy theorem holds for all spaces. Let D: X , W be defined as D(x)= wx, where wx is the constant map of A into x. Clearly D covers the diagonal map. Again let f be the deformations which arise because cat(X) < k, where 0 < ? _ k-i. Now define a deformation of the diagonal map, d,: X , Xk+l, by d,(x) = (x, f ?(x), ..., fk-,(X)). By the covering homotopy theorem, dt is covered by a homotopy Dt: X-) W. One sees easily that D1(X) c Wk, and r1D, is the identity map of X.

Consider the commutative diagram

Wkl-1 W

D1 /n DI\ ZTE1 X

where i is the inclusion map. As r, is a homotopy equivalence (for A is contractible to y*) and 1ziD, = identity, it follows from the induced diagram of homology groups that i*: H( Wkl) > H( W) is an epimorphism. Since for p _ k-i, the diagram, where all maps are induced by inclusion,

H( Wp) /, \j* Aj*

H(Wk-l) * H(W)

commutes, j* is an epimorphism for all p ? kI--i. Theorem 2.2 now follows from the definition of E-.

3. Relations between Er and other spectral sequences

Let A be DGA algebra over the ring Z of integers, B(A) and B(A) the "total space " and " base space ", respectively, of the bar construction on A (see [2] or [3]). Recall that if

A = A/Z A? = Z Ak= AOgAO ...O(&A k times, k > 1,

then -3~Z Ak = B(A) and A 0 B(A) = B(A). Letting :Ak =

the B(A)i form a filtration of the chain complex B(A). The resulting




spectral sequence is denoted Er(A). Let R+ denote the non-negative real numbers, I, the interval [0, r] for

r C R+. The (Moore) path and loop spaces of X, EX and s2X respectively, are defined by (see [1] or [2])

EX = [(a, r) I a: I, > X, a(O) = x0, r e R+], S2X = [(a, r) C EX I a(r) x0].

The map w: EX - X defined by w(a, r) = a(r) is a fibre map with fibre &2X, and &2X acts on the left of EX by multiplication. Let C(Y) denote the normalized singular complex of a space Y. We will adopt the con- vention that if Y is arcwise connected, C( Y) denotes chains with vertices at the base point. C(Q2X) is a DGA algebra and C(EX) is a DGA module over C(f2X).

THEOREM 3.1. If w1(X) = 0 and A= C(f2X), then the spectral sequences Er(A) and Er are isomorphic for r > 1.

Let X be a countable simplicial complex with the weak topology, G(X) the group of simplicial loops on X (see [6]). Following Milnor [7], we let Boo be the universal bundle of G(X), XO. the classifying space for G(X), and Xi, i = 0, 1, ***, the filtering subspaces of XO,. The Xi give rise to a spectral sequence denoted Er(M).

THEOREM 3.2. If w1(X)- 0, the spectral sequences Et(M) and Er are isomorphic for r ? 1.

These theorems will follow from the general theorem we now state. Let p: M - N be a fibre map with arcwise connected fibre F, and suppose F operates on itself and on M in such a way that

1. C(F) is a DGA algebra; 2. C(M) is a DGA module over C(F); 3. p(am) = p(m) for a C F, m C M;

suppose furthermore that there are subspaces Ni c N, i = 0, 1, * ,

Mi = p-'(Nj), such that 4. Ni c Ni +; 5. there exist maps gt: Mj > Mi, where Mi is the cone over Mi, which

extend the inclusion maps of Mi into Mj+1; 6. p. gi : H(Mi, Mj P H(Nj + , Nj) 7. No is contractible.

THEOREM 3.3. Let A= C(F). There exists a filtration preserving chain map f: B(A) - C(N). Letting hi = ft B(A)i, pi: B(A)i--C(Ni) is a chain




equivalence for each i. Theorem 3.3 is just a conglomeration of statements in [2]. Let B(A) be

filtered by the subcomplexes B(A)i = A 0 B(A)i. One constructs a filtration preserving map of DGA A-modules It: B(A) C(M), which then in a standard way induces the desired map ft. fe is constructed by induction on i: assuming a chain equivalence fti: B(A)i - C(Mi) already constructed (this is trivial for i = 0, using condition 7), one extends ft' to a chain map

P : B(A)i QC(MJ, where B(A)i = B(A)i + Ai+', by a standard technique which uses the fact that C(Mi) is acyclic. Then gtfr is extended by A-linearity to give the map bej?'. Comparing the exact sequences of the pairs (B(A)i, B(A)i) and (C(M), C(MJ), since B(A)i is acyclic and Iti is a chain equivalence by the induction assumption, one verifies that fr induces an isomorphism of H(B(A)i, B(A)i) with H(Mi, Mi). From condition 6 and the fact that H(B(A)i, B(A)i) H(B(A)i+?, B(A)i), it follows that fti+l

induces an isomorphism of H(B(A)i+1, B(A)i) with H(Ni+1, Ni). Applying the five lemma and a theorem of J. Moore, one finally verifies that eai+1 and fti?1 induce isomorphisms of homology groups and are therefore chain equivalences. The details and tedious verifications are contained in [2] and will not be repeated here.

PROOF OF THEOREM 3.1. This theorem follows immediately from Theo- rem 3.3 if the hypotheses of that theorem are satisfied. For this, let w1: W , X be the map defined in the last section as 71(w) = w(y*), and let M be the fibre space induced from EX by r1. Thus there is a fibre map p: M-) W with arewise connected fibre F = Q2X. Then, with W and W. replacing N and Ni in Theorem 3.3, it is trivial to show that conditions 3, 4, and 7 hold. Conditions 1 and 2 are verified in [2]. We now show conditions 5 and 6 hold as well.

LEMMA 3.1. Mk is deformable to a point in Mkl.

PROOF. By definition, Mk = [(w, a, r) e Wk x EX I w(y*) = a(r)]. Thus an element in Mk resembles a kite with tail a(O) at x0. We will deform these kites over themselves so that one of the vertices yo, I*, Yk+1 is at x0 at each stage of the deformation.

Let Kr = A v Ir, r e R+, where the points identified are y* and r. Define ar K1 ) Kr by

rx x A aqr(, ) l rx x e I.

Define a homotopy at K1- K1 for 0 ? t ? 1/2 by




U0 = identity

at(Yi) = Yi O < i < k

at(y*) = 1-2t

(Yi) = 0 k < i

U112(Y*Yk+1) = 0,

where Y*Yk,+ denotes the edge of A between y* and Yk+1, and extend a, arbitrarily to the rest of K1 x [0, 1/2], which is possible since K1 is contractible. Let Ur = arat

For each element m = (w, a, r) e Mk, define m*: KrX by m* = wVa. Let j: A , K1 and j8: I, - Irs <r, be the inclusion maps. We can now define a contracting homotopy Gt: Mk - Mk,+.

(a) forO<t?1/2: Foranym=(w,a,r)eMkdefine

Gt(m) = (m*tj, aj(-2t)r, (1-2t)r). During this part of the deformation

(1) Gt(Mk) c Mk since aor(yj) = yi for all r, t, and 0 < i < k; (2) pGll2(Mk) c Fk+i

0 Wk since m*Ur/2 (Yk?) = 1n*(O) = a(O) = x0 for all m e Mk.

(b) for 1/2 < t < 1: Let ht be any strong deformation retraction of A into its edge Y*Yk+i, this deformation taking place in the range 1/2 < t < 1. Such a map exists by Proposition 2.5 of [6]. Define, for m = (w, a, r) G Mk,

Gt(m) = (m*1/,2jht, ao, 0)

where ao is the unit of EX. Then (1) pGt(Mk) c Fk+1 c Wk+1, since for m = (w,a,r) e Mk,

pG,(m)(yk+1) = m*112jht(yk+l) = m* 1/2(Yk+1) = m*(O) = a(O) =

(2) Gl(Mk) = mo, the base point: for if x e A,

pG1(m)(x) = m*a12jhj(x) e m*a12(y*yk+l) = m*(O) = a(O) = x0;

thus GQ(m) = (wo, ao, 0) = mo. It is straightforward to check that G, is continuous, so we have proven Lemma 3.1.

Condition 5 of Theorem 3.3 is an immediate consequence of Lemma 3.1, for let Mk be realized as the identification space obtained from Mk x I by identifying (m, 0), for all m e Mk, to a point. Then gk: Mk - Mk+1 is defined by gk(m, t) = G1jt(m). Notice that by construction of Gt,

{p'(Fk+l) 0 < t < 1/2. LEMMA 3.2. p*gk : H(Mk, Mk) H(Wk-l, Wk) for all k.




PROOF. Consider the diagram

Hq(Mk I Mk) 81>Hq-l(Mk)

Ik l k* l ok

Hq(Mk+ly Mk) * Hq(p-1(Fk+l), P-'(Fk+l n Wk.)) HiH 1(p-i(F11n w0 ) P* ~ P* {P*

Hq( Wk+l, Wk) Hq(Fk+l, Fk+l n Wk) 3 Hql(Fk~l 0 Wk.)

which we now explain. &1, 82, and &3 are the boundary operators in the exact sequences of the appropriate pairs; i and j are inclusion maps; p' and p" are induced by p; while 8k is now to be defined.

Let 8l: Mew Mk be the deformation which slides all points of the cone towards the vertex and stops half way there: for (m, s) e Mk, Ot(m, s) (m, s - (ts/2)). The composite gkok is a homotopy of the pairs (Mk, Mk) and (Mk+l, Mk) such that g ?1(Mk, Mk) c (P-1(Fk+l), P-1(Fk+i 0 Wk)), the last statement following immediately from the properties of gk stated above. Define Ok =gkQ.

Commutativity of the above diagram follows for standard reasons every- where but in the upper left triangle. Here jok* = gk* for jOk is homotopic as a map of pairs to gk by construction of Ok.

Since Mk and Fk+l are both contractible, it follows from the exact homology sequences of the respective pairs that a, and 83 are isomorphisms, while j* is an isomorphism because j is an excision (here one needs x0 to be a non-degenerate base point, as shown in [11]). Since pg* 3 **

to prove the lemma we need to show p'*SkO: H(Mk) H(Fkl n Wk) is an isomorphism in all dimensions. We will do this by showing pfOk is a weak homotopy equivalence, that is pfOk induces isomorphisms of homotopy groups in all dimensions.

Since w1(X) = 0, Mk is arewise connected, for it is a fibre space with arewise connected base and fibre; also it follows easily from the definitions that Fk+l n Wk is arewise connected if w1(X) = 0. For the remainder of the proof replace all the homology groups in the above diagram by homotopy groups, retaining the original notation for the homomorphisms involved. This new diagram is well defined, and commutative for the same reasons as the old one was; 81 and 83 are still isomorphisms for the same reason as before, while p* is an isomorphism, since p is a fibre map.

Consider the commutative diagram




WUq (Mk, Mk) Uq- l(Mk) k I'll,

Wq(Mk+l, Mk)

where 84 is the boundary operator of the exact sequence of (Mk+l, Mk).

Since a, is an isomorphism, gk is a monomorphism. But this means

j a-lip' ok = p gk a-1

is a monomorphism; hence so is pIod*.

To complete the proof of Lemma 3.2, we show p ok* is onto by exhibit- ing a map C: Fk+l 0 wk Mk such that pOkU is homotopic to the identity map of Fk+l 0 Wk. Let a: I - A be the simplicial map defined by the vertex mapping (O) =Yk+1, a(1) =Y*. is then defined by C(w) = (w,wa,1).

Referring to the notation of Lemma 3.1, define q: K1 - A by q = id V a, where id is the identity map of A. Then by definition of p, Ok, and C,

pOk(w) =wquai12j; in other words pOkU is just composition with qoll/2j: A- A. By definition of a1/2 and q, qf/2j(YJ) = Yi for 0 < i < k+1. Hence qofl2j is homotopic, modulo Yo,* , Yk+l, to the identity map of A. Let ht be any such homotopy. Then Ht(w) = wh, is a homotopy H,: Fkk+l n Wk -

Fk+l 0 Wk between pOkC and the identity. This proves Lemma 3.2, and finishes the proof of Theorem 3.1.

PROOF OF THEOREM 3.2. In the statement of Theorem 3.3, let F, M, N, and Ni be G(X), Boo XOO, and Xi respectively. Conditions 1 through 7 of this theorem are now immediate (5 and 6 are proven in [7]), so it follows that Er(M) Er(C(G(X))) for r > 1. Since a simplicial complex in the weak topolopy is paracompact, it follows from [1, Theorem 1.5] that there is a weak homotopy equivalence f: &2X - G(X) which preserves multiplication. Thus f induces a map of the DGA algebra C(f2X) into the DGA algebra C(G(X)) which is an isomorphism on the homology level. But it is shown in [3] that under such conditions f induces an isomorphism of Er(C(f2X)) with Er(C(G(X))) for r > 1. Theorem 3.2 now follows from Theorem 3.1.

4. Massey products and an example

Let H*(f7X) be the Pontrjagin ring of f2X. Massey triple products are defined in H*(f2X), using Pontrjagin multiplication, in the same way they are defined in the cohomology ring of a space using the cup product. Similarly, Massey n-products can be defined for n elements of H*(f2X)




(see [5] for the details in the case of cohomology). Such an n-product, < a,, *.., an >, is a set in H*(72X).

It has been noticed (for instance [9]) that the differentials dr in the spectral sequence of the bar construction Er(A), where A = C(Q2X), are modifications of these r + 1-products in the following sense: if <a1, ***, an> is defined, where ai are homogeneous elements of H*(f2X), then

(1) a= al).. *ane H,([&2X,b]n) = En,. for s = E degree (ai); (2) dr(a) = O for r < n-1, so a EnEs'; (3) dn-1(a) = < a,, an > E En-2 = H?n_#2(&X)/some subgroup. We will call these differentials dr modified Massey r + 1-products. The

following theorem is then an immediate consequence of Theorems 2.1 and 3.1.

THEOREM 4.1. If 71(X) = 0 and cat(X) < k, then all modified Massey n-products in H*(&2X) vanish for n > k + 1.

The following example is included to show, as we remarked in ? 2, that Theorem 2.1 is a best possible result. It also shows that 72CPn and S' x f2S2n+l , though homotopically equivalent, are not equivalent as H- spaces. (Cpn is complex projective n-space and S' is the i sphere.) We will show that if a generates H1(f2CPn) = Z, then < a.*, a>, the n + 1 Massey product, generates H2n(f2CPn) = Z.

Since f7CPn is weakly homotopically equivalent to S' x &2S2'+l (see [10]), a simple computation, using the Kunneth formula and the fact that H* (2S2f+l') is a polynomial ring with one generator in dimension 2n, shows that

Hi (&7CP n) =fZ i = 2sn, 2sn + 1, for s =0,1,.. 10 otherwise.

Hence <a, * , a> is defined. Since E, q = Hq([&2CPn, b]P), repeated use of the Kunneth formula implies that for p > 1,

E f, Z q p P.Qlo q # p and 0< q < 2n.

Thus, in the portion of the p, q plane given by 1 < p, 0 < q < 2n, the only non-zero groups are isomorphic to Z along the diagonal and at the point p = 1, q = 2n. As the differentials are transverse to the diagonal, one sees that En?1,~ l F and Ej2 n E H2 (2CPn). Hence, in this case there is no indeterminacy, and d(a... (*a) = <a,.**, a>.

If <a,.**, a> does not generate El then El n + . Again, since the only non-vanishing groups in the pq plane for 1 < p, q < 2n, are on the diagonal, El2- E n+ # 0. But this is impossible since El2- is a subgroup of H2n+1(CPn) 0.




Thus, first of all, dn ? 0. Since cat(CPn) < n + 1, Theorem 2.1 is a best possible result. Secondly, if there were a weak homotopy equivalence f between &7CPn and S1 x &2S2n+1 which also preserved products, the induced isomorphism of the spectral sequences of the bar constructions would imply <a, * * *, a> # 0 in H*(S1 x 2S2fn+l) for a e H1(S1 x f2S2n+1). But this is impossible.

UNIVERSITY OF CHICAGO

REFERENCES

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Colloque de Topologie Algebrique, Louvain, 1957. 6. J. MILNOR, Construction of universal bundles: I, Ann. of Math., 63 (1956), 272-284. 7. , Construction of universal bundles: II, Ann. of Math., 63 (1956), 430-436. 8. D. PUPPE, Homotopiemengen und ihre induzierten Abbildungen: I, Math. Z., 69(1958),

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