Lusternik Schnirelmann category of skeleta

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1 Topology and its Applications 125 (2002) Lusternik Schnirelmann category of skeleta Yves Felix a,, Steve Halperin b, Jean-Claude Thomas c a Université Catholique de Louvain, 1348 Louvain-La-Neuve, Belgium b University of Maryland, College Park, MD , USA c Université d Angers, Bd Lavoisier, Angers, France Received 31 May 2000; received in revised form 16 August 2001 Abstract In this paper we make explicit the relationship between the category of a connected CW complex and the category of its skeleta. In particular we prove that if Y is a connected non-contractible CW complex, then cat Y k cat Y for each k-skeleton Y k. On the other hand, we prove that cat 0 (Y ) = lim k cat 0 (Y k ), where cat 0 denotes the rational category of the space Elsevier Science B.V. All rights reserved. MSC: 55P62; 55P50 Keywords: Rational homotopy theory; Lusternik Schnirelmann category; Rational category 1. Introduction In this note we consider the relations between the category of a CW complex and the category of its skeleta k. Several authors have recently give examples of simply connected spaces such that cat k < cat < for any k. In [8], Roitberg shows that the mapping cone Z of an essential phantom map ϕ : ΣK(Z, 5) S 4 satisfies catz k = 1 and catz = 2. Another example by D. Stanley, using the same argument as in his paper [9, Theorem 2.21], works as follows: Let f : S 2n 1 be an essential phantom map that is not a suspension and let w : S 2n 1 CP n 1 denote the attaching map of the top cell in CP n, then the cofiber Z of w f satisfies catz k = n 1, k 2n 1, while catz = n. On the other hand in [5], K. Hardie gives an upper bound for the category of in terms of the categories of the skeleta cat 2sup k {cat k }. * Corresponding author. addresses: felix@agel.ucl.ac.be (Y. Felix), shalper@deans.umd.edu (S. Halperin), thomas@tonton.univ-angers.fr (J.-C. Thomas) /01/$ see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S (01)

2 358 Y. Felix et al. / Topology and its Applications 125 (2002) In this paper we give two relations between the category of a CW complex and the category of its skeleta, the first one being an improvement of a result of Cornea [1]. Theorem 1. Suppose f : Y is a map between (r 1)-connected CW complexes, some r 1. Assume that for some q 0: (i) π i (f ) is an isomorphism, i<qand π q (f ) is surjective; (ii) H i (; ) vanishes for i rcaty + q for all ( possibly twisted) coefficients. Then cat caty. Corollary. If Y is a connected noncontractible CW complex then caty k caty for each k-skeleton Y k. The hypothesis of the corollary are necessary as shown for instance by the space Y = S. In this particular case, we have indeed cat Y = 0 and cat Y k = cat S k = 1for all k 1. The corollary implies that if Y is a connected noncontractible CW complex, then for sufficiently large N, caty N caty N+1 caty N+2 caty. Nonetheless it may happen that caty k < caty for all k as shown by the examples of Roitberg and Stanley quoted above. By contrast, the rational category of the limit is the limit of the rational category. To make this assertion precise, recall first that the rational category of a 1-connected space Y is the category of its rationalization Y Q. cat 0 Y = caty Q. Theorem 2. Suppose Y is a CW complex such that Y 1 ={pt}, cat 0 Y>0, and each Y k has finite Betti numbers. Then for sufficiently large N, cat 0 Y N cat 0 Y k cat 0 Y k+1 cat 0 Y, and cat 0 (Y ) = lim cat 0(Y k ). k 2. Proof of Theorem 1 The proof of Theorem 1 uses the Ganea characterization of category: cat n if and only if the Ganea fibration pn admits a section [7]. These fibrations F i n n G n () p n are the natural fibrations defined inductively by the following process. The fibration p0 is the path space fibration P with fiber Ω.Givenpn, it extends naturally to G n () in CF n. Convert this to a fibration to obtain pn+1 : G n+1(). The continuous map f : Y induces maps G n (f ) : G n () G n (Y ) such that pn Y G n(f ) = fpn.

3 Y. Felix et al. / Topology and its Applications 125 (2002) Denote the restriction of G n (f ) to the fibres by ϕ n.thefibresofpn and py n have the weak homotopy types of the (n + 1)st joins (Ω) n+1 and (ΩY ) n+1 respectively. This identifies up to homotopy ϕ n with (Ωf ) n+1. The natural homotopy equivalences (Ω ) n+1 = Σ n (Λ n+1 Ω ) then identify ϕ n with Σ n (Λ n+1 Ωf ). Because and Y are CW complexes Ω and ΩY have the homotopy type of CW complexes. Since π i (f ) is an isomorphism for i<qand surjective for i = q,wemayuse homotopy equivalences to replace Ωf by the inclusion i : Z W of a subcomplex of a CW complex in which Z r 2 ={pt} if r 2, and the additional cells of W have dimension at least q. Sinceϕ n is identified with the inclusion of the subcomplex Σ n (Λ n+1 Z) in Σ n (Λ n+1 W) and since the additional cells have dimension at least n + n(r 1) + q = nr + q it follows that π i (ϕ n ) is an isomorphism for i<nr+ q 1 and surjective for i = nr + q 1. Denote by q n : Z(n) the pullback of pn Y along f. We have then the commutative diagram of fibrations (Ω) n+1 j G n () p n ϕ n h (ΩY ) n+1 k Z(n) q n Evidently π i (h) is an isomorphism for i<nr+q 1and is surjective for i = nr +q 1. If caty = n then pn Y admits a section, which pull backs to a section σ of q n.the obstructions to lifting this to a map τ : G n () lie in the groups H i (; π i 1 (F )), where F is the homotopy fibre of h. But since π i 1 (F ) = 0, i rn + q 1, and H i (; ) = 0, i rn + q, these all vanish. Thus τ may be constructed and pn τ = q n ϕ n τ = id ; i.e., cat n. 3. Proof of Theorem 2 We begin by recalling some definitions. If (M, d) is a module over a differential graded algebra (A, d) then (M, d) is semifree if it is the union of an increasing sequence of submodules (M(0), d) (M(1), d) such that M(0) and each M(i)/M(i 1) is A-free on a basis of cycles [3]. If (N, d) (M, d) is an inclusion of (A, d)-modules such that (M/N, d) is semifree then (M,N,d) is a relative semifree module. Next suppose Y is a simply connected space with finite Betti numbers. Its rational category, cat 0 Y, can be calculated as follows [2] from its Sullivan model ( V,d).Let N 1 and consider the commutative diagram ( V,d) p ( V/ >N V,d) i ϕ ( V H,d)

4 360 Y. Felix et al. / Topology and its Applications 125 (2002) where ( V H,d) is a ( V,d)-semifree module, d(h) + V H,andϕ is a quasi-isomorphism. Then cat 0 Y is the least integer N such that i admits a ( V,d)-linear retraction σ : ( V H,d) ( V,d) [2,6]. We proceed now to the proof. Theorem 1 applies to the inclusions (Y k ) Q (Y k+1 ) Q Y Q with q = k and r = 2, provided that (Y k+1 ) Q is not contractible. This is true for k N sufficiently large, and establishes the inequalities in the first assertion. In particular, if the sequence cat 0 Y k is unbounded then cat 0 Y = = lim cat 0Y k. k Suppose then that cat 0 Y k m for every k. Denote by ( V k,d) the Sullivan minimal model of Y k. Then the graded vector space V k has finite type; i.e., each Vn l is finite dimensional. Hence in the commutative diagram ( V k,d) ( V k / >m V k,d) i k ϕ k ( V k H k,d) each H k has finite type. By hypothesis each i k admits a retraction r k. The semifree modules ( V k H k,d) together with the sequence of inclusions Y k Y k+1 Y give a commutative diagram of differential modules: ( V k,d) i k ( Vk H k,d) ( V k / >m V k,d) ( V k+1,d) i k+1 ( V k+1 H k+1 ) ( V k+1 / >m V k+1,d) i ( V,d) ( V H,d) ( V/ >m V,d) Since π i (Y k ) = π i (Y k+1 ) = =π i (Y ), i<k, the vertical arrows are isomorphisms in degrees <k. Thus without loss of generality we may identify Vp i = V i and Hp i = H i for i<kand p k and we may suppose the vertical arrows are the identity in these degrees. By hypothesis, cat 0 Y k m, k N, and so each i k admits a retraction r k, k N. Denote by S(k) the set of V k -linear retractions ( V k H k <k,d) ( V k,d).then S(k). S(k) is a finite dimensional affine space. Restriction defines a map S(k + 1) S(k). Let S(k,l) be the image of the composite S(k + l) S(k + l 1) S(k). Then S(k,l) S(k,l 1) and so for dimension reasons there is some l k such that S(k,l) = S(k,l k ), l l k.putt(k)= S(k,l k ). Then by construction every element in T(k) is the restriction of an element in T(k+1). This gives a sequence of retractions σ k : ( V k H k <,d) ( V k,d) such that σ k+1 restricts to σ k. Define a retraction σ : ( V H) ( V,d) by setting σ = σ k in H <k = H k <k and conclude that cat 0 Y m.

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