Implications of the Ganea Condition Norio Iwase, Donald Stanley and Jeffrey Strom September 18, 2003 Abstract Suppose the spaces X and XxA have the same Lusternik-Schnirelmann category: cat(X x A) = cat(X). Then there is a strict inequality cat(X x (A o B)) < cat(X) + cat(A o B) for every space B, provided the connectivity of A is large enough (depending only on X). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres. MSC Classification 55M30 Keywords Lusternik-Schnirelmann category, Ganea conjecture, prod- uct formula, cone length Introduction The product formula cat(X x Y ) cat(X) + cat(Y ) [1] is one of the most basic relations of Lusternik-Schnirelmann category. Taking Y = Sr, it implies that cat(X xSr) cat(X)+1 for any r > 0. In [5], Ganea asked whether the inequality can ever be strict in this special case. The study of the `Ganea condition' cat(X x Sr) = cat(X) + 1 has been, and remains, a formidable challenge to all techniques for the calculation of Lusternik-Schnirelmann category. In fact, it was only recently that techniques were developed which were powerful enough to identify a space which does not satisfy the Ganea condition [8] (see also [9, 12]. It is still not well understood exactly which spaces X do not satisfy the Ganea condition, although it has been conjectured that they are precisely the spaces for which cat(X) is not equal to the related invariant Qcat(X) (see [14, 17]). 1 Since the failure of the Ganea condition appears to be a strange property for a space to have, it is reasonable to expect that such failure would have useful and interesting implications. In this paper we explore some of the implications of the equation cat(X x A) = cat(X) for general spaces A, and for A = Sr in particular. A brief look at the method of the paper [8] will help to put our results into proper perspective. The new techniques begin with the following question: if Y = X[fet+1, the cone on f : St ! X, then how can we tell if cat(Y ) > cat(X)? It is shown (see [9, Thm. 5.2] and [12, Thm. 3.6]) that, if t dim(X), then cat(Y ) = cat(X) + 1 if and only if a certain Hopf invariant Hs(f) (which is a set of homotopy classes) does not contain the trivial map *. It is also shown [9, Thm. 3.8] that if * 2 rHs(f), then cat(Y x Sr) cat(X) + 1. Thus Y does not satisfy Ganea's condition if * 62 Hs(f), but there is at least one h 2 Hs(f) such that rh ' *. Of course, if rh ' *, then r+1h ' * as well, and this suggests the following conjecture (formulated in [8, Conj. 1.4]): Conjecture If cat(X xSr) = cat(X), then cat(X xSr+1) = cat(X). In this paper we prove that this conjecture is true, provided r is large enough. Theorem 1 Suppose X is a (c - 1)-connected space and let r > dim(X) - c . cat(X) + 2. If cat(X x Sr) = cat(X), then cat(X x St) = cat(X) for all t r. The conjecture remains open for small values of r. Our main result is much more general: it shows how the equation cat(X x A) = cat(X) governs the Lusternik-Schnirelmann category of products of X with a vast collection of other spaces. Theorem 2 Let X be a (c - 1)-connected space and let A be (r - 1)- connected with r > dim(X) - c . cat(X) + 2. If cat(X x A) = cat(X) then cat(X x (A o B)) < cat(X) + cat(A o B) for every space B. 2 When A is a suspension, the half-smash product decomposes as A o B ' A _ (A ^ B) (see, for example [12, Lem. 5.9]), so we obtain the following. Corollary Under the conditions of Theorem 2, if A is a suspension, then cat(X x (A ^ B)) = cat(X) for every space B. Our partial verification of the conjecture is an immediate consequence of this corollary: it the special case A = Sr and B = St-r. Organization of the paper. In Section 1 we recall the necessary back- ground information on homotopy pushouts, cone length and Lusternik- Schnirelmann category. We introduce an auxiliary space and establish its important properties in Section 2. The proof of Theorem 2 is pre- sented in Section 3. 1 Preliminaries In this paper all spaces are based and have the pointed homotopy type of CW complexes; maps and homotopies are also pointed. We denote by * the one point space and any nullhomotopic map. Much of our exposition uses the language of homotopy pushouts; we refer to [11] for the definitions and basic properties. 1.1 Homotopy Pushouts We begin by recalling some basic facts about homotopy pushout squares. We call a sequence A ! B ! C a cofiber sequence if the associated square f A __________//_B | | | | fflffl| fflffl| * __________//_C is a homotopy pushout square. The space C is called the cofiber of the map f. One special case that we use frequently is the half-smash product A o B, which is the cofiber of the inclusion B ! A x B. Finally, we recall the following result on products and homotopy pushouts. 3 Proposition 3 Let X be any space. Consider the squares A _____//B X x A _____//X x B | | | | | | and | | fflffl|fflffl| fflffl| |fflffl C _____//D X x C _____//X x D. If the first square is a homotopy pushout, then so is the second. Proof This follows from Theorem 6.2 in [15]. 2 1.2 Cone Length and Category A cone decomposition of a space Y is a diagram of the form L0 L1 Lk-1 | | | | | | fflffl| fflffl| fflffl| Y0 ____//_Y1___//_._._.//_Yk-1___//_Yk in which Y0 = *, each sequence Li! Yi! Yi+1 is a cofiber sequence, and Yk ' Y ; the displayed cone decomposition has length k. The cone length of Y , denoted cl(Y ), is defined by 8 < 0 if Y ' * cl(Y ) = 1 if Y has no cone decomposition, and : k if the shortest cone decompositionYofhas lengthk. The Lusternik-Schnirelmann category of X may be defined in terms of the cone length of X by the formula cat(X) = inf{cl(Y ) | X is a homotopy retract of}Y. Berstein and Ganea proved this formula in [3, Prop. 1.7] with cl(Y ) replaced by the strong category of Y ; the formula above follows from another result of Ganea _ strong category is equal to cone length [7]. It follows directly from this definition that if X is a homotopy retract of Y , then cat(X) cat(Y ). The reader may refer to [10] for a survey of Lusternik-Schnirelmann category. The category of X can be defined in another way that is es- p0 sential to our work. Let F0(X) _____//G0(X)___//_Xbe the familiar path-loop fibration sequence (X) _____//P(X)____//X. Given the nth 4 pn __ Ganea fibration sequence Fn(X) _____//Gn(X)____//X, let_G n+1(X) = Gn(X)[CFn(X) be the cofiber of pn and define __pn+1: Gn+1 (X) ! X by sending the cone to the base point of X. The (n + 1)stGanea fi- bration pn+1 : Gn+1(X) ! X results from converting the map __pn+1 to a fibration. The following result is due to Ganea (cf. Svarc). Theorem 4 For any space X, (a) cl(Gn(X)) n, (b) the map pn : Gn(X) ! X has a section if and only if cat(X) n, and (c) Fn(X) ' ( (X))*(n+1), the (n + 1)-fold join of X with itself. Proof Assertion (a) follows immediately from the construction. For parts (b) and (c), see [6]; these results also appear, from a different point of view, in [16]. 2 2 An Auxilliary Space Let eGndenote the homotopy pushout in the square " i1 Gn-1(X) O__________//_Gn-1(X) x A | | | | fflffl| fflffl| Gn(X) _______________//_eGn. The maps pn : Gn(X) ! X and 1A : A ! A piece together to give a map epn: eGn! X xA. The space eGnand the map epnplay key roles in the forthcoming constructions; this section is devoted to establishing some of their properties. 2.1 Category Properties of Gen We begin by estimating the category of eGn. Proposition 5 For any noncontractible A and n > 0, cat(Gen) < n + cat(A). 5 Proof Let cat(A) = k. The space A is a retract of a space A0which has cl(A0) = k. Let eG0n= Gn [ Gn-1 x A0; clearly eGnis a homotopy retract of eG0nand so it suffices to show that cl(Ge0n) < n + k. Let L0 L1 Lk-1 | | | | | | fflffl| fflffl| fflffl| A00____//_A01__//_._._.//_A0k-1__//_Ak be a cone decomposition of A0. According to a result of Baues [2] (see also [13, Prop. 2.9]), there are cofiber sequences Fi-1* A0j-1____//Gix A0j-1[ Gi-1x A0j___//_Gix A0j. S Now let Ws = Gi+1(X) [ i+j=s,i 0) factors through Gi(X) x A and these factorizations are compatible because pi+1 ex- tends pi. So h factors as bG0n(X x A) ! eGn! X x A. Therefore, if cat(X x A) = n, then h, and hence epn, has a section. 2 6 2.2 Comparison of Gen with Gn(X) x A Let j : eGn! Gn(X) x A denote the natural inclusion map. Proposition 7 Assume that X is (c - 1)-connected and that A is (r - 1)-connected. Then the homotopy fiber F of the map j is (nc + r - 2)-connected. Proof There is a cofiber sequence eGn__j_//_Gn(X) x A___//_ Fn-1(X) ^ A. Therefore the homotopy fiber of j and the space ( Fn-1(X) ^ A) ' ( (X)*n * A) have the same connectivity, namely nc + r - 2. 2 Corollary 8 Assume dim (Z) < nc + r - 2 and let f, g : Z ! Gen. Then f ' g if and only if jf ' jg. The proof is standard, and we omit it. 2.3 New Sections from Old Ones Suppose that cat(X) = cat(X x A) = n. By Proposition 6 there is a section oe : X x A ! eGnof the map epn: eGn! X x A. Define a new map oe0: X ! Gn(X) by the diagram 0 X _____________oe_____________//_Gn(X) | OO i1|| |pr1| fflffl| oe O " j | X x A __________//_eGn_______//_Gn(X) x A. We need the following basic properties of oe0. Proposition 9 If cat(X x A) = cat(X) = n, then (a) oe0 is a homotopy section of the projection pn : Gn(X) ! X, and (b) if X is (c - 1)-connected and A is (r - 1)-connected with r > dim(X) - nc + 2, then the diagram 0 X _______oe___//_Gn(X)" | ` i1|| k|| fflffl| oe fflffl| X x A ____________//eGn commutes up to homotopy. 7 Proof First consider the diagram 0 pn X ______oe____//_Gn(X)_____________Gn(X) ______________//_XOO | _____ OO i1|| k ______ |pr1| |pn| fflffl| oe fflffl___j | pr1 | X x A ____________//XXXXXeGn_____//_Gn(X) x A__________//_Gn(X) XXXXXX XXXXXXX |pnx1A |pn 1XxAXXXXXXXXXXX | | XXX,,X fflffl| pr1 fflffl| X x A _______________//X. The diagram of solid arrows is evidently commutative. Therefore, we have pn O oe0' pr1O 1XxA O i1 ' 1X , proving (a). To prove (b) we have to show that two maps X ! eGnare homo- topic. Since dim(X) < nc + r - 2, it suffices by Corollary 8 to show that j O(oe Oi1) ' j O(kOoe0). Since pr2Oj O(oe Oi1) ' * ' pr2Oj O(kOoe0), it remains to show that pr1O j O (oe O i1) ' pr1O j O (k O oe0). But both of these maps are homotopic to oe0. 2 3 Proof of the Main Theorem Proof of Theorem 2 We have n = cat(X) = cat(X x A) by hypothesis. It follows from Proposition 6 that there is a section oe : X x A ! Gen of the map epn: Gen ! X x A. We then get the section oe0: X ! Gn(X) that was constructed and studied in Section 2.3. Consider the following diagram and the induced sequence of maps on the homotopy pushouts of the rows i1x1B pr1 (X x A) x B oo__________X_x B ______________//_X Y | oex1B's|| |oe0x1B| oe0|| || fflffl| kx1 fflffl| pr fflffl| homotopy fflffl| eGnx B oo______B____Gn(X) x B _____1____//_Gn(X) pushout//_____________* *_P | epnx1B|| |pnx1B| pn|| || fflffl| i1x1B fflffl| pr1 fflffl| fflffl| (X x A) x B oo__________X_x B ______________//_X Y. Proposition 9 implies that the upper left square commutes up to homo- topy. Since i1x 1B is a cofibration, we can apply homotopy extension and replace the map oe x1B : (X xA)xB ! eGnxB with a homotopic 8 map s which makes that square strictly commute. All other squares are strictly commutative as they stand. Since the composites (epnx 1B ) O (oe0x 1B ) and pn O oe0 are the identity maps and (epnx1B )Os is a homotopy equivalence, each vertical composite in the modified diagram is a homotopy equivalence. Thus Y is a homotopy retract of P , and consequently cat(Y ) cat(P ). The space Y is the homotopy pushout of the top row in the dia- gram, which is the product of the homotopy pushout diagram B ______________//_* | | | | fflffl| fflffl| A x B __________//_A o B with the space X. Therefore Y ' X x(AoB) by Proposition 3. Since Y is a homotopy retract of P , it follows that cat(X x (A o B)) cat(P ), the proof will be complete once we establish that cat(P ) < cat(X) + cat(AoB). This is accomplished in Lemma 10, which is proved below. 2 Lemma 10 The space P constructed in the proof of Theorem 2 sat- isfies cat(P ) cl(P ) n. Proof The space eGnis defined by the homotopy pushout square Gn-1(X) ____________//_Gn(X) | | | | fflffl| fflffl| Gn-1(X) x A ___________//_eGn. Take the product of this square with the space B and adjoin the homotopy pushout square that defines P to obtain the diagram Gn-1(X) x B ____________//_Gn(X) x B__________//_Gn(X) | | | | | | |fflffl fflffl| fflffl| Gn-1(X) x A x B ____________//eGnx B______________//P. 9 By [11, Lem. 13], the outer square Gn-1(X) x B ____________//_Gn(X) | | | | fflffl| fflffl| Gn-1(X) x A x B ____________//_P is also a homotopy pushout square. The top map is the composite pr1 O " Gn-1(X) x B ____//_Gn-1(X)___//_Gn(X), and so we have a new factorization into homotopy pushout squares: pr1 Gn-1(X) x B ____________//_Gn-1(X)__________//_Gn(X) | | | | | | fflffl| fflffl| fflffl| Gn-1(X) x A x B ______________//L_______________//P. To identify the space L, observe that the left square is simply the product of the space Gn-1(X) with the homotopy pushout square B ______________//_* | | | | fflffl| fflffl| A x B __________//_A o B. By Proposition 3, L ' Gn-1(X) x (A o B). Hence the right-hand square is the homotopy pushout square Gn-1(X) _______________//Gn(X) | | | | fflffl| fflffl| Gn-1(X) x (A o B) ____________//_P. Therefore cl(P ) cat(A) + cat(A o B) by Proposition 5. 2 References [1]A. Bassi, Su alcuni nuovi invarianti della variet'a topoligische, Annali Mat. Pura Appl. 16 (1935), 275-297. 10 [2]H. Baues, Iterierte Join-Konstruktion, Math. Zeit. 131 (1973), 77-84. [3]I. Berstein and T. 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