Categorical Sequences Rob Nendorf, Nick Scoville and Jeff Strom December 9, 2004 Abstract We define and study the categorical sequence of a space, which is a new fo* *r- malism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction on its CW skelta. The kthterm in the categorical sequence of a CW complex X, oeX (k), is the least integer n for which catX(Xn) k. We show that oeX is a well-defined homo- topy invariant of X. We prove that oeX (k + l) oeX (k) + oeX (l), which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomol- ogy algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most impor- tant of the many examples that we offer is a simple proof of a theorem of Ghienne: if X is a member of the Mislin genus of the Lie group Sp(3), then cat(X) = cat(Sp(3)). MSC Classification 55M30, 55P62 Keywords Categorical sequence, Lusternik Schnirelmann category, CW skeleta, rational homotopy Introduction The Lusternik-Schnirelmann category of a topological space X is the least integer k for which X has an open cover X = X0 [ X1 [ . .[.Xk with the property that each inclusion map Xj ,! X is homotopic to a constant map; it is denoted cat(X). This homotopy invariant of topological spaces was first introduced by Lusternik and Schnirelmann in 1934 as a tool to use in studying functions on (compact) manifolds: a smooth function f : M ! R must have at least cat(M) + 1 critical points. If X is a CW complex, then Xn = Xn-1[ff(n-cells), and therefore cat(Xn) cat(Xn-1) + 1. Berstein and Hilton asked what conditions must be placed on the attaching map ff in order to guarantee that equality holds in this upper 1 bound [3]; the answer is that equality holds when a certain set of generalized Hopf invariants does not contain the trivial map *. Thus it is possible, at least in principle, to compute the Lusternik-Schnirelmann category of a finite- dimensional CW complex inductively up its skeleta. It was shown in [34] that the Hopf sets for lower-dimensional skeleta partia* *lly determine the Hopf sets for high-dimensional skeleta. In actual computations, this makes it possible to `bootstrap' up from relatively simple low-dimensional results to (apparently) difficult high-dimensional calculations. Our goal in th* *is paper is to establish a convenient formalism for doing category calculations wh* *ile making use of all low-dimensional information. This is done via the categorical sequence of a space X, which is a function oeX : N ! N [ {1} defined by oeX (k) = inf{n | catX(Xn) k} where catX(Xn) is the category of Xn in X (see Definition 4). It is shown in Propositions 8 and 9 that oeX is a well-defined homotopy invariant of X; i.e., that, when n is larger than the connectivity of X, catX(Xn) depends only on n and the homotopy type of X, and not on any choices made in constructing a CW decomposition of X. If X is finite-dimensional, then oeX determines cat(X); examples due to Roitberg [26] show that this is not true for infinite-dimension* *al spaces. In any case, the categorical sequence of X holds a wealth of useful information. Though we are not directly concerned with the applications of Lusternik- Schnirelmann category to critical point theory in this paper, categorical se- quences could play a useful role there. For example, in the study of the n-body problem, one is often interested in infinite-dimensional Sobolev spaces W ; in order to apply the Lusternik-Schnirelmann method in this situation, it is neces- sary to find compact subsets K W such that the relative category catW(K) is large (see [1, Rmks. 2.15 & 3.5], [6, Thm. 4.6], or [25], for example). The * *cat- egorical sequence of W gives lower estimates on the dimension of such subsets. If oeW (k) = n, then catW(Wn-1) < k; if dim(K) < n then K can be deformed into Wn-1, and so catW(K) < k. Our theoretical results establish formulas for calculation with categorical sequences. Some of the statements make use of another sequence, the product length sequence of a nonnegatively graded commutative algebra A, defined by setting oeA (k) to be the least dimension n for which An contains a nontrivi* *al k-fold product. Proposition 16 For any space X and any ring R, oeX oeH*(X;R). We also estimate the categorical sequence of a rational space in terms of any of its models. Proposition 21 If X is a simply-connected rational space and A is any model for X, then oeX oeA . Recall that a simply-connected rational space X is formal if its cohomology algebra H*(X), with trivial differential, is a model for X. Thus we have the following computation for formal spaces. 2 Proposition 30 If X is a simply-connected formal rational space, then oeX = oeH*(X). More generally, we completely determine the sequences oe which can arise as the categorical sequences of simply-connected formal rational spaces. Theorem 31 The following conditions on a sequence oe with oe(1) > 1 are equivalent: (a)oe = oeA for some commutative graded algebra A, (b)oe(k + 1) k+1_koe(k)Wfor each k,Q (c)oe = oeW where W = Pi and Pi= Snj is a product of spheres, and (d)oe = oeX for some formal rational space X. The keys to the computational power of categorical sequences, though, are the three properties listed in the following theorem. In order to prove parts (* *b) and (c) for all spaces (and not just spaces of finite type, say), we have to use a set-theoretical framework in which Whitehead's problem (which asks: does Ext (A, Z) = 0 imply A is free?) has a positive solution. See x1.1 and Lemma 3 for details. Theorem 18 For any space X, (a)oeX (k + l) oeX (k) + oeX (l), (b)if X is simply-connected and oeX (k) = n, then Hn(X; A) 6= 0 for some coefficient group A, and (c)if equality occurs in (a) and X is simply-connected, then the cup prod- uct Hk(X; A) Hl(X; B) ! Hk+l(X; A B) is nontrivial for some choice of coefficients. The point we hope to make in this paper is that calculation with sequences is no harder than calculation of category; indeed, the extra information contained in the sequence, together with Theorem 18, can greatly facilitate computations. To illustrate this point, let X be any simply-connected space such that H*(X; Z) ~=H*(Sp(3); Z) = (x3, x7, x11). The categorical sequence oeX clearly has oeX (1) = 3 and oeX (2) 7 by Theorem 18(b). By Theorem 18(a), oeX (4) oeX (2) + oeX (2) 14. Furthermore, oeX (4) > 14 by Theorem 18(c), because the cup product H7(X) H7(X) ! H14(X) is trivial. Now we have oeX (4) 18 by Theorem 18(b), and hence oeX (5) oeX (4) + oeX (1) = 21 by Theorem 18(a). Since X is 21-dimensional (up to homotopy), this proves that cat(X) 5 by Theorem 18(b). We will see in Theorem 37 below that this calculation con- stitutes a simple proof of a result of Ghienne [16] about the Mislin genus of Sp(3). Theorem 18 can also be used to prove a generalization of a somewhat obscure result of Ganea [14]. Corollary 20 Let X be simply-connected and of finite type with oeX (k) = n. If there are integers 0 < a1 < a2 < . .<.al such that {n | eHn(X; G) 6= 0 for someG} I1 [ I2 [ . .[.Il 3 where Ij = [aj, aj + (n - 1)] (brackets denote closed intervals in R), then cat(X) < k(l + 1). The importance of Corollary 20 is not the result as such. Rather, it is the fact that, since it simply encodes an elementary computation with sequences, the result can be safely disregarded without losing computational power. Our proof is completely different from the one given in [14]. Ganea's proof makes use of the Blakers-Massey theorem: certain cofiber sequences are treated as fibration sequences. Our argument uses Theorem 18, which in turn rests on a much more elementary fact: the factorization k+l= ( kx l)O 2 of diagonal maps. One of our most pleasing general results gives formulas relating the categor- ical sequences of the spaces in a fibration sequence. Theorem 24 Let F __q__//E_p__//Bbe a fibration sequence and write a = cat(q) cat(F ) and b = cat(p) cat(B). Then (a)oeE (k(a + 1)) oeB (k), and (b)oeE (k(b + 1)) oeF (k). As a corollary to Theorem 24 we obtain the following elaboration of the cele- brated Mapping Theorem from the rational theory of Lusternik-Schnirelmann category. Proposition 29 Let f : X ! Y be a map between simply-connected rational spaces which induces an injective map f* : ss*(X) ! ss*(Y ). Then oeX oeY . Finally, we address products. To state our result (and our conjectures), we construct, for sequences oe and o, a `product sequence' oe*o defined by oe*o(k)* * = min {oe(i) + o(j) | i + j = k}. It is not hard to see, using Proposition 21, th* *at if X and Y are simply-connected formal rational spaces, then oeXxY = oeX * oeY . We conjecture that this equation holds in general for simply-connected rational spaces. So far however, the best we have been able to do is an inequality. Theorem 35 If X and Y are simply-connected rational spaces, then oeXxY oeX * oeY . This inequality is certainly not true in general for nonrational spaces, as the examples of Iwase [20] show. However, we conjecture that the reverse inequality oeX * oeY oeXxY is valid, not only for rational spaces, but for all spaces. Acknowledgments. We thank Don Stanley, Martin Arkowitz, Yves F'elix, Daniel Tanr'e and Terrell Hodge for their interest and advice at various stages of this project. 1 Preliminaries In this section we establish the basic notation and concepts that will be used * *in the body of the paper. 4 1.1 Basics We work with pointed spaces and maps; we use * to denote the one point space or any trivial map. We use idX : X ! X to denote the identity map and k : X ! Xk to denote the diagonal map k(X) = (x, x, . .,.x). The symbol ' denotes homotopy equivalence of spaces or homotopy of maps. All solid arrow diagrams in this paper are (homotopy) commutative. If S is a set of real numbers, then inf(S) is the infimum of S. We adopt the convention that inf(?) = 1. Set_Theory__ Whitehead's problem asks: if A is an abelian group such that Ext (A, Z) = 0, does it follow that A is free? The answer is `yes' if A is fini* *tely generated. Shelah has shown that the general problem is undecidable in ordinary ZFC set theory, but the answer is `yes' if G"odel's Axiom of Constructibility is assumed [30]. In order to avoid (unnecessary) hypotheses in Lemma 3 and Theorem 18 below, o we work in a set theory where Ext(A, Z) = 0 implies that A is free. For those uncomfortable with this assumption, we emphasize that ordinary ZFC set theory is sufficient to prove Lemma 3 and Theorem 18 when Hn(X; Z) is finitely generated for each n. 1.2 Skeleta We are concerned with the Lusternik-Schnirelmann category of the CW skeleta of a space. It will simplify some of our later work to use the following slight* *ly abstract notion of skeleton. Definition 1 An n-skeleton for a space X is a map i : Xn ! X, where Xn is a CW complex such that (a)Xn is n-dimensional (up to homotopy), and (b)i is an n-equivalence. This definition is justified by the observation that an n-skeleton i : Xn ! X can be taken as the nthCW skeleton of a CW replacement for X. The following result will help us to recognize skeleta. We omit the proof. Lemma 2 Let i : A ! X where A and X are simply-connected. Then i is an n-skeleton for X if and only if (a)H*(A) = 0 for * > n in all coefficients, and (b)the induced map i* : H*(X) ! H*(A) is an isomorphism for * < n and is injective for * = n in all coefficients. When we are working with rational spaces [11] we will want our skeleta to also be rationalWspaces. Unfortunately, this won't always happen; for example, the inclusion 1n=1Sn ,! SnQis an n-skeleton for the rational n-sphere. We avoid this problem by defining a rational n-skeleton of a simply-connected rational space X to be a map i : Xn ! X where Xn is a simply-connected rational space such that 5 (a)H*(Xn; Q) = 0 for * > n, and (b)the induced map i* : H*(X; Q) ! H*(Xn; Q) is an isomorphism for * < n and is injective for * = n. Rational n-skeleta are plentiful: if X is a rational space and Xn is an (integr* *al) n-skeleton of X, then (Xn)Q is a rational n-skeleton of X. In view of the discussion above, we make a standing convention that if a space X is assumed to be rational, then whenever we refer to an n-skeleton of X, we actually mean a rational n-skeleton. The proof of part (b) of our next result in full generality depends on a positive solution to Whitehead's problem. Lemma 3 Let X be a simply-connected space. (a)If Hn(X; Z) is free abelian, then X has an n-skeleton i : Xn ! X such that i* : H*(X) ! H*(Xn) is an isomorphism for * n, (b)If Hn(X; A) = 0 for all coefficient groups A, then X has an (n - 1)- dimensional n-skeleton. The corresponding statements also hold for all simply-connected rational spaces. Proof Write M(G, n) for the Moore space with Hn(M(G, n); Z) ~= G and Hi(M(G,Wn); Z) = 0 for i 6= 0, n. When G is free abelian, we take M(G, n) = Sn. According to [4], any simply-connected space X admits a homology de- composition, i.e., a sequence of CW complexes X(n) which are related to one another by cofiber sequences Mn-1 ! X(n - 1) ! X(n) (where Mn-1 = M(Hn(X; Z), n - 1)) and satisfy X ' hocolimnX(n). The inclusion map X(n) ! X induces isomorphisms on integral homology through dimension n, and Hk(X(n); Z) = 0 for k > n. With the CW decomposition inherited from the colimit, Xn X(n) Xn+1 for each n. Furthermore, if Hn(X; Z) is free abelian then X(n) = X(n - 1) [ (n-cells) so X(n) is n-dimensional and hence Xn = X(n). Thus X(n) is the desired n-skeleton of X. To prove (b), assume that Hn(X; A) = 0 for all A. Using the Universal Coefficient isomorphism [35, Cor. 13.11], we obtain Hom (Hn(X; Z), A) = 0 and Ext(Hn-1(X; Z); A) = 0 for all A. We claim that (i) Hn(X; Z) = 0 and (ii) Hn-1(X; Z) is free abelian. * *To prove (i), we let A = Hn(Z; Z); if A were nonzero, then Hom (Hn(Z; Z), A) would be nonzero (since it contains the identity map), thereby contradicting the as- sumption. For the second statement, we set A = Z; now Ext(Hn-1(Z; Z), Z) = 0, and by Whitehead's problem, we conclude that Hn-1(X; Z) is free. Now apply part (a) to conclude that X(n-1) is an (n-1)-dimensional (n-1)- skeleton and X(n) is an n-skeleton of X. Furthermore, since Hn(M(G, n); Z) = 0, Mn-1 ' *, and so X(n) ' X(n - 1) showing that X(n) is an is (n - 1)- dimensional n-skeleton for X. 2 6 1.3 Lusternik-Schnirelmann Category We make use of three equivalent definitions of the Lusternik-Schnirelmann cat- egory of maps and spaces. Definition 4 The Lusternik-Schnirelmann category of a map f : X ! Y is the least integer k for which X has a cover by open sets X = X0 [ X1 [ . .[.Xk such that f|Xi ' * for each i. When f = idX, we write cat(X) = cat(idX) and when i : A ,! X, we write catX(A) = cat(i). If X is a CW complex, then it is equivalent to require each Xito be a subcomplex of X in some CW decomposition. The category of f : X ! Y can also be defined in terms of the Ganea fibrations pk : Gk(Y ) ! Y with fiber Fk(Y ). The inductive definition of these fibrations begins by defining F0(Y )____//G0(Y_)p0_//Yto be the familiar path- loop fibration sequence (Y )____//_P(Y_)__//Y. Given the kthGanea fibration __ sequence Fk(Y )_____//Gk(Y_)pk//_Y,_let G k+1(Y ) = Gk(Y ) [ CFk(Y ) be the cofiber of pk and define _pk+1: Gk+1(Y ) ! Y by sending the cone to the base point of Y . The (k + 1)stGanea fibration pk+1 : Gk+1(Y ) ! Y results from converting the map _pk+1to a fibration. A result of Ganea [14] implies that cat(f) k if and only if there is a lift ~ of f in the diagram Gk(Y7)7__ ______ _~________|pk|____ ______f_____ fflffl| X _____________//Y. Our third definition is due to G. W. Whitehead. According to [36, p. 458], cat(f) k if and only if the composition of f with the diagonal map of pairs k+1 (X, *)_f__//(Y, *)___//(Y, *)k+1 factors, up to homotopy of pairs, through the trivial pair (X, X). We will make use of a related invariant, called Qcat, which is defined in te* *rms of the fibrations that result from applying a fiberwise version of the infinite suspension functor Q to the Ganea fibrations. Let qk : eGk(Y ) ! Y denote the fiberwise infinite suspension of the k-th Ganea fibration. Then Qcat(f) is the least integer k for which f lifts through qk [27]. 1.4 Rational Homotopy and L-S Category We briefly recall some key elements of the rational theory of L-S category. The reader is encouraged to consult [5, Ch. 5] or [11, Part V] for details. 7 A (simply-connected) Sullivan algebra is a commutative differential graded algebra (CDGA) A over Q such that: (a) A0 ~=Q and A1 = 0; (b) as a Q- algebra, A ~= (V ) where V is a graded vector space; and (c) the differential d is decomposable in the sense that d(A) A~2, where A~is the augmentation ideal of A. Every simply-connected space X has a Sullivan minimal model, M(X), which is a Sullivan algebra such that H*(M(X)) ~=H*(X; Q). A model for X is any CDGA for which there is a map OE : M(X) ! A which induces an isomorphism in cohomology (OE is a quasi-isomorphism). Definition 5 Let A be an augmented CDGA and write A~for the augmentation ideal. The nilpotency of A, denoted nil(A), is the greatest integer k such that (A~)k 6= 0. The algebraic study of the Lusternik-Schnirelmann category of rational spaces can be developed from the following result, which can be found in [5, Cor. 5.16] (though, historically, it was not [5, Rem. 5.15]). Theorem 6 If X is a rational space, then the following are equivalent (a)cat(X) k, and (b)M(X) is a retract (up to chain homotopy) of a Sullivan algebra B which is quasi-isomorphic with another CDGA A with nil(A) k. It follows immediately from Theorem 6 that if u 2 H*(Y ) = H*(M(Y )) can be represented by a cocycle which is a k-fold product, then f*(u) = 0 for any map f : X ! Y with cat(X) < k. In this case, the (rational) category weight of u is at least k. We write wgt(u) k and observe that cat(X) wgt (u) whenever u 6= 0 2 H*(X). The maximum value of wgt(u) for u 2 H*(X) is known as the Toomer invariant of X, and is denoted e0(X). There is a related invariant, denoted Mcat [7]. It is known that Mcat(X) = cat(X) for simply-connected rational spaces [19, Thm. 0]. The equality of Mcat and catis known to fail for maps: according to Parent [24, Thms. 2&11] Mcat (f xg) = Mcat(f)+Mcat (g); on the other hand, Stanley [33] has produced examples of maps f and g between simply-connected rational spaces such that cat(f xg) < cat(f)+cat(g). It is also known that Mcat(X) = Qcat(X) when X is a simply-connected rational space [28] (but see also [5, Thm. 5.49]). A simp* *le adaptation of the proof of [5, Thm. 5.49] yields the following generalization to maps; we omit the proof. Proposition 7 If f : X ! Y is a map between simply-connected rational spaces, then Qcat(f) = Mcat(f). 2 Categorical Sequences In this section we will define our object of study, the categorical sequence as- sociated to a space X. To ensure that our sequences are well-defined, we must first prove some results concerning the relative category of an n-skeleton. 8 2.1 Relative Category of Skeleta Since we usually think of an n-skeleton as a subspace of X, we will sometimes write catX(Xn) instead of cat(i) when i : Xn ! X is an n-skeleton. Proposition 8 For fixed n, the integer catX(Xn) depends only on the homotopy type of X, and not on the choice of n-skeleton. Proof Let i : A ! X and j : B ! X be two n-skeleta of X and consider the diagram __8B8_____ ______ _l_______j||___ _________ fflffl| A _____i____//_X. Since j is an n-equivalence and A is n-dimensional, there is a lift l : A ! B such that j O l ' i [35, Thm. 6.31]. It follows that catX(A) = cat(i) cat(j) * *[2, 1.4]. Since the situation is symmetrical, we also have cat(j) cat(i). 2 It can be conceptually easier to work with the Lusternik-Schnirelmann cate- gory of spaces rather than of maps. Happily, there is no difference between the two for skeleta. Proposition 9 If X is (c - 1)-connected and i : Xn ! X is an n-skeleton with n c, then (a)cat(Xn) = cat(i), (b)Qcat (Xn) = Qcat(i), and (c)if X is a rational space and i : Xn ! X is a rational n-skeleton for X, then cat(Xn) = Mcat(i). Proof We begin by proving (a). It is trivial that catX(Xn) cat(Xn); we wish to prove the reverse inequality. Assume that catX(Xn) = k; we will show that cat(Xn) k. Since n c, the map i* : ssn(Xn) ! ssn(X) is nontrivial, and hence k 1. Now consider the diagram Fk(Xn) _____l____//_Fk(X)___________Fk(X) | | | | | | fflffl| j fflffl| fflffl| Gk(Xn) ____________//_P____________//Gk(X)66kk__ _______ II___ mm | __oe___ _|__ ~mmmm | | ___o__|___ mmmm | fflffl| ___fflffl|mmmmi__ fflffl| Xn ________________Xn______________//X. in which the bottom right square is a pullback. Since catX(Xn) = k there is a lift ~ of i. By the pullback property, there is a section o : Xn ! P . According to [5, Lem. 6.26], the map l : Fk(Xn) ! Fk(X) is an (n + kc - 1)- equivalence since k 1, and it follows that j is also a (n + kc - 1)-equivalen* *ce. 9 Since n n + kc - 1, it follows that there is a (unique) map oe : Xn ! Gk(Xn) with j O oe = o [35, Thm. 6.31]. This oe is a section (up to homotopy) of the fibration Gk(Xn) ! Xn, and so cat(Xn) k. The key to the proof of part (a) is the fact that l : Fk(Xn) ! Fk(X) is an (n + kc - 1)-equivalence. But this implies that Ql : QFk(Xn) ! QFk(X) is also an (n + kc - 1)-equivalence, and so the proof of (a) can be used again to show Qcat (i) = Qcat(Xn). It remains to prove (c). For this we simply compute cat(Xn) = Mcat(Xn) by [19, Thm. 0] = Qcat(Xn) by [28] = Qcat(i) by part (b) = Mcat(i) by Proposition7. 2 Remark 10 The proof of Proposition 9(a) is an adaptation of the proof of [12, Thm. 1]. The argument actually works equally well with i : Xn ! X replaced by any n-equivalence f : Z ! X with dim(Z) n + kc - 1. The conclusion in this case is that cat(f) = cat(Z) = k. 2.2 Sequences from Topology and Algebra We will be concerned with sequences whose values are either nonnegative inte- gers or 1; thus a sequence is a function oe : N ! N [ {1}. We say that oe o if oe(k) o(k) for each k 0. We write oe < o if oe o and oe 6= o (oe < o d* *oes not mean that oe(k) < o(k) for every k). If oe is increasing, then the length of oe is sup{k | oe(k) < 1}. In view of Propositions 8 and 9, we may make the following definition. Definition 11 The catgorical sequence of a CW complex X is the sequence oeX : N ! N [ {1} defined by oeX (k) = inf{n | catX(Xn) k}. The following elementary observations about categorical sequences will be used constantly in what follows. Remark 12 (a)oeX is an invariant of the weak homotopy type of X. (b)If X is (c - 1)-connected but not c-connected, then oeX (0) = 0 and oeX (1) = c. (c)The finite values of oeX are strictly increasing. (d)If oeX (k) = n, then Xn 6= Xn-1 in every cellular decomposition of X. In particular, if X is simply-connected and oeX (k) = n, then Hn(X) 6= 0 for some coefficients (see Theorem 18(b) below). (e)If X is finite-dimensional, then oeX = length(oeX ); if X is infinite- dimensional, then length(oeX ) cat(X) 2 . length(oeX ) [18]. 10 (f)If oeX (k) < 1 for all k, then cat(X) = 1, even for infinite-dimensional X. (g)If oeX oeY and Y is finite-dimensional, then cat(X) cat(Y ). If Y is infinite-dimensional and cat(Y ) = 1, then cat(X) = 1. Before proceeding further, we give some examples. Example 13 (a)As is well-known, the integral cohomology of the sym- plectic group Sp(2) is H*(Sp(2)) = (x3, x7), an exterior algebra on generators in dimensions 3 and 7. It follows from Theorem 18(b) that the only possible finite values for oeX (k) are 0, 3, 7 and 10. Since i* *t is known [29, Ex. 4.4] that cat(Sp(2)) = 3, oeSp(2)(3) < 1, and hence oeSp(2)= (0, 3, 7, 10, 1, 1, . .).. (b)Define a Sullivan algebra M = (x3, y3, z5) with d(z5) = x3y3, and let X be a rational space whose minimal model is isomorphic to M (this algebra and space appear in [11, p. 387]). The nontrivial cohomology of X is H3(X) = Q[x] Q[y] H8(X) = Q[xz] Q[yz] H11(X) = Q[xyz] where brackets indicate cohomology classes. Thus cat(X) 3, and since cat(X) wgt([xyz]) = 3, we have cat(X) = 3. This forces oeX = (0, 3, 8, 11, 1, 1, . .).. (c)The `finite-dimensional' hypothesis in Remark 12(g) cannot be re- moved. Roitberg has shown that the cofibers C of certain (phantom) maps f : K(Z, 5) ! S4 have the property that Cn is a suspen- sion for all n, but cat(C) = 2. Thus oeC = (0, 4, 1, 1, . .)., and so cat(C) = 2 > 1 = length(oeC ). We will often abbreviate a sequence by deleting any terms known to be infinite. Thus, for example, we could summarize the results of Example 13(a,b) by writing oeSp(2)= (0, 3, 7, 10) and oeX = (0, 3, 8, 11). If we were unsure of* * the later values of the sequence, we would write, for example, oeSp(2)= (0, 3, 7, .* * .).; knowing that cat(Sp(2)) 3, we might write oeSp(2)= (0, 3, 7, a), where a = 10 or a = 1. We will also make use of the algebraic product length sequence of a nonnegatively graded augmented CGA A, defined by oeA (k) = inf{n | 9 nontrivialk-fold products inAn}. If each of P and Q is either a space or a graded algebra, then it may happen that oeP = oeQ . If so, then we say that P and Q are isosequential. z_n_factors_"_____- Example 14 (a)The spaces S2 x . .x.S2and CPn are easily seen to be isosequential. 11 (b)It is easy to verify that oeCP1 = (0, 2, 4, 6, . .).; it is even easie* *r to check that if A = H*(CP1 ; Q), oeA = (0, 2, 4, 6, . .).. Thus the space CP1 and the graded algebra H*(CP1 ; Q) are isosequential. In this case, CP1 is formal [5, Ex. 5.4], so A ~=M(CP1 ), and hence CP1 is isosequential with its minimal model. (c)Let X be the rational space of Example 13(b). Then oeX = (0, 3, 8, 11), but oeH*(X) = (0, 3, 11), and oeM(X) = (0, 3, 6, 11), so X is not isos* *e- quential with either H*(X) or M(X). Instead, these sequences are related by the string of strict inequalities oeM(X) < oeX < oeH*(X). 3 Inequalities Between Sequences One of our goals is to develop techniques for computing categorical sequences oeX . As with formulas for the calculation of cat(X), many our results for se- quences come in the form of inequalities. 3.1 Inequalities for General Spaces We begin by dispensing with wedges and retracts. Proposition 15 Let X and Y be any two spaces. Then (a)oeX_Y (k) = min{oeX (k), oeY (k)}, and (b)if X is a homotopy retract of Y , then oeX oeY . Proof Part (a) follows from the formula cat(f _ g) = max{cat(f), cat(g)}. For (b), we consider the homotopy commutative diagram Xn-1 _____s_____//___________________Yn-1 i|| j|| fflffl| fflffl| X _____________//_____________88____________________________* *_________Y//_X ________________________________________________________* *_____________________________________________________________________________* *_____________________________ __________________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *________________ idX in which the map s exists by cellular approximation. It follows that catX(Xn-1)* * = cat(i) cat(j) = catY(Yn-1). Now oeY (k) = n implies that catY(Yn-1) < k and hence that catX(Xn-1) < k. Therefore oeX (k) n = oeY (k). 2 Our next result recasts the classical cup length lower bound for Lusternik- Schnirelmann category in terms of sequences. Proposition 16 For any space X and any ring R, oeX oeH*(X;R). Proof If oeH*(X)(k) = n, then there is a nontrivial k-fold cup product u1. .u.k2 Hn(X). Let i : Xn ! X be an n-skeleton. Thus i induces an injection 12 i* : Hn(X) ! Hn(Xn), so i*(u1. .u.k) = i*(u1) . .i.*(uk) 6= 0 2 Hn(Xn). Therefore cat(Xn) k [5, Prop. 1.5] and so oeX (k) n. 2 Proposition 16 can be used to determine the categorical sequence of a prod- uct of spheres. This simple corollary will play an important role in our charac- terization of the categorical sequences of formal rational spaces (x5). Corollary 17 If X = Sn1 x . .x.Snr with n1 n2 . . .nr, then oeX is given by the formula oeX (k) = oeH*(X)(k) = n1 + n2 + . .+.nk for k r and oeX (k) = 1 for k > r. Proof Clearly oeH*(X)(k) = n1+ n2+ . .+.nk, and Proposition 16 implies that oeX oeH*(X). For the reverse inequality, let X(k) = {(x1, . .,.xr) | at leastr - k entries are*} X. It is well-known that X(0), X(1), . .,.X(r) constitute a cone decomposition of X. Furthermore, X(k - 1) contains the cellular (n1+ n2+ . .+.nk- 1)-skeleton of X, and so cat(Xn1+n2+...+nk-1) cat(X(k - 1)) < k. Therefore oeX (k) n1 + n2 + . .+.nk = oeH*(X)(k). 2 The following theorem gives surprisingly strong algebraic control over cat- egorical sequences. The proofs of parts (b) and (c) in full generality depend on the positive solution to Whitehead's problem; but they are valid in ordinary ZFC set theory if X is of finite type. Theorem 18 For any space X, (a)oeX (k + l) oeX (k) + oeX (l), (b)if X is simply-connected and oeX (k) = n, then Hn(X; A) 6= 0 for some coefficient group A, and (c)if equality occurs in (a) and X is simply-connected, then the cup prod- uct Hk(X; A) Hl(X; B) ! Hk+l(X; A B) is nontrivial for some choice of coefficients. Proof Write oeX (k) = a and oeX (l) = b. Then cat(Xa-1) = k - 1 and cat(Xb-1) = l - 1, which means that there are factorizations (X, *) ! (X, Xa-1) ! (X, *)k and (X, *) ! (X, Xb-1) ! (X, *)l of k and l, up to homotopy of pairs. Putting these together using cellular approximation and the factorization k+l = ( k x l) O 2, we obtain the homotopy-commutative diagram 13 (Xn, *)________//(X, *)___2_____//_(X, *) x_(X,_*)kx/l/_(X, *)k x (X, *)l | ff __|_________ | || | ___________|____ ______________|_ || fflffl|_____ fflffl| __((_fflffl|__ || (Xn, *)________//(X, *)_____//_(X, Xa-1)OxO(X, Xb-1)//_(X, *)k x (X, *)l | | | | | | fflffl| fflffl| | (Xn, Xa+b-1)____//(X, Xa+b-1)__//(X x X, (X x X)a+b-1). Taking n = a + b - 1 we see that k+l|Xa+b-1factors, up to homotopy of pairs, through (Xa+b-1, Xa+b-1), and so catX(Xa+b-1) < k + l by the Whitehead definition and Proposition 9. Therefore oeX (k + l) a + b, proving (a). Now we prove part (b). If oeX (k) = n, then catX(Xn) > catX(Xn-1), so X does not have an (n - 1)-dimensional n-skeleton. By Lemma 3(c), then, it cannot be that Hn(X; A) = 0 for all A. To prove the statement (c) about cup products, we first recall that by Theo- rem 18(b), if oeX (i) = m, then Hm (X; A) 6= 0 for some coefficient group G. Let u 2 Hm (X; A) be nonzero, and interpret it as a map u : X ! K(A, m). This map factors X NNN________u_____________________________________________* *_____________________________________________________________________________* *____________________________________ | NNN~mNNN_______________________________________________* *_____________________________________________________________________________* *_____ | _________________________________________* *__________________________________________ fflffl| NN&&N __))________________________________* *____ X=Xm-1 _~m_//_K(ssm ,_m)__//K(A, m), where ssm = ssm (X=Xm-1 ). Since u 6' *, ~m 6' * as well. Note also that K(ssm , m) may be constructed from X=Xm-1 by attaching cells of dimension m + 2 and higher, so ~m is an (m + 1)-equivalence. Since (X, Xa-1)x(X, Xb-1) = (XxX, XxXb-1[Xa-1xX) is an (a+b-1)- connected pair and X x Xb-1[ Xa-1 x X is 1-connected, we apply the Blakers- Massey theorem [35, Cor. 6.22] to conclude that the collapse map (X, Xa-1) x (X, Xb-1) ! (X=Xa-1 ^ X=Xb-1, *) is an (a + b + 1)-equivalence. Assuming oeX (k + l) = oeX (k) + oeX (l) = a + b, we may set n = a + b in the diagram of part (a) and conclude that the composite map (Xa+b, *) ! (X, Xa-1) x (X, Xb-1) is nontrivial. Because the collapse map is an (a + b + 1)- equivalence and Xa+b is (a + b)-dimensional, we see that the composition (Xa+b, *) ! (X, Xa-1) x (X, Xb-1) ! (X=Xa-1 ^ X=Xb-1, *) is also nontrivial. Now the desired cup product is Xa+b ________________________________________________________________* *_____________________________________________________________________________* *___________________________________________________________ _________~a.~b_________________________________________* *_____________________________________________________________________________* *_____________________________________________________________________________* *_______________ | _____________________________________________* *_____________________________________________________________________________* *_____________________________________________________ | ______________________________________* *_____________________________________________________________________________* *________________________________________ fflffl| ___++__________________________* *_____________________________________________ X=Xa-1 ^ X=Xb-1 _____//K(ssa, a) ^ K(ssb,_b)//_K(ssa ssb, a + b), 14 and it is nontrivial because the horizontal maps are all (a + b + 1)-equivalenc* *es and Xa+b is (a + b)-dimensional. 2 The following elementary computation illustrates the use of Theorem 18. Example 19 Let us consider the exceptional Lie group G2. It is known [23] that H*(G2; Z=2) ~= Z=2[x3]=(x43) (x5). Therefore oeG2 oeH*(G2;Z=2)= (0, 3, 6, 9, 14, 1, . .). by Proposition 16. On the other hand, we know oeG2(1) = 3 by Remark 12(b), so oeG2 (0, 3, 6, 9, 12, 1, . .).by Theorem 18(b,c); this determines oeG2 exc* *ept for oeG2(4). However, H*(G2; A) = 0 for * = 12, 13 and any abelian group A, so oeG2(4) 6= 12, 13 by Theorem 18(b). We conclude that oeG2 = (0, 3, 6, 9, 14). Theorem 18 implies the well-known result cat(X) _dimension(X)_connectivity* *(X). In [15], Ganea generalized this familiar upper bound to obtain an upper bound for the category of X in terms of the set of dimensions in which H*(X) is nontrivial. We now prove a further generalization by a completely different method. For a space X, let h(X) = {n | eHn(X; G) 6= 0 for someG}. Corollary 20 Let X be simply-connected and of finite type with oeX (k) = n. If there are integers 0 < a1 < a2 < . .<.al such that h(X) I1 [ I2 [ . .[.Il where Ij = [aj, aj + (n - 1)] (brackets denote closed intervals in R), then cat(X) < k(l + 1). Proof Consider the integers oeX (kj), j = 1, 2, . ...We show by induction that oeX (kj) aj. If oeX (kj) = 1 we are done, so we assume that this value is finite, and hence is an element of h(X) by Theorem 18(b). Since n 2 h(X), a1 n a1 + (n - 1). Now assume that oeX (k(j - 1)) aj-1. By Theorem 18(a), oeX (kj) oeX (k(j - 1)) + oeX (k) aj-1 + n, S S which implies that oeX (kj) 62 t oeX (kl)+oeX (k) = al+n * *by Theorem 18(a). Thus oeX (k(l +1)) 62 h(X), and so oeX (k(l +1)) = 1. Therefore cat(X) < k(l + 1) by Remark 12(e). 2 Ganea's theorem is the special case k = 1 when X is (n - 1)-connected. It should be noted, though, that Ganea's result applies for strong category (i.e., cone length), where ours only applies for ordinary Lusternik-Schnirelmann cat- egory. It would be interesting to know whether our generalization holds with cone length in place of category. 15 3.2 Rational Spaces The categorical sequence oeX for a rational space X can be easily bounded above in terms of any one of its models. Proposition 21 For any simply-connected rational space X, and any model A for X, oeX oeA . Proof Write oeA (k) = n and let B be the quotient of A by the differential ideal consisting of all elements of dimension n or greater. Then nil(B) < k and the quotient q : A ! B induces an isomorphism on cohomology in dimensions < n - 1 and an injection in dimension n - 1. Let N be the Sullivan minimal model for B and let r : M(X) ! N cover the map q. Then r has a spatial realization i : Z ! X such that q* = i* : H*(X) ! H*(Z) [11, Ch. 17]. It follows that i : Z ! X is a rational (n - 1)- skeleton. Since M(Z) ~ B by construction and nil(B) < k, we conclude using Theorem 6 that cat(Z) < k. It follows that oeX (k) n. 2 Example 22 Let (A, d) be the CDGA with generators xn, ym and wn+m-1 (subscripts indicate dimension; 2 n m) subject to the relations x2 = y2 = w2 = 0 and with differential determined by dx = dy = 0 and dw = xy. This is not a Sullivan algebra, but it does have a Sullivan model, M, and M has a spatial realization, X. Then A is a model for X, and according to Proposition 21, oeX oeA = (0, n, n + m, 2(n + m) - 1, 1, 1, . .).. But we can say even more, because the nonzero cohomology of X occurs in dimensions n, m, 2n + m - 1, n + 2m - 1 and 2(n + m) - 1. Since X is indistin- guishable from Sn _Sm through dimension n+m, we know that oeX (2) > n+m, and therefore oeX (2) 2n + m - 1. Thus oeX (0, n, 2n + m - 1, 2(n + m) - 1, 1, 1, . .).. Since A is finite-dimensional, so is H*(X), and we conclude that cat(X) 3. 4 Sequences and Fibrations In this section we study the relationship between the sequences oeF , oeE and o* *eB when F ! E ! B is a fibration sequence. Our general result is the key to a `mapping theorem' for categorical sequences of rational spaces. 4.1 General Spaces Our first result is proved by a slight generalization of the method Hardie used to prove the main result of [17]. 16 Proposition 23 Consider the diagram X NN NNN f|| NpOfNNNN q fflffl|p NN&&N F ___________//E__________//_B in which the bottom row is a fibration sequence. Then cat(f) + 1 (cat(p O f) + 1) . (cat(q) + 1). Proof Suppose cat(p O f) = k and that cat(q) = l. Then X has a cover X = A0[A1[. .[.Ak by subcomplexes such that (pOf)|Ai ' * for each i. Since p is a fibration with fiber F , f|Ai factors (up to homotopy) as jOgi, where gi: Ai! F* * . Therefore cat(f|Ai) cat(q) = lSand so we can write Ai= Ai0[ Ai1[ . .[.Ail where (q O gi)|Aij' *. Thus X = i,jAijwhere 0 i k and 0 j l and f|Aij' * for all i and j. Therefore cat(f) + 1 (k + 1)(l + 1). 2 Hardie's result is the special case in which f = idE. We are interested in t* *he more general situation in which f : X ! E is an n-skeleton. Theorem 24 Let F__q__//E_p__//Bbe a fibration sequence and write a = cat(q) cat(F ) and b = cat(p) cat(B). Then (a)oeE (k(a + 1)) oeB (k), and (b)oeE (k(b + 1)) oeF (k). Proof Let oeB (k) = n. Thus cat(Bn-1) < k and we have to show that cat(En-1) < k(a + 1). Consider the homotopy-commutative diagram En-1 ____//_________Bn-1 i|| || q fflffl|p fflffl| F ______//_E_______//B in which the dotted arrow exists by cellular approximation. According to Propo- sition 23, cat(i) (cat(p O i) + 1) . (cat(q) + 1) - 1 < (cat(Bn-1) + 1) . (a + 1) < k(a + 1), proving (a). For part (b), we let oeF (k) = n, so cat(Fn-1) = k - 1. Choose an (n - 1)- skeleton i : En-1 ! E. Since cat(p O i) cat(p) = b, we can write En-1 = A0 [ A1 [ . .[.Ab where Aj is a subcomplex of En-1 (so dim(Aj) < n) and (pOi)|Aj ' * for each j. Thus i|Aj factors (up to homotopy) through the F ! E, and so we have the diagram Fn-1 oo____Aj_________ | i|| | | Aj |fflffl fflffl| F ___q__//_E_p__//_B 17 in which the dotted arrow exists by cellular approximation. This proves that cat(i|Aj) cat(Fn-1) = k - 1, and so cat(i) < (b + 1)k, which implies oeE ((b + 1)k) n. 2 Remark 25 These inequalities are not the best possible. A quick look at the proof of Theorem 24 shows that, in studying the category of En, for example, the estimate cat(pOi) b can be improved to cat(pOi) cat(Bn), and similarly for the second formula. We leave the cumbersome formulation of the sharper results to the reader. Since the reverse formulas expressing oeE in terms of oeB and oeF are not entirely obvious, we record them here. Corollary 26 In the situation of Theorem 24, il mj (a)oeE (k) oeB k-a_a+1, and il mj (b)oeE (k) oeF k-b_b+1. In [6], Fadell and Husseini studied the Lusternik-Schnirelmann category of free loop spaces using a general result that relates the category of the fiber * *and the total space in a fibration sequence with a section. This result generalizes* * to a statement about categorical sequences. Corollary 27 Let F _____//Ep__//_Bbe a fibration sequence. If p has a section s, then oeE oeF . Proof Extend the given fibration sequence to the left to obtain tt_s___________________________________________________* *________________ E ___p_// B__@__//F____//_E. Since p has a section, the map @ : B ! F is trivial. Thus cat(@) = 0, and Theorem 24(a) implies oeF (k) = oeF ((cat(@) + 1)k) oeE (k). 2 We can now expand upon the main homotopy-theoretical result of [6]. Example 28 Let (X) = map (S1, X) denote the free loop space on X. Eval- uation at the basepoint determines a fibration p : (X) ! X with fiber X, and the map s : x 7! lx, where lx is the constant map lx(S1) = x, is a section of p. Therefore Corollary 27 shows that oe (X) oe X . In particular, cat( (X)) = 1 if cat( X) = 1. 18 4.2 A Mapping Theorem for Sequences One of the most powerful early results concerning the Lusternik-Schnirelmann category of rational spaces is the Mapping Theorem [8]; the `book proof' of this result [9] uses Proposition 23 in the special case cat(j) = 0. We use exactly t* *he same argument to get an inequality for categorical sequences. Proposition 29 Let f : X ! Y be a map between rational spaces which induces an injective map f* : ss*(X) ! ss*(Y ). Then oeX oeY . Proof Let q : F ! X be the homotopy fiber of f. According to the proof of the standard Mapping Theorem, the injectivity hypothesis on f* implies that q ' * and so cat(q) = 0 [5, Thm. 4.11]. It now follows from Theorem 24 that oeX (k) oeY (k) for all k. 2 5 Formal Sequences A simply-connected space X is formal if its cohomology algebra, with trivial differential, is a model for X [11, p. 156]. In this section we characterize t* *he categorical sequences of formal rational spaces in several ways. First we show that formal rational spaces and their cohomology algebras are isosequential. Proposition 30 If X is a simply-connected formal rational space, then oeX = oeH*(X). Proof By definition, H*(X) is a model for X. Propositions 21 and 16 show that oeH*(X) oeX oeH*(X), which proves the result. 2 Our main result in this section completely characterizes the sequences which can occur as categorical sequences of simply-connected rational formal spaces. Theorem 31 The following conditions on a sequence oe with oe(1) > 1 are equiv- alent: (a)oe = oeA for some CGA A, (b)oe(k + 1) k+1_koe(k)Wfor each k,Q (c)oe = oeW where W = Pi and Pi= Snj is a product of spheres, and (d)oe = oeX for some formal space X. Before proceeding to the proof of Theorem 31 we need to establish a technical result about sequences. Let 0 < k n be integers, write n = kx + r with 0 r < k and let r + s = k. Define o to be the sequence whose finite values are o = (0, x, 2x, . .,.sx, sx + (x + 1), sx + 2(x + 1), . .,.sx_+_r(x_+-1)z_* *___"). n We call o the optimal k-term sequence with o(k) = n. 19 Lemma 32 Assume that oe is a sequence satisfying condition (b) of Theorem 31, and that oe(k) < 1. Let o be the optimal k-term sequence with o(k) = oe(k). Then oe(j) o(j) for all j. Proof This is clearly true for j > k, because o(j) = 1 for such j. If oe(j) > o* *(j) for some j k, then oe(j) o(j) + 1, and so oe(j + 1) 1_joe(j) + oe(j) 1_joe(j) + (o(j) + 1) Now oe(j) > o(j) jx, so 1_joe(j) > x. Therefore oe(j + 1) > o(j) + (x + 1) o(j + 1). Inductively, we see that oe(l) > o(l) for all i l k, which contradicts the hypothesis oe(k) = o(k). 2 Proof of Theorem 31 We begin by proving that (a) implies (b). Let A be a CGA such that oe = oeA . If oe = (0, n) has length 1, then there is nothing to prove, so we proceed by induction, assuming that the implication is valid for sequences of length k. Write n = oe(k + 1) = oeA (k + 1). Then there is a nontrivial product x1x2. .x.k+12 An, where we write the terms in order so that |x1| |x2| . . .|xk+1|. For j k + 1we have x1x2. .x.j6= 0 2 A|x1|+|x2|+...+|xj|, so oeA (j) |x1| + |x2| + . .+.|xj| for each j. Since oeA (k + 1) = |x1| + |x2* *| + . .+.|xk+1| by construction, we have oeA (k + 1) - oeA (k)(|x1| + . .+.|xk+1|) - (|x1| + . .+.|xk|) = |xk+1| z_________k_terms_"_____________- = 1_k(|xk+1| + |xk+1| + . .+.|xk+1|) 1_k(|x1| + |x2| + . .+.|xk|) 1_koeA (k), which proves the result. Next we prove that (b) implies (c) by induction on the length k of the sequence oe. If oe = (0, n), then oe = oeSn and the result holds. Suppose now t* *hat the result is known for all sequences with length k, and let oe be a sequence with length k + 1. Write ~oefor the sequence ae ~oe(j) = oe(j)1ifjifjk> k. Since length(~oe) k, we can apply the inductive hypothesis, to find a wedge of products of spheres W such that oeW = ~oe. Let o be the optimal (k + 1)-term sequence with o(k + 1) = oe(k + 1), and define z____s_factors_"________-z_r_factors_"___________- P = Sx x Sx x . .x.Sxx Sx+1 x Sx+1 x . .x.Sx+1. 20 Then oeP = o by Corollary 17, and Proposition 15 shows that oeW_P (j) = min{oeW (j), oeP (j)} for all j. For j k, we have oeW (j) = oe(j) o(j) = oeP (j) by Lemma 32, so oeW_P = oe(j) for j < k by Proposition 15(a). Also oeP (k + 1) = oe(k + 1) < 1 = oeW (k + 1), so oeW_P (k + 1) = oe(k + 1). The implication (c) ) (d) follows from the fact that the rationalization of a wedge of products of spheres is formal. According to Proposition 30, if X is a formal rational space, then oeX = oeH*(X). Thus (d) implies (a). 2 In view of Theorem 31, we define a formal sequence to be any sequence oe which satisfies the condition oe(k + 1) k_+_1_koe(k) for allk. It is not true that every formal space is isosequential with its minimal mod* *el. For example, the minimal model of S4 is (x4, x7), so oeS4 = (0, 4) > (0, 4, 8, 12, . .).= oeM(S4). Our study of formal sequences grew out of a simple question: is every simply- connected rational space isosequential with a product of spheres, or a wedge of products of spheres, or a product of wedges of products of spheres, etc? Any space constructed from spheres by repeatedly taking products and wedges is automatically formal [5, Ex. 5.4]. Using Theorem 31, we see that any such space is isosequential with a simple wedge of products of spheres. Furthermore, Theorem 31 reveals that our original question reduces to asking whether or not oeX is a formal sequence whenever X is a rational space. We have already seen that this is not the case! Example 33 The space X of Example 13(b) is a rational space whose categor- ical sequence is oeX = (0, 3, 8, 11). Since 11 < 3_2. 8, oeX is not a formal se* *quence. By Theorem 31, X is not isosequential with any wedge of products of spheres. 6 Products For two sequences oe and o, we define a new sequence oe * o by oe * o(k) = min{oe(i) + oe(j) | i + j = k} Our goal in this section is to prove a result linking the sequences oeXxY and oeX * oeY . When the spaces in question are formal, this is not hard to do. Proposition 34 Let A and B be simply-connected CGAs and let X and Y be simply-connected formal rational spaces. Then 21 (a)oeA B = oeA * oeB, and (b)oeXxY = oeX * oeY . Proof We omit the easy proof of (a), and use it to prove (b) as follows: since X, Y and X x Y are each formal and rational, oeXxY = oeH*(XxY ) = oeH*(X) H*(Y ) = oeH*(X) * oeH*(Y ) = oeX * oeY by Proposition 30. 2 The following conjecture seems quite plausible. Conjecture 1 For simply-connected rational spaces X and Y , oeXxY = oeX * oeY . Unfortunately, we have been unable to prove this. However, we can prove that there is an inequality relating these sequences. Theorem 35 For simply-connected rational spaces X and Y , oeXxY oeX *oeY . Proof Let oeX * oeY (k) = n. Thus there are i and j with i + j = k, oeX (i) = * *a, oeY (j) = b, and a + b = n. Now let ia x ib : Xa x Yb ! X x Y and compute cat((X x Y )n) catXxY (Xa x Yb) Mcat(ia x ib) by Proposition9(c) = Mcat(ia) + Mcat(ib) by [24, Thm. 2] = cat(Xa) + cat(Xb) by Proposition 9(c) = k, which means that oeXxY (k) n = oeX * oeY (k), 2 The inequality of Theorem 35 fails when the spaces are not rational, as the following example demonstrates. Example 36 Iwase [20] has constructed a space X = S2 [ D10 with the prop- erty that cat(X x Sk) = cat(X) = 2 for all k 2. The categorical sequences for X and S2 are oeX = (0, 2, 10, 1, . .).and oeS2 = (0, 2, 1, . .)., respectively.* * Now we have oeX * oeS2 = (0, 2, 4, 12, 1, . .).< (0, 2, 4, 1, . .).= oeXxS2 . Nevertheless, the following conjecture seems reasonable. Conjecture 2 For general spaces X and Y , oeXxY oeX * oeY . Observe that Conjecture 2, together with Theorem 35, implies Conjecture 1. 22 7 The Mislin Genus of Sp(3) In this section we use categorical sequences to give a simple proof of a theorem of Ghienne [16]. The Mislin genus of a nilpotent space X is the set G(X) of homotopy types of nilpotent spaces Y such that the p-localizations X(p)and Y(p)are homotopy equivalent for every prime p. McGibbon [22, x 8] asked whether Lusternik- Schnirelmann category is an invariant of Mislin genus; that is, if X 2 G(Y ), d* *oes it follow that cat(X) = cat(Y )? This is known to be false for certain infinite- dimensional spaces [26], but the question remains open for finite complexes Y . In [16], Ghienne proved that McGibbon's conjecture holds in the special case Y = Sp(3). We use a sequence computation to give a simple alternative proof of this result. Theorem 37 (Ghienne) If X 2 G(Sp(3)), then cat(X) = 5. Proof According to [13, 21], wcat(Sp(3)) = cat(Sp(3)) = 5. Since weak category is a genus invariant, we have cat(X) wcat(X) = wcat(Sp(3)) = 5 for any space X 2 G(Sp(3)). It remains to show that cat(X) 5 for ev- ery X 2 G(Sp(3)). In fact, we prove the following stronger statement: any simply-connected space X whose cohomology ring H*(X; Z) is isomorphic to H*(Sp(3); Z) must have cat(X) 5. The categorical sequence oeX clearly has oeX (1) = 3 and oeX (2) 7 by Theorem 18(b). By Theorem 18(a), oeX (4) oeX (2)+oeX (2) 14. Furthermore, oeX (4) > 14 by Theorem 18(c), because the cup product H7(X) H7(X) ! H14(X) is trivial. Now we have oeX (4) 18 by Theorem 18(b), and hence oeX (5) oeX (4)+oeX (1) = 21. From this we immediately conclude that cat(X) = cat(X21) 5. 2 McGibbon's conjecture for finite complexes is equivalent to the following conjecture for finite type spaces. Conjecture 3 If X is a nilpotent space of finite type, then oeY = oeX for every Y 2 G(X). Conjecture 3 is easily seen to be valid for X = Sp(2). 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