The Lusternik-Schnirelmann Category of S1Qx S1 and S1Qx S1Q Jeffrey Strom Dartmouth College Email: Jeffrey.Strom@Dartmouth.edu Abstract We answer a question of Rudyak by showing that cat(S1QxS1) = cat(S1QxS1Q) = 3. The second formula shows that X = S1Qis an example of a space for which cat(X x X) < 2 cat(X). These calculations are derived from a general formula for the category weight of elements of H*(BG; ß) that is of independent interes* *t. AMS Classification numbers Primary: 55M30 Secondary: 55P62 Keywords: Lusternik-Schnirelmann category, Ganea's conjecture The Lusternik-Schnirelmann category of a map f : X -! Y between CW complexes is the least integer n for which X has a cover {A0, . .,.An} by subcomplexes with the property that f|Ai ' * for each i [1]. The category of a space X , cat(X), is the category of the identity map idX . A classical result due to Bassi [4, Thm. 9] shows that, for two CW complexes X and Y , cat(X x Y ) cat(X) + cat(Y ). Until recently, the only known cases in which this formula was not an equality involved torsion phenomena in homology. In [5], Ganea asked whether cat(X x Sk) = cat(X) + 1 for every space X . That this is true has come to be known as Ganea's conjecture. The rational version of the conjecture, which can be stated cat(XQ x SkQ) = cat(XQ) + 1 when X x Sk is simply connected, was proved for simply connected spaces by Jessup and Hess in [10, 7] (we denote by XQ the rationalization of the space X ). More recently, counterexamples to the conjecture have been constructed by Iwase [8], by Stanley [17], and others. In the wake of these counterexamples, there 1 has been some interest in the related problem of finding space X such that cat(X x X) < 2 cat(X) [12]. In this note we use category weight techniques to compute the Lusternik- Schnirelmann category of the spaces S1Qx S1 and S1Qx S1Q, thereby answering a question asked by Y. Rudyak [15]. The calculation shows that S1Qsatisfies the Ganea conjecture, and also has the property that cat(X x X) < 2 cat(X). Recall that the category weight of a map f : X -! Y is the least integer n such that f O g ' * for every g : Z -! X with cat(g) < n. We write wgt (f) = n if the category weight of f is n, and wgt (f) = 1 if there is no such integer. The category weight of a cohomology class is obtained from the isomorphism Hn(X; ß) ~= [X, K(ß, n)]. Clearly wgt (f) is a lower bound for cat(X), provided f is nontrivial. This concept has also been called strict category weight (see Rudyak [13, 14, 11]) or essential category weight [18, 19]. We will make use of a well-known alternative characterization of the category of a map f : X -! Y . For each space Y and n 0 there is a fibration pn : Gn(Y ) -! Y with the property that cat(f) n if and only if f has a lift into Gn(Y ) [6]; these are known as the Ganea fibrations. If X ' BG for some discrete group G then the map pn is homotopically equivalent to the inclusion BnG ,! BG ' X [20], see also [18]. The following result is a basic property of category weight [13, 14, 18, 19]. Theorem 1 Let f : X -! Y , and let pn : Gn(X) -! X be the nth Ganea fibration. Then wgt(f) n if and only if f O pn ' *. This follows immediately from the definitions. The following corollary, which can be found in [18, Cor. 79] and implicitly in the proof of [11, Thm. 4.1], has proved useful in differential geometry; in fact, it is an essential ingredient * *in the proof of the Arnold conjecture for certain special symplectic manifolds [11, 16* *]. Corollary 2 If G is a discrete group and u 2 Hn(BG; ß), then wgt(u) = n. Proof Since G is a discrete group, Gn(BG) ' BnG which is (n-1)-dimensional, and hence has trivial cohomology in dimensions n. 2 This corollary generalizes to finite-dimensional groups: if G is a d-dimensional topological group and u 2 Hn(BG; ß), then wgt (u) _n_d+1. This shows, for example, that every nonzero class u 2 H4n(HPm ; ß) has wgt (u) = n, even without a cup product structure. Corollary 2 immediately implies a result of Eilenberg and Ganea: if G is a discrete group then the category of BG is bounded below by its cohomological 2 dimension [2]. Since S1Qis a K(Q, 1) and H2(S1Q; Z) ~= Ext(Q, Z) which is isomorphic to R as rational vector spaces, it follows that cat(S1Q) 2; since S1Qis homotopy equivalent to a 2-dimensional space we see that cat(S1Q) = 2. Therefore, Ganea's conjecture predicts cat(S1Qx S1) = cat(S1Q) + 1 = 3. Since H*(S1Q) is torsion free, the general product formula which motivated Ganea's conjecture predicts that cat(S1Qx S1Q) is equal to 4. With these preliminaries in place, we state and prove our main theorem. Theorem 3 cat(S1Qx S1) = cat(S1Qx S1Q) = 3. Proof Notice first that it follows from Bassi's formula that cat(S1Qx S1) 3, and since S1Qx S1Q' (S1 x S1)Q can be constructed as a 3-dimensional CW complex, cat(S1Qx S1Q) 3 as well. Now we have from the universal coefficient formula H3(S1Qx S1; Z) ~=Ext(Q, Z) Z ~=Ext(Q, Z) ~=R 6= 0 and H3(S1Qx S1Q; Z) ~=Ext(Q, Z) ~=R 6= 0. Since S1Qx S1 ' K(Q x Z, 1) and S1Qx S1Q' K(Q x Q, 1), the result follows immediately from Corollary 2. 2 Remark In fact, a similar argument shows that cat((S1Q)n) = n + 1 < 2n. References [1]I. Berstein and T. Ganea, The category of a map and of a cohomology class, Fund. Math. 50 (1961/1962), 265-279. [2]S. Eilenberg and T. Ganea, On the Lusternik-Schnirelmann category of abst* *ract groups, Ann. of Math. (2) 65 (1957), 517-518. [3]Y. F'elix, S. Halperin and J.-M. 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