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
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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.
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