"Ganea's conjecture on LS category"
N.Iwase (Kyushu and Aberdeen)

The LS (Lyusternik-Schnirelmann) category of a manifold was originally defined to be the lower bound of the number of the critical points of Morse functions of the manifold, and is the least number of the closed sets which cover the manifold and are contractible in the manifold. There are various definitions of LS categories. The "normalised LS category" cat(X) of a general CW complex X is defined to be the least number m so that the diagonal map of X to X^{m+1} is compressible into the fat wedge X^{[m+1]}. For a manifold, the new definition differs from the original definition by 1, so that cat{*} = 0.

The Rumanian mathematician Tudor Ganea died 1971 just after a conference on topology. He contributed the conference by giving a list of problems. But several problems in the list are still unknown: e.g., Problem 10 is a fundamental conjecture on a co-H-space... My talk concerns Ganea's Problem 2: is  cat(XxS^n) = cat(X)+1?

There has been some supporting evidence: no counterexamples could be found, the rational version of the conjecture has been verified by Hess and Jessup and the original conjecture for some manifolds has recently be verified by Rudyak.

I consider LS category by using the quasi-fibring associated with the loop space of the given space: let cat(X) = m. Then
cat(XxS^n) =< m+1.  Here the obstruction to that cat(X) = m-1 is obtained as a map to the total space of the quasi-fibring whose n-fold suspension gives essentially the one to that cat(XxS^n) = m, while some maps induce the trivial stable map. This suggests the existence of counterexamples for the conjecture. In my talk, I will construct a series of counter examples to the conjecture.