Two Special Cases of Ganea's Conjecture
Jeffrey A. Strom
University of Wisconsin-Madison
strom@math.wisc.edu
Ganea conjectured that for any finite CW complex and any
k>0, cat(X x S^k) = cat(X) + 1. In this paper we prove
two special cases of this conjecture. The main result is
the following. Let X be a (p-1) connected n dimensional
CW complex (not necessarily finite). If cat(X) = [n/p] + 1
and n is not congruent to -1 mod p (which implies p > 1),
then cat(X x S^k) = cat(X) + 1. This is proved by showing
that wcat(X x S^k) = wcat(X) + 1 in a much larger range,
and then showing that under the conditions imposed,
cat(X) = wcat(X). The second special case is an extension
of Singhof's earlier result for manifolds.