Two Special Cases of Ganea’s Conjecture
نویسنده
چکیده
Ganea conjectured that for any finite CW complex and any k > 0, cat(X×Sk) = 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). We show that if cat(X) = ⌊ n p ⌋ + 1 and n 6≡ −1 mod p (which implies p > 1), then cat(X×Sk) = cat(X) + 1. This is proved by showing that wcat(X×Sk) = 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.
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