On the Lusternik-Schnirelmann category of maps
نویسنده
چکیده
We give conditions when cat(f × g) < cat(f) + cat(g). We apply our result to show that under suitable conditions for rational maps f , mcat(f) < cat(f) is equivalent to cat(f) = cat(f×idSn). Many examples with mcat(f) < cat(f) satisfying our conditions are constructed. We also resolve one open case of Ganea’s conjecture by constructing a space X such that cat(X × S1) = cat(X) = 2. In fact for every Y , cat(X × Y ) ≤ cat(Y ) + 1 < cat(Y ) + cat(X). We show that this same X has the property cat(X) = cat(X ×X) = cl(X ×X) = 2.
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