A minimum dimensional counterexample to Ganea's conjecture
Let us recall that the Lusternik-Schnirelmann category of a space \(X\), \(\text{cat}\,X\), is one less than the minimal number of open subspaces of \(X\) contractible in \(X\) that are necessary to cover \(X\). In 1971, Ganea conjectured that \(\text{cat}(X\times S^n)=\text{cat}\,X + 1\) for any finite CW-complex and \(n\geq 1\). By results of Jessup and Hess this is true for rational spaces, but not in general. The first counterexample has been given by Iwase in 1998. The lowest dimension of all the known counterexamples was 10, while Vandembroucq proved that there is no counterexample in dimension \(\leq 5\). The main result of this paper is the production of a counterexample \(T\) in dimension 7. The space \(T\) is quite simple, \(T = S^2\vee S^6\cup_f e^7\) for some particular \(f\). We have \(\text{cat}\,T=2\) and \(\text{cat}(T\times S^r)= 2\) for \(r\geq 2\). The proof is based on the possibility of factoring a Hopf invariant stably.
scientific article; zbMATH DE number 5775747
scientific article