Lusternik-Schnirelmann category of 3-manifolds (Q1209387)

For any space \(X\) let \(\text{cat}(X)\) (resp. \(\text{cat}_{\pi_ 1}(X)\)) be the smallest integer such that \(X\) can be covered by \(n+1\) open sets \(U\) such that \(U\) is contractible in \(X\) (resp. each loop in \(U\) is contractible in \(X\)). The authors prove that for a closed 3-manifold \(M\) \(\text{cat}(M)\) depends only on the fundamental group. It is 1, 2, 3, if \(\pi_ 1(M)\) is trivial, free nontrivial, not free resp., and \(\text{cat}_{\pi_ 1}(M)=0,1,3\) correspondingly. The method consists in combining the behaviour of \(\text{cat}_{\pi_ 1}\) with that of the homomorphism of homology induced by the canonical map \(M\to BG\), \(G=\pi_ 1(M)\). It also leads to another proof (in the spirit of algebraic topology) of Krasnoselski's theorem that \(\text{cat}_{\pi_ 1}(\Sigma^ n/G)= \text{cat}(\Sigma^ n/G)= n+1\), where \(G\) is a finite nontrivial group acting freely on the homotopy sphere \(\Sigma^ n\).