Manifolds with \(S ^{1}\)-category 2 have cyclic fundamental groups (Q604218)

A closed topological \(n\)-manifold is of \(S^1\)-category 2 if it can be covered by 2 open subsets \(W_1, W_2\) such that the inclusions \(W_i\to M^n\) factor homotopically through maps \(W_i\to S^1\). The authors show: Theorem. If \(M\) is a closed \(n\)-manifold for \(n>3\) with \(S^1\)-category 2 then its fundamental group is trivial or infinite cyclic.