Manifolds with S1-category 2 have cyclic fundamental groups

J. C. Gomez-Larranaga, F. Gonzalez-Acuna, W. Heil

A closed topological $n$-manifold $M^n$ is of $S^1$-category $2$ if it can be covered by two open subsets $W_1$, $W_2$ such that the inclusions $W_i \to M^n$ factor homotopically through maps $W_i \to S^1$. We show that for $n>3$ the fundamental group of such an $n$-manifold is either trivial or infinite cyclic.