Fundamental groups of manifolds with S1-category 2
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 -> M^n factor homotopically through maps W_i -> S^1 -> M^n. We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.