Higher dimensional 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 $W_{i}\rightarrow M^{n}$ factor homotopically through maps $W_{i}\rightarrow S^{1}$. We show that for $n>3$, if $cat_{S^1}(M^n )=2$ then $M^n\approx S^n$ or $M^n \approx S^{n-1}\times S^1$ or the non-orientable $S^{n-1}$-bundle over $S^1$.