Homotopy groups, focal points, and totally geodesic immersions Sergio MendoncaHeudson Mirandola 53C20 (53C42)homotopy groupsfocal pointsimmersions In this paper, on a complete Riemannian manifold $M$ we consider an immersed totally geodesic hypersurface $\Sigma$ existing together with an immersed submanifold $N$ without focal points. No curvature condition is needed. We obtain several connectedness results relating the topologies of $M$ and $\Sigma$ which depend on the codimension of $N$. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5097 10.1512/iumj.2013.62.5097 en Indiana Univ. Math. J. 62 (2013) 1075 - 1103 state-of-the-art mathematics http://iumj.org/access/