TITLE: Converse of Poincaré-Hopf theorem
QUESTION [14 upvotes]: Let $M$ be a connected, compact, oriented manifold of dimension $n<7$. If any two maps $M \to M$ having equal degrees are homotopic, must $M$ be diffeomorphic to the $n$-sphere?

REPLY [8 votes]: To complete @RyanBudney's  suggestion (and @Misha's comment), there are 3-manifolds in the snappea census which have trivial isometry group, so in particular have no non-trivial degree 1 self-maps, and which have trivial first betti number. If the map is degree $>0$, then this follows from Gromov's proof of Mostow rigidity (using the Gromov norm, the degree must $=\pm 1$, and the map is homotopic to an isometry).  If a degree zero map from a 3-manifold to itself is homotopically non-trivial, then it must surject the fundamental group of some covering space. If this covering space is finite-index, then since the group is co-finitely Hopfian, this map must induce an isomorphism on fundamental group, and thus must be degree $\pm 1$, a contradiction. Thus, the image group must be infinite index. These groups have positive betti number, a contradiction.