# Contractible manifolds

Classify the closed contractible manifolds (Recall that a manifold is closed if it is compact and its boundary is empty.)

1. There are no closed contractible manifolds of positive dimension! If such manifold is orientable, then there is a nonzero top integral homology class by deRham and Stokes. If it is non-orientable, there is a nonzero $\Bbb Z_2$ top cohomology class, and by universal coefficients, there is a nonzero top integral homology class.