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

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# Contractible manifolds

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

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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 top cohomology class, and by universal coefficients, there is a nonzero top integral homology class.

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