Homology sphere

An integral homology $d$-sphere is a space with the same integral homology as $S^d$. Removing one point from a closed oriented integral homology sphere produces an acyclic space by the long exact sequence of the pair and the local orientation isomorphism in degree $d$.

A simply connected closed three-dimensional manifold is orientable, since its orientation monodromy factors through its trivial fundamental group. Its first integral homology is the abelianization of that fundamental group and is zero. The universal coefficient theorem for cohomology gives $H^1(M;\mathbb{Z})=0$, and Poincare duality gives $H_2=0$, $H_3=\mathbb{Z}$. Connectedness gives $H_0=\mathbb{Z}$, and duality gives zero homology above dimension three. Thus it is an integral homology sphere, without needing any homeomorphism classification.

A homotopy sphere is a closed manifold with the homotopy type of a sphere of the same dimension. In dimensions at least two, a closed simply connected integral homology sphere is a homotopy sphere: successive applications of the Hurewicz theorem produce a map from the sphere inducing an isomorphism on integral homology, and the homological Whitehead theorem makes it a homotopy equivalence. The topological generalized Poincare theorem identifies its topological type, while prescribed smooth structures require a separate question.

Every closed topological manifold with the homotopy type of $S^d$ is homeomorphic to $S^d$. This is a classification theorem about homeomorphisms, and does not assert a diffeomorphism between prescribed smooth structures. It is the precise final step needed after proving that a cone link is a simply connected integral homology sphere in the manifold criterion for a suspension.