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 is homeomorphic to . 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.
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A **homotopy sphere** is a mathematical concept in the field of topology, specifically in geometric topology. It refers to a manifold that is homotopically equivalent to a sphere. This means that, while a homotopy sphere may not be geometrically the same as a standard sphere (such as the 2-sphere \( S^2 \) in three-dimensional space), it shares the same topological properties related to how paths can be continuously deformed within it.