= Homotopy sphere
{title2=$M^d\simeq S^d$}
{wiki}
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.
Back to article page
