An integral homology -sphere is a space with the same integral homology as . 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 .
A homology sphere is a topological space that behaves like a sphere in terms of its homological properties, even if it is not actually a sphere in the classical sense. More formally, an \( n \)-dimensional homology sphere is a manifold that is homotopy equivalent to the \( n \)-dimensional sphere \( S^n \), and, importantly, it has the same homology groups as \( S^n \).
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