nLab homology sphere

Idea

A homology sphere is a topological space which need not be homeomorphic to an $n$-sphere, but which has the same ordinary homology as an $n$-sphere.

Properties

Corollary

Every homology sphere is a rational homology sphere.

Corollary

Every homotopy sphere is a homology sphere.

Theorem

Every simply connected homology sphere is a homotopy sphere.

Proposition

There are simply connected homology-$n$-spheres not homeomorphic to the $n$-sphere iff $n\geq 5$.

(Ruberman 94, Example 7, Barden 65 and MO/214727)

Related concepts

• rational homology sphere

• homotopy sphere

• rational homotopy sphere

References

• Daniel Ruberman: Null-homotopic embedded spheres of codimension one, in: Tight and Taut Submanifolds, Math. Sci. Res. Inst. Publ. 32, Cambridge University Press (1998) 229-232 [pdf, ISBN:9780521620475]

• D. Barden, Simply connected five-manifolds. Ann. of Math. 82 3 (1965) 365-385 [jstor:1970702, doi:10.2307/1970702]

See also:

• Wikipedia, Homology sphere