nLab homotopy sphere

Idea

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

Properties

Corollary

Every homotopy sphere is a rational homotopy sphere.

Corollary

Every homotopy sphere is a homology sphere.

Theorem

Every simply connected homology sphere is a homotopy sphere.

Proposition

A path-connected, $n$-connected, orientable and closed $2n$-manifold is a homotopy-$2n$-sphere.

Proposition

A path-connected, $n$-connected, orientable and closed $2n+1$-manifold is a homotopy-$2n+1$-sphere.

• Every smooth manifold homotopy-4-sphere is homeomorphic to the actual 4-sphere.

  (due to Michael Freedman, see Siebenmann 1982)

Related concepts

• rational homotopy sphere
• homology sphere
• rational homology sphere

Redirects

• Wikipedia, Homotopy sphere

References

• Laurent Siebenmann: La conjecture de Poincaré topologique en dimension 4, Séminaire Bourbaki : volume 1981/82, exposés 579-596, Astérisque, no. 92-93 (1982), Talk no. 588 [numdam:SB_1981-1982__24__219_0]

• Laurent Siebenmann (translation by M. H. Kim & M. Powell): Topological Poincaré conjecture in dimension 4 (the work of M. H. Freedman), Celebratio Mathematica [webpage]