nLab homotopy sphere

Contents

Context

Spheres

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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 nn-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, nn-connected, orientable and closed 2n2n-manifold is a homotopy-2n2n-sphere.

Proposition

A path-connected, nn-connected, orientable and closed 2n+12n+1-manifold is a homotopy-2n+12n+1-sphere.

Redirects

References

  • Laurent Siebenmann, Topological PoincarΓ© conjecture in dimension 4 (the work of M. H. Freedman) (pdf)

Last revised on March 21, 2024 at 16:32:47. See the history of this page for a list of all contributions to it.