nLab rational homology 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 rational homology sphere is a topological space which need not be homeomorphic to an n-sphere, but which has the same rational homology as an nn-sphere.

Properties

Corollary

Every homology sphere is a rational homology sphere.

Examples

Example

The Wu manifold W=SU(3)/SO(3)W=SU(3)/SO(3) is a simply connected rational homology 55-sphere (with non-trivial homology groups H 0(W)β‰…β„€H_0(W)\cong\mathbb{Z}, H 2(W)β‰…β„€ 2H_2(W)\cong\mathbb{Z}_2 and H 5(W)β‰…β„€H_5(W)\cong\mathbb{Z}), but isn’t a homotopy 5 5 -sphere.

Last revised on May 31, 2024 at 15:04:15. See the history of this page for a list of all contributions to it.