# Homology of spheres

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology group and the topological space/family is sphere

Get more specific information about sphere | Get more computations of homology group

In this article, we briefly describe the homology groups of spheres, a proof using the Mayer-Vietoris homology sequence, and explanations in terms of cellular and simplicial homology.

## Contents

## Statement

### Reduced version over integers

For a nonnegative integer, we have the following result for the reduced homology groups:

:

and:

### Unreduced version over integers

We need to make cases based on whether or is a positive integer:

- case: and is trivial for .
- case: and is trivial for .

### Reduced version over a module over a ring

For a nonnegative integer, we have the following result for the reduced homology groups:

:

and:

### Unreduced version over a module over a ring

We need to make cases based on whether or is a positive integer:

- case: and is trivial for .
- case: and is trivial for .

## Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant | General description | Description of value for spheres |
---|---|---|

Betti numbers | The Betti number is the rank of the homology group. | For , , all other are ; for , , all other s are . |

Poincare polynomial | Generating polynomial for Betti numbers | for , for |

Euler characteristic | for even, for odd. |

## Facts used

## Proof using singular homology

### Equivalence of reduced and unreduced version

The equivalence follows from the fact that reduced and unreduced homology groups coincide for and for , the unreduced homology group is obtained from the reduced one by adding a copy of (or, if working over another ring or module, the base ring or module).

### Proof of reduced version

The case is clear: the space is a discrete two-point space, hence it has two single-point path components, so the zeroth homology group is . Higher homology groups are trivial because the cycle and boundary groups both coincide with the group of *all* functions to , so the homology group is trivial.

In general, we use induction, starting with the base case . The inductive step follows from fact (1) and the fact that each is the suspension of .

## Proof using cellular homology

See (2): CW structure of spheres.

## Proof using simplicial homology

See (3): Simplicial structure of spheres.