# Product of two 2-spheres

From Topospaces

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## Contents

## Definition

This topological space is defined as the Cartesian product of two copies of the 2-sphere , equipped with the product topology. It is denoted .

## Topological space properties

## Algebraic topology

### Homology groups

`Further information: homology of product of spheres`

The homology groups with coefficients in integers are as follows:

The homology groups with coefficients in any module over a ring are as follows:

### Cohomology groups

`Further information: cohomology of product of spheres`

The cohomology groups with coefficients in integers are as follows:

The homology groups with coefficients in any module over a ring are as follows:

### Homology-based invariants

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

Betti numbers | The Betti number is the rank of the torsion-free part of the homology group. | , , , all higher s are zero. | |

Poincare polynomial | Generating polynomial for Betti numbers, i.e., the polynomial | See also Poincare polynomial of product is product of Poincare polynomials. The Poincare polynomial for is | |

Euler characteristic | , obtained by evaluating Poincare polynomial at -1. | 4 | Follows from Euler characteristic of product is product of Euler characteristics. In particular, the Euler characteristic of a product space is zero if any of the factor spaces has Euler characteristic zero. |