# Product of 3-sphere and circle

From Topospaces

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

## Contents

## Definition

This topological space is defined as the Cartesian product of the 3-sphere and the circle , equipped with the product topology. It is denoted or .

## 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 a module are as follows:

### Cohomology groups

`Further information: cohomology of product of spheres`

The cohomology groups with coefficients in integers are as follows:

The cohomology groups with coefficients in a module 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. | , and 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 | 0 | 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. In this case, both the factor spaces have Euler characteristic zero. |