# Three-dimensional lens space

## Contents

## Definition

Suppose are relatively prime integers (it turns out that the choice of is relevant only modulo ).

Consider the 3-sphere . View this as the following set:

Denote by a primitive root of unity (explicitly, we can take . Consider the continuous map given by:

Note that iterating times gives the identity map, so we get the action of a cyclic group of order on where the generator is . The lens space is defined as the quotient of under the equivalence relation of being in the same orbit under this group action.

## Facts

### Determination up to homeomorphism

The spaces and are homeomorphic if and only if or .

### Determination up to homotopy type

The spaces and are of the same homotopy type if and only if either or is a quadratic residue modulo . *Note: Despite the choice of notation, is not here assumed to be a prime number.*

## Particular cases

Value of | Value of | Cyclic group of order | Quotient of by this as the subgroup of roots of unity |
---|---|---|---|

1 | 1 | trivial group | 3-sphere |

2 | 1 | cyclic group:Z2 | real projective three-dimensional space . Also can be identified as a Lie group with . |

3 | 1 | cyclic group:Z3 | lens space:L(3,1) |

### Number of lens spaces for values of

Value of | Number of homeomorphism classes of lens spaces | List of -values grouped by homeomorphism class of | Number of homotopy types of lens spaces | List of -values grouped by homotopy type of |
---|---|---|---|---|

1 | 1 | 1 | ||

2 | 1 | 1 | ||

3 | 1 | 1 | ||

4 | 1 | 1 | ||

5 | 2 | 2 | ||

6 | 1 | 2 | ||

7 | 2 | 1 | ||

8 | 2 | 2 | ||

9 | 2 | 1 |