# Three-dimensional lens space

## Definition

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

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

$\{ (z_1,z_2) \in \mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1 \}$

Denote by $\zeta$ a primitive $p^{th}$ root of unity (explicitly, we can take $\zeta = \cos(2\pi/p) + i\sin(2\pi/p) = \exp(2\pi i/p)$. Consider the continuous map $f_{p,q}:S^3 \to S^3$ given by:

$\! f_{p,q}(z_1,z_2) = (\zeta z_1, \zeta^q z_2)$

Note that iterating $f_{p,q}$ $p$ times gives the identity map, so we get the action of a cyclic group of order $p$ on $S^3$ where the generator is $f_{p,q}$. The lens space $L(p,q)$ is defined as the quotient of $S^3$ under the equivalence relation of being in the same orbit under this group action.

## Facts

### Determination up to homeomorphism

The spaces $L(p,q)$ and $L(p,q')$ are homeomorphic if and only if $q \equiv \pm q' \pmod p$ or $qq' \equiv \pm 1 \pmod p$.

### Determination up to homotopy type

The spaces $L(p,q)$ and $L(p,q')$ are of the same homotopy type if and only if either $qq'$ or $-qq'$ is a quadratic residue modulo $p$. Note: Despite the choice of notation, $p$ is not here assumed to be a prime number.

## Particular cases

Value of $p$ Value of $q$ Cyclic group of order $p$ Quotient of $S^3$ by this as the subgroup of $m^{th}$ roots of unity
1 1 trivial group 3-sphere $S^3$
2 1 cyclic group:Z2 real projective three-dimensional space $\R\mathbb{P}^3$. Also can be identified as a Lie group with $SO(3,\R)$.
3 1 cyclic group:Z3 lens space:L(3,1)

### Number of lens spaces for values of $p$

Value of $p$ Number of homeomorphism classes of lens spaces List of $q$-values grouped by homeomorphism class of $L(p,q)$ Number of homotopy types of lens spaces List of $q$-values grouped by homotopy type of $L(p,q)$
1 1 $\{ 1 \}$ 1 $\{ 1 \}$
2 1 $\{ 1 \}$ 1 $\{ 1 \}$
3 1 $\{ 1,2 \}$ 1 $\{ 1,2 \}$
4 1 $\{ 1,3 \}$ 1 $\{ 1,3 \}$
5 2 $\{ 1,4 \}, \{ 2,3 \}$ 2 $\{ 1,4 \}, \{ 2,3 \}$
6 1 $\{ 1,5 \}$ 2 $\{ 1,5 \}$
7 2 $\{ 1,6 \}, \{ 2,3,4,5 \}$ 1 $\{ 1,2,3,4,5,6 \}$
8 2 $\{ 1, 7\}, \{ 3,5 \}$ 2 $\{ 1,7 \}, \{ 3,5 \}$
9 2 $\{ 1,8 \}, \{ 2,4,5,7 \}$ 1 $\{1,2,3,4,5,6,7,8 \}$