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 $L(p,q^{\prime })$ are homeomorphic if and only if $q\equiv \pm q^{\prime }(\mathrm{mod}p)$ or $q{q}^{\prime }\equiv \pm 1(\mathrm{mod}p)$.

Determination up to homotopy type

The spaces $L(p,q)$ and $L(p,q^{\prime })$ are of the same homotopy type if and only if either $q{q}^{\prime }$ or $-q{q}^{\prime }$ 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 $\mathbb{R}{\mathbb{P}}^3$. Also can be identified as a Lie group with $SO(3,\mathbb{R})$.
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	$\{1,2,3,4,5,6,7,8\}$