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.

It turns out that the spaces $L(p,q)$ and $L(p,q^{\prime })$ are homeomorphic if and only if $q\equiv q^{\prime }(\mathrm{mod}p)$. In particular, if we choose $q$ to be in the set $\{1,2,3,\dots ,p-1\}$, then all the spaces $L(p,q)$ are pairwise non-homeomorphic.

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)
3	2	cyclic group:Z3	lens space:L(3,2)