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 defining ingredient::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.

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.