Three-dimensional lens space

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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 \}