Three-dimensional lens space
Suppose are relatively prime integers (it turns out that the choice of is relevant only modulo ).
Consider the 3-sphere . View this as the following set:
Denote by a primitive root of unity (explicitly, we can take . Consider the continuous map given by:
Note that iterating times gives the identity map, so we get the action of a cyclic group of order on where the generator is . The lens space is defined as the quotient of under the equivalence relation of being in the same orbit under this group action.
Determination up to homeomorphism
The spaces and are homeomorphic if and only if or .
Determination up to homotopy type
The spaces and are of the same homotopy type if and only if either or is a quadratic residue modulo . Note: Despite the choice of notation, is not here assumed to be a prime number.
|Value of||Value of||Cyclic group of order||Quotient of by this as the subgroup of roots of unity|
|2||1||cyclic group:Z2||real projective three-dimensional space . Also can be identified as a Lie group with .|
|3||1||cyclic group:Z3||lens space:L(3,1)|
Number of lens spaces for values of
|Value of||Number of homeomorphism classes of lens spaces||List of -values grouped by homeomorphism class of||Number of homotopy types of lens spaces||List of -values grouped by homotopy type of|