Lens spaces
Contents |
1 Introduction
A lens space is the orbit spaces of a free linear action of a finite cyclic group on a sphere. The importance of lens spaces stems from the fact that they provide examples of peculiar phenomena. For example, there are pairs of lens spaces where both lens spaces have the same homotopy and homology groups but are not homotopy equivalent, and there are also pairs where both lens spaces are homotopy equivalent, but not homeomorphic. The lens spaces also play a role in Milnor's disproof of Hauptvermutung for polyhedra, that means they were used to find two polyhedra which are homeomorphic but combinatorially distinct.
2 Construction and examples
Let , for be natural numbers such that for all . The lens space is defined to be the orbit space of the free action of the cyclic group on the sphere given by the formula
3 Invariants
Abbreviate .
- , for
- , , for , for all other values of .
- Let be natural numbers satisfying mod for all . Then the Reidemeister torsion is defined by
4 Classification/Characterization
Abbreviate and .
Homotopy classification:
See also [Cohen1973].
Homeomorphism classification:
Theorem 4.2 [Franz1935]. if and only if for some permutation and some we have for all .
See for example [Milnor1966]
5 Further discussion
More details and a discussion of fake lens spaces are planned. This includes the -invariant.
6 References
- [Cohen1973] M. M. Cohen, A course in simple-homotopy theory, Springer-Verlag, New York, 1973. MR0362320 (50 #14762) Zbl 0261.57009
- [Franz1935] W. Franz, Über die Torsion einer Überdeckung., Journ. f. Math. 173 (1935), 245-254. Zbl 61.1350.01
- [Milnor1966] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR0196736 (33 #4922) Zbl 0147.23104
- [Olum1953] P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458–480. MR0058212 (15,338a) Zbl 0052.19901
This page has not been refereed. The information given here might be incomplete or provisional. |
3 Invariants
Abbreviate .
- , for
- , , for , for all other values of .
- Let be natural numbers satisfying mod for all . Then the Reidemeister torsion is defined by
4 Classification/Characterization
Abbreviate and .
Homotopy classification:
See also [Cohen1973].
Homeomorphism classification:
Theorem 4.2 [Franz1935]. if and only if for some permutation and some we have for all .
See for example [Milnor1966]
5 Further discussion
More details and a discussion of fake lens spaces are planned. This includes the -invariant.
6 References
- [Cohen1973] M. M. Cohen, A course in simple-homotopy theory, Springer-Verlag, New York, 1973. MR0362320 (50 #14762) Zbl 0261.57009
- [Franz1935] W. Franz, Über die Torsion einer Überdeckung., Journ. f. Math. 173 (1935), 245-254. Zbl 61.1350.01
- [Milnor1966] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR0196736 (33 #4922) Zbl 0147.23104
- [Olum1953] P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458–480. MR0058212 (15,338a) Zbl 0052.19901
This page has not been refereed. The information given here might be incomplete or provisional. |