Fake lens spaces
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Contents |
1 Introduction
A fake lens space is the orbit space of a free action of a finite cyclic group on a sphere . It is a generalization of the notion of a lens space which is the orbit space of a free action which comes from a unitary representation.
2 Construction and examples
- ...
3 Invariants
- , for
- , , for , for all other values of .
- , , ...
4 Homotopy Classification
All the results are taken from chapter 14E of [Wall1999].
Notation
Recall the arithmetic (Rim) square:
where with be the group ring of and is the ideal generated by the norm element of . The maps , are the augmentation maps. We also suppose that a generator of is chosen.
Recall that the Reidemeister torsion is a unit in where .
The homotopy classification is stated in terms of a certain unit in . These invariants also suffice for the homotopy and simple homotopy classification of finite CW-complexes with and with the universal cover homotopy equivalent to of which fake lens spaces are obviously a special case. It is convenient to make the following definition.
The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in
Definition 4.1.
A polarization of a CW-complex as above is a pair where is a choice of a generator of and is a choice of a homotopy equivalence .
Recall the classical lens space . By is denoted its -skeleton with respect to the standard cell decomposition. If is odd this is a lens space, if is even this is a CW-complex obtained by attaching an -cell to the lens space of dimension .
Proposition 4.2.
Let be a finite CW-complex with and universal cover polarized by . Then there exists a simple homotopy equivalence
preserving the polarization. It is unique up to homotopy and the action of . The chain complex differential on the right hand side is given by for some which maps to a unit . Then is a simple Poincare complex with Reidemeister torsion .
- The polarized homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by . The invariant can be identified with the first non-trivial -invariant of (in the sense of homotopy theory) .
- The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
See Theorem 14E.3 in [Wall1999]
The existence of a fake lens space in the homotopy type of such is addressed in [Theorem 14E.4] of [Wall(1999)].
Since the units are exhausted by the lens spaces we obtain the following corollary.
Corollary 4.3. For any fake lens space there exists and a homotopy equivalence
5 Homeomorphism classification
There is the following commutative diagram of abelian groups and homomorphisms with exact rows
where is the homomorphism induced by (see [Macko&Wegner2010, Proposition 3.5]).
The map is injective if with odd (compare [Wall1999, Corollary on page 222]?).
The following theorem is taken from [Wall1999, Theorem 14E.7].
Theorem 5.1. Let and be oriented fake lens spaces with fundamental group cyclic of odd order . Then there is an orientation preserving homeomorphism inducing the identity on if and only if and .
Given and , there exists a corresponding fake lens space if and only if the following four statements hold:
- and are both real ( even) or imaginary ( odd).
- generates ...
- ...
- ...
Theorem 5.2. Let be a fake lens space with where with , odd and . Then we have
where is a free abelian group. If is odd then its rank is . If is even then its rank is if and if . In the torsion summand we have .
6 Further discussion
- ...
7 References
- [Macko&Wegner2010] Template:Macko&Wegner2010
- [Wall(1999)] Template:Wall(1999)
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
This page has not been refereed. The information given here might be incomplete or provisional. |
2 Construction and examples
- ...
3 Invariants
- , for
- , , for , for all other values of .
- , , ...
4 Homotopy Classification
All the results are taken from chapter 14E of [Wall1999].
Notation
Recall the arithmetic (Rim) square:
where with be the group ring of and is the ideal generated by the norm element of . The maps , are the augmentation maps. We also suppose that a generator of is chosen.
Recall that the Reidemeister torsion is a unit in where .
The homotopy classification is stated in terms of a certain unit in . These invariants also suffice for the homotopy and simple homotopy classification of finite CW-complexes with and with the universal cover homotopy equivalent to of which fake lens spaces are obviously a special case. It is convenient to make the following definition.
The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in
Definition 4.1.
A polarization of a CW-complex as above is a pair where is a choice of a generator of and is a choice of a homotopy equivalence .
Recall the classical lens space . By is denoted its -skeleton with respect to the standard cell decomposition. If is odd this is a lens space, if is even this is a CW-complex obtained by attaching an -cell to the lens space of dimension .
Proposition 4.2.
Let be a finite CW-complex with and universal cover polarized by . Then there exists a simple homotopy equivalence
preserving the polarization. It is unique up to homotopy and the action of . The chain complex differential on the right hand side is given by for some which maps to a unit . Then is a simple Poincare complex with Reidemeister torsion .
- The polarized homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by . The invariant can be identified with the first non-trivial -invariant of (in the sense of homotopy theory) .
- The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
See Theorem 14E.3 in [Wall1999]
The existence of a fake lens space in the homotopy type of such is addressed in [Theorem 14E.4] of [Wall(1999)].
Since the units are exhausted by the lens spaces we obtain the following corollary.
Corollary 4.3. For any fake lens space there exists and a homotopy equivalence
5 Homeomorphism classification
There is the following commutative diagram of abelian groups and homomorphisms with exact rows
where is the homomorphism induced by (see [Macko&Wegner2010, Proposition 3.5]).
The map is injective if with odd (compare [Wall1999, Corollary on page 222]?).
The following theorem is taken from [Wall1999, Theorem 14E.7].
Theorem 5.1. Let and be oriented fake lens spaces with fundamental group cyclic of odd order . Then there is an orientation preserving homeomorphism inducing the identity on if and only if and .
Given and , there exists a corresponding fake lens space if and only if the following four statements hold:
- and are both real ( even) or imaginary ( odd).
- generates ...
- ...
- ...
Theorem 5.2. Let be a fake lens space with where with , odd and . Then we have
where is a free abelian group. If is odd then its rank is . If is even then its rank is if and if . In the torsion summand we have .
6 Further discussion
- ...
7 References
- [Macko&Wegner2010] Template:Macko&Wegner2010
- [Wall(1999)] Template:Wall(1999)
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
This page has not been refereed. The information given here might be incomplete or provisional. |