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.
Recall that the Reidemeister torsion is a unit in where .
The homotopy classification is stated in the a priori broader context 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.
Definition 4.1.
Let be a CW-complex with and with universal cover homotopy equivalent to .
A polarization of 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 . Furthermore, 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 [Wall1999, Theorem 14E.4].
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 ?).
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 ...
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The following theorem is proved in [Macko&Wegner2010, Theorem 1.2]).
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
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7 References
- [Macko&Wegner2010] Template:Macko&Wegner2010
- [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003
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