Fake lens spaces
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* $\Delta$ generates $I_G^n$, $\rho \in I_{\widehat G}^{-n}$. | * $\Delta$ generates $I_G^n$, $\rho \in I_{\widehat G}^{-n}$. | ||
− | * The classes of $\rho \mod I_{\widehat G}^{-n+1}$ and $(-2)^n | + | * The classes of $\rho \mod I_{\widehat G}^{-n+1}$ and $(-2)^n \Delta \mod I_G^{n+1}$ correspond under |
$$I_{\widehat G}^{-n} / I_{\widehat G}^{-n+1} \cong {\widehat H}^{2n}({\widehat G};\Zz) \cong {\widehat H}^{-2n}(G;\Zz) \cong I_G^n / I_G^{n+1}.$$ | $$I_{\widehat G}^{-n} / I_{\widehat G}^{-n+1} \cong {\widehat H}^{2n}({\widehat G};\Zz) \cong {\widehat H}^{-2n}(G;\Zz) \cong I_G^n / I_G^{n+1}.$$ | ||
Revision as of 10:49, 8 June 2010
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 15:18, 25 April 2013 and the changes since publication. |
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 such a free action which comes from a unitary representation.
The classification of fake lens spaces can be seen as one of the basic questions in topology of manifolds. It is systematically obtained in three stages. First, homotopy classification using classical homotopy theory. Second, simple homotopy classification using Reidemeister torsion. Finally, surgery theory is employed to obtain a classification within the respective simple homotopy types. In fact, this classification was one of the early spectacular applications of surgery theory.
2 Definition
Throughout this page we use the following notation. By is denoted the finite cyclic group of order .
Let be a free action of on the sphere . By is denoted the orbit space of . Sometimes, when the dimension and the action are clear, we leave them from notation and simply write .
3 Invariants
For we have
- , for
- , , for , for all other values of .
Interesting invariants for fake lens spaces are
- the Reidemeister torsion and
- the -invariant .
4 Homotopy Classification and simple 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 .
Theorem 4.2 Wall.
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 [Wall1999, Theorem 14E.3].
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
The homeomorphism classification, as already noted, is an excellent application of the non-simply connected surgery theory. Recall that for a topological manifold the surgery theoretic homeomorphism classification of manifolds wihin the homotopy type of is stated in terms of the surgery structure set and that the primary tool for its calculation is the surgery exact sequence.
For a fake lens space there is enough information about the normal invariants, the L-groups and the surgery obstruction in the surgery exact sequence so that one is left with just an extension problem. The strategy to proceed further is to relate the surgery exact sequence to representation theory of . This is done via the following commutative diagram of abelian groups and homomorphisms with exact rows
where is the homomorphism induced by (see ?).
Theorem 5.1 Wall. If is odd, then the map
is injective.
See [Wall1999, Theorem 14E.7].
For odd Wall managed to obtain an even beter result, namely the complete classification of fake lens spaces of a given dimension with the fundamental group which goes as follows:
Theorem 5.2 Wall. 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 , .
- The classes of and correspond under
- .
The following theorem is taken from [Wall1999, Theorem 14E.7].
For general the following theorem is proved in [Macko&Wegner2008, Theorem 1.2]).
Theorem 5.3. 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 Construction and examples
- ...
7 Further discussion
- ...
8 References
- [Macko&Wegner2008] T. Macko and C. Wegner, On the classification of fake lens spaces, to appear in Forum. Math. Available at the arXiv:0810.1196.
- [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. |
Let be a free action of on the sphere . By is denoted the orbit space of . Sometimes, when the dimension and the action are clear, we leave them from notation and simply write .
3 Invariants
For we have
- , for
- , , for , for all other values of .
Interesting invariants for fake lens spaces are
- the Reidemeister torsion and
- the -invariant .
4 Homotopy Classification and simple 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 .
Theorem 4.2 Wall.
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 [Wall1999, Theorem 14E.3].
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
The homeomorphism classification, as already noted, is an excellent application of the non-simply connected surgery theory. Recall that for a topological manifold the surgery theoretic homeomorphism classification of manifolds wihin the homotopy type of is stated in terms of the surgery structure set and that the primary tool for its calculation is the surgery exact sequence.
For a fake lens space there is enough information about the normal invariants, the L-groups and the surgery obstruction in the surgery exact sequence so that one is left with just an extension problem. The strategy to proceed further is to relate the surgery exact sequence to representation theory of . This is done via the following commutative diagram of abelian groups and homomorphisms with exact rows
where is the homomorphism induced by (see ?).
Theorem 5.1 Wall. If is odd, then the map
is injective.
See [Wall1999, Theorem 14E.7].
For odd Wall managed to obtain an even beter result, namely the complete classification of fake lens spaces of a given dimension with the fundamental group which goes as follows:
Theorem 5.2 Wall. 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 , .
- The classes of and correspond under
- .
The following theorem is taken from [Wall1999, Theorem 14E.7].
For general the following theorem is proved in [Macko&Wegner2008, Theorem 1.2]).
Theorem 5.3. 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 Construction and examples
- ...
7 Further discussion
- ...
8 References
- [Macko&Wegner2008] T. Macko and C. Wegner, On the classification of fake lens spaces, to appear in Forum. Math. Available at the arXiv:0810.1196.
- [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. |