Fake lens spaces
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Tibor Macko (Talk | contribs) (→Homotopy Classification) |
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− | where $R_G = \Zz G / \langle Z \rangle$ with $\Zz G$ be the group ring of $G$ and $\langle Z \rangle$ is the ideal generated by the norm element $Z$ of $G$. The maps $\varepsilon$, $\varepsilon'$ are the augmentation maps | + | where $R_G = \Zz G / \langle Z \rangle$ with $\Zz G$ be the group ring of $G$ and $\langle Z \rangle$ is the ideal generated by the norm element $Z$ of $G$. The maps $\varepsilon$, $\varepsilon'$ are the augmentation maps. |
Recall that the Reidemeister torsion is a unit in $\Qq R_G$ where $\Qq R_G = \Qq \otimes R_G$. | Recall that the Reidemeister torsion is a unit in $\Qq R_G$ where $\Qq R_G = \Qq \otimes R_G$. | ||
− | The homotopy classification is stated in | + | The homotopy classification is stated in the a priori broader context of finite CW-complexes $L$ with $\pi_1 (L) \cong \Zz_N$ and with the universal cover homotopy equivalent to $S^{2d-1}$ of which fake lens spaces are obviously a special case. It is convenient to make the |
− | + | ||
− | finite CW-complexes $L$ with $\pi_1 (L) \cong \Zz_N$ and with the | + | |
− | universal cover homotopy equivalent to $S^{2d-1}$ of which fake lens | + | |
− | spaces are obviously a special case. It is convenient to make the | + | |
following definition. | following definition. | ||
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− | |||
− | |||
{{beginthm|Definition}} \label{def-pol-lens-spc} | {{beginthm|Definition}} \label{def-pol-lens-spc} | ||
− | A polarization of a CW-complex $L$ | + | A polarization of a CW-complex $L$ with $\pi_1 (L) \cong \Zz_N$ and with the universal cover homotopy equivalent to $S^{2d-1}$ is a pair $(T,e)$ where $T$ is a choice of a generator of $\pi_1 (L)$ and $e$ is a choice of a homotopy equivalence $e \colon \widetilde L \rightarrow S^{2d-1}$. |
{{endthm}} | {{endthm}} | ||
− | Recall the classical lens space $L^{2d-1}(N,k,1,\ldots,1)$. By $L^i(N,k,1,\ldots,1)$ is denoted its $i$-skeleton with respect to the standard cell decomposition. If $i$ is odd this is a lens space, if $i$ is even this is a CW-complex obtained by attaching an $i$-cell to the | + | Recall the classical lens space $L^{2d-1}(N,k,1,\ldots,1)$. By $L^i(N,k,1,\ldots,1)$ is denoted its $i$-skeleton with respect to the standard cell decomposition. If $i$ is odd this is a lens space, if $i$ is even this is a CW-complex obtained by attaching an $i$-cell to the lens space of dimension $i-1$. |
− | lens space of dimension $i-1$. | + | |
{{beginthm|Proposition}} \label{prop-simple-htpy-class} | {{beginthm|Proposition}} \label{prop-simple-htpy-class} | ||
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action of $G$. The chain complex differential on the right hand side | action of $G$. The chain complex differential on the right hand side | ||
is given by $\partial_{2d-1} e^{2d-1} = e_{2d-2} (T-1) U$ for some | is given by $\partial_{2d-1} e^{2d-1} = e_{2d-2} (T-1) U$ for some | ||
− | $U \in \Zz G$ which maps to a unit $u \in R_G$. Then $L$ is a simple Poincare complex with Reidemeister torsion $\Delta (L) = (T | + | $U \in \Zz G$ which maps to a unit $u \in R_G$. Then $L$ is a simple Poincare complex with Reidemeister torsion $\Delta (L) = (T-1)^d \cdot u$. |
* The polarized homotopy types of such $L$ are in one-to-one correspondence with the units in $\Zz_N$. The correspondence is given by $\varepsilon' (u) \in \Zz_N$. The invariant $\varepsilon' (u)$ can be identified with the first non-trivial $k$-invariant of $L$ (in the sense of homotopy theory) $k_{2d-1} (L) \in H^{2d} (B \Zz_N ; \Zz)$. | * The polarized homotopy types of such $L$ are in one-to-one correspondence with the units in $\Zz_N$. The correspondence is given by $\varepsilon' (u) \in \Zz_N$. The invariant $\varepsilon' (u)$ can be identified with the first non-trivial $k$-invariant of $L$ (in the sense of homotopy theory) $k_{2d-1} (L) \in H^{2d} (B \Zz_N ; \Zz)$. | ||
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Since the units $\varepsilon' (u) \in \Zz_N$ are exhausted by the lens spaces | Since the units $\varepsilon' (u) \in \Zz_N$ are exhausted by the lens spaces | ||
$L^{2d-1}(\alpha_k)$ we obtain the following corollary. | $L^{2d-1}(\alpha_k)$ we obtain the following corollary. | ||
+ | |||
{{beginthm|Corollary}} \label{lens-spaces-give-all-htpy-types} | {{beginthm|Corollary}} \label{lens-spaces-give-all-htpy-types} | ||
For any fake lens space $L^{2d-1}(\alpha)$ there exists $k \in \Nn$ | For any fake lens space $L^{2d-1}(\alpha)$ there exists $k \in \Nn$ | ||
and a homotopy equivalence | and a homotopy equivalence | ||
$$ | $$ | ||
− | h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(\ | + | h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(N,k,1,\ldots,1). |
$$ | $$ | ||
{{endthm}} | {{endthm}} |
Revision as of 16:55, 7 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 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.
A polarization of a CW-complex with and with the universal cover homotopy equivalent to 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.
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.
A polarization of a CW-complex with and with the universal cover homotopy equivalent to 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. |