Fake lens spaces
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− | \xymatrix{ 0 \ar[r] & \ | + | \xymatrix{ 0 \ar[r] & {\widetilde L}^s_{2d} (G) \ar[r]^(0.4){\partial} |
− | \ar[d]_{\cong}^{ | + | \ar[d]_{\cong}^{G-sign} & {\mathcal S}^s (L^{2d-1}(\alpha)) \ar[r]^{\eta} |
\ar[d]^{\widetilde \rho}& | \ar[d]^{\widetilde \rho}& | ||
− | \widetilde \ | + | \widetilde {\mathcal N} (L^{2d-1}(\alpha)) \ar[r] \ar[d]^{[\widetilde \rho]}& 0 \\ |
− | 0 \ar[r] & 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & \ | + | 0 \ar[r] & 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} |
− | R^{(-1)^d}_{\widehat G} \ar[r] & \ | + | R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G}/ 4 |
\cdot R^{(-1)^d}_{\widehat G} \ar[r] & 0 } | \cdot R^{(-1)^d}_{\widehat G} \ar[r] & 0 } | ||
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Revision as of 15:59, 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
We cite mainly from [Wall(1999), chapter 14E].
We start by introducing some notation for {\it lens spaces} which are a special sort of fake lens spaces. Let , , where are such that . When define a representation of on by . Any free representation of on a -dimensional complex vector space is isomorphic to some . The representation induces a free action of on which we still denote .
\begin{defn} A {\it lens space} is a manifold obtained as the orbit pace of a free action of the group on for some as above.\footnote{In the notation of [Wall(1999), chapter 14E] we have .} \end{defn}
The lens space is a -dimensional manifold with . Its universal cover is , hence for . There exists a convenient choice of a CW-structure for with one cell in each dimension . Moreover, we have when , when is odd and when is even.
The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in [Milnor(1966)]. The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in . This ring is defined as with where be the group ring of and is the ideal generated by the norm element of . We also suppose that a generator of is chosen. There is also an augmentation map [Wall(1999), page 214]. 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.
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 .Denote further by the lens space with . By is denoted the -skeleton of the lens space . 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 as above 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, the complex is a Poincar\'e complex.
- The polarized homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
- The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
{{cite|Wall(1999)
The existence of a fake lens space in the homotopy type of such is addressed in [Wall(1999), Theorem 14E.3]}. Since the units are exhausted by the lens spaces we obtain the following corollary. \begin{cor} For any fake lens space there exists and a homotopy equivalence
\end{cor}
5 Homeomorphism classification
There is the following commutative diagram of abelian groups and homomorphisms with exact rows
where is the homomorphism induced by .
6 Further discussion
- ...
7 References
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
We cite mainly from [Wall(1999), chapter 14E].
We start by introducing some notation for {\it lens spaces} which are a special sort of fake lens spaces. Let , , where are such that . When define a representation of on by . Any free representation of on a -dimensional complex vector space is isomorphic to some . The representation induces a free action of on which we still denote .
\begin{defn} A {\it lens space} is a manifold obtained as the orbit pace of a free action of the group on for some as above.\footnote{In the notation of [Wall(1999), chapter 14E] we have .} \end{defn}
The lens space is a -dimensional manifold with . Its universal cover is , hence for . There exists a convenient choice of a CW-structure for with one cell in each dimension . Moreover, we have when , when is odd and when is even.
The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in [Milnor(1966)]. The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in . This ring is defined as with where be the group ring of and is the ideal generated by the norm element of . We also suppose that a generator of is chosen. There is also an augmentation map [Wall(1999), page 214]. 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.
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 .Denote further by the lens space with . By is denoted the -skeleton of the lens space . 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 as above 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, the complex is a Poincar\'e complex.
- The polarized homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
- The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
{{cite|Wall(1999)
The existence of a fake lens space in the homotopy type of such is addressed in [Wall(1999), Theorem 14E.3]}. Since the units are exhausted by the lens spaces we obtain the following corollary. \begin{cor} For any fake lens space there exists and a homotopy equivalence
\end{cor}
5 Homeomorphism classification
There is the following commutative diagram of abelian groups and homomorphisms with exact rows
where is the homomorphism induced by .
6 Further discussion
- ...
7 References
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
We cite mainly from [Wall(1999), chapter 14E].
We start by introducing some notation for {\it lens spaces} which are a special sort of fake lens spaces. Let , , where are such that . When define a representation of on by . Any free representation of on a -dimensional complex vector space is isomorphic to some . The representation induces a free action of on which we still denote .
\begin{defn} A {\it lens space} is a manifold obtained as the orbit pace of a free action of the group on for some as above.\footnote{In the notation of [Wall(1999), chapter 14E] we have .} \end{defn}
The lens space is a -dimensional manifold with . Its universal cover is , hence for . There exists a convenient choice of a CW-structure for with one cell in each dimension . Moreover, we have when , when is odd and when is even.
The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in [Milnor(1966)]. The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in . This ring is defined as with where be the group ring of and is the ideal generated by the norm element of . We also suppose that a generator of is chosen. There is also an augmentation map [Wall(1999), page 214]. 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.
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 .Denote further by the lens space with . By is denoted the -skeleton of the lens space . 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 as above 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, the complex is a Poincar\'e complex.
- The polarized homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
- The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
{{cite|Wall(1999)
The existence of a fake lens space in the homotopy type of such is addressed in [Wall(1999), Theorem 14E.3]}. Since the units are exhausted by the lens spaces we obtain the following corollary. \begin{cor} For any fake lens space there exists and a homotopy equivalence
\end{cor}
5 Homeomorphism classification
There is the following commutative diagram of abelian groups and homomorphisms with exact rows
where is the homomorphism induced by .
6 Further discussion
- ...
7 References
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
We cite mainly from [Wall(1999), chapter 14E].
We start by introducing some notation for {\it lens spaces} which are a special sort of fake lens spaces. Let , , where are such that . When define a representation of on by . Any free representation of on a -dimensional complex vector space is isomorphic to some . The representation induces a free action of on which we still denote .
\begin{defn} A {\it lens space} is a manifold obtained as the orbit pace of a free action of the group on for some as above.\footnote{In the notation of [Wall(1999), chapter 14E] we have .} \end{defn}
The lens space is a -dimensional manifold with . Its universal cover is , hence for . There exists a convenient choice of a CW-structure for with one cell in each dimension . Moreover, we have when , when is odd and when is even.
The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in [Milnor(1966)]. The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in . This ring is defined as with where be the group ring of and is the ideal generated by the norm element of . We also suppose that a generator of is chosen. There is also an augmentation map [Wall(1999), page 214]. 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.
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 .Denote further by the lens space with . By is denoted the -skeleton of the lens space . 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 as above 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, the complex is a Poincar\'e complex.
- The polarized homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
- The polarized simple homotopy types of such are in one-to-one correspondence with the units in . The correspondence is given by .
{{cite|Wall(1999)
The existence of a fake lens space in the homotopy type of such is addressed in [Wall(1999), Theorem 14E.3]}. Since the units are exhausted by the lens spaces we obtain the following corollary. \begin{cor} For any fake lens space there exists and a homotopy equivalence
\end{cor}
5 Homeomorphism classification
There is the following commutative diagram of abelian groups and homomorphisms with exact rows
where is the homomorphism induced by .
6 Further discussion
- ...
7 References
This page has not been refereed. The information given here might be incomplete or provisional. |