Fake lens spaces
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− | == Homotopy Classification == | + | == Homotopy Classification == |
− | <wikitex>; We cite mainly from | + | |
− | \cite[chapter 14E]{Wall(1999)}. | + | <wikitex>; |
+ | |||
+ | We cite mainly from \cite[chapter 14E]{Wall(1999)}. | ||
We start by introducing some notation for {\it lens spaces} which | We start by introducing some notation for {\it lens spaces} which | ||
− | are a special sort of fake lens spaces. Let $N \in \ | + | are a special sort of fake lens spaces. Let $N \in \Nn$, $\bar k = |
− | (k_1, \ldots k_d)$, where $k_i \in \ | + | (k_1, \ldots k_d)$, where $k_i \in \Zz$ are such that $(k_i,N)=1$. |
− | When $G = \ | + | When $G = \Zz_N$ define a representation $\alpha_{\bar k}$ of $G$ on |
− | $\ | + | $\Cc^d$ by $(z_1 \ldots , z_n) \mapsto (z_1 e^{2\pi i k_1/N}, |
\ldots, z_n e^{2\pi i k_d/N})$. Any free representation of $G$ on a | \ldots, z_n e^{2\pi i k_d/N})$. Any free representation of $G$ on a | ||
$d$-dimensional complex vector space is isomorphic to some | $d$-dimensional complex vector space is isomorphic to some | ||
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A {\it lens space} $L^{2d-1}(\alpha_{\bar k})$ is a manifold | A {\it lens space} $L^{2d-1}(\alpha_{\bar k})$ is a manifold | ||
obtained as the orbit pace of a free action $\alpha_{\bar k}$ of the | obtained as the orbit pace of a free action $\alpha_{\bar k}$ of the | ||
− | group $G = \ | + | group $G = \Zz_N$ on $S^{2d-1}$ for some $\bar k = (k_1, \ldots |
k_d)$ as above.\footnote{In the notation of \cite[chapter | k_d)$ as above.\footnote{In the notation of \cite[chapter | ||
14E]{Wall(1999)} we have $L(\alpha_{\bar k}) = | 14E]{Wall(1999)} we have $L(\alpha_{\bar k}) = | ||
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The lens space $L^{2d-1}(\alpha_{\bar k})$ is a $(2d-1)$-dimensional | The lens space $L^{2d-1}(\alpha_{\bar k})$ is a $(2d-1)$-dimensional | ||
− | manifold with $\pi_1 (L^{2d-1}(\alpha_{\bar k})) \cong \ | + | manifold with $\pi_1 (L^{2d-1}(\alpha_{\bar k})) \cong \Zz_N$. Its |
universal cover is $S^{2d-1}$, hence $\pi_i (L^{2d-1}(\alpha_{\bar | universal cover is $S^{2d-1}$, hence $\pi_i (L^{2d-1}(\alpha_{\bar | ||
k})) \cong \pi_i (S^{2d-1})$ for $i \geq 2$. There exists a | k})) \cong \pi_i (S^{2d-1})$ for $i \geq 2$. There exists a | ||
convenient choice of a CW-structure for $L^{2d-1}(\alpha_{\bar k})$ | convenient choice of a CW-structure for $L^{2d-1}(\alpha_{\bar k})$ | ||
with one cell $e_i$ in each dimension $0 \leq i \leq 2d-1$. | with one cell $e_i$ in each dimension $0 \leq i \leq 2d-1$. | ||
− | Moreover, we have $H_i (L^{2d-1}(\alpha_{\bar k})) \cong \ | + | Moreover, we have $H_i (L^{2d-1}(\alpha_{\bar k})) \cong \Zz$ when |
− | $i = 0,2d-1$, $H_i (L^{2d-1}(\alpha_{\bar k})) \cong \ | + | $i = 0,2d-1$, $H_i (L^{2d-1}(\alpha_{\bar k})) \cong \Zz_N$ when $0 |
< i < 2d-1$ is odd and $H_i (L^{2d-1}(\alpha_{\bar k})) \cong 0$ | < i < 2d-1$ is odd and $H_i (L^{2d-1}(\alpha_{\bar k})) \cong 0$ | ||
when $i \neq 0$ is even. | when $i \neq 0$ is even. | ||
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simple homotopy equivalence is presented for example in | simple homotopy equivalence is presented for example in | ||
\cite{Milnor(1966)}. The simple homotopy classification is stated in | \cite{Milnor(1966)}. The simple homotopy classification is stated in | ||
− | terms of Reidemeister torsion which is a unit in $\ | + | terms of Reidemeister torsion which is a unit in $\Qq R_G$. This |
− | ring is defined as $\ | + | ring is defined as $\Qq R_G = \Qq \otimes R_G$ with $R_G = \Zz G / |
− | \langle Z \rangle$ where $\ | + | \langle Z \rangle$ where $\Zz G$ be the group ring of $G$ and |
$\langle Z \rangle$ is the ideal generated by the norm element $Z$ | $\langle Z \rangle$ is the ideal generated by the norm element $Z$ | ||
of $G$. We also suppose that a generator $T$ of $G$ is chosen. There | of $G$. We also suppose that a generator $T$ of $G$ is chosen. There | ||
− | is also an augmentation map $\varepsilon' \ | + | is also an augmentation map $\varepsilon' \colon R_G \rightarrow |
− | \cite[page 214]{Wall(1999)}. The homotopy classification is | + | \Zz_N$ \cite[page 214]{Wall(1999)}. The homotopy classification is |
− | in terms of a certain unit in $\ | + | stated in terms of a certain unit in $\Zz_N$. These invariants also |
− | for the homotopy and simple homotopy classification of | + | suffice for the homotopy and simple homotopy classification of |
− | CW-complexes $L$ with $\pi_1 (L) \cong \ | + | finite CW-complexes $L$ with $\pi_1 (L) \cong \Zz_N$ and with the |
− | cover homotopy equivalent to $S^{2d-1}$ of which fake lens | + | universal cover homotopy equivalent to $S^{2d-1}$ of which fake lens |
− | are obviously a special case. It is convenient to make the | + | spaces are obviously a special case. It is convenient to make the |
− | definition. | + | following definition. |
− | + | ||
− | A | + | {{beginthm|Definition}} \label{def-pol-lens-spc} A polarization of a |
− | where $T$ is a choice of a generator of $\pi_1 (L)$ and $e$ is a | + | CW-complex $L$ as above is a pair $(T,e)$ where $T$ is a choice of a |
− | choice of a homotopy equivalence $e \ | + | generator of $\pi_1 (L)$ and $e$ is a choice of a homotopy |
− | + | equivalence $e \colon \widetilde L \rightarrow S^{2d-1}$. {{endthm}} | |
− | + | ||
+ | Denote further by $L^{2d-1}(\alpha_k)$ the lens space | ||
$L^{2d-1}(\alpha_{\bar k})$ with $\bar k = (1,\ldots,1,k)$. By | $L^{2d-1}(\alpha_{\bar k})$ with $\bar k = (1,\ldots,1,k)$. By | ||
$L^i(\alpha_1)$ is denoted the $i$-skeleton of the lens space | $L^i(\alpha_1)$ is denoted the $i$-skeleton of the lens space | ||
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even this is a CW-complex obtained by attaching an $i$-cell to the | even this is a CW-complex obtained by attaching an $i$-cell to the | ||
lens space of dimension $i-1$. | lens space of dimension $i-1$. | ||
− | + | ||
− | \label{prop-simple-htpy-class} Let $L$ be a finite CW-complex as | + | {{beginthm|Proposition}} \label{prop-simple-htpy-class} |
− | above polarized by $(T,e)$. Then there exists a simple homotopy | + | |
− | equivalence | + | Let $L$ be a finite CW-complex as above polarized by $(T,e)$. Then |
− | + | there exists a simple homotopy equivalence | |
− | h \ | + | $$ |
− | + | h \colon L \rightarrow L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} | |
+ | $$ | ||
preserving the polarization. It is unique up to homotopy and the | preserving the polarization. It is unique up to homotopy and the | ||
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 \ | + | $U \in \Zz G$ which maps to a unit $u \in R_G$. Furthermore, the |
complex $L$ is a Poincar\'e complex. | complex $L$ is a Poincar\'e complex. | ||
− | + | ||
− | + | * The polarized homotopy types of such $L$ are in one-to-one correspondence with the units in $\Zz_N$. The correspondence is given by $\varepsilon' (u) \in \Zz_N$. | |
− | one-to-one correspondence with the units in $\ | + | |
− | correspondence is given by $\varepsilon' (u) \in \ | + | * The polarized simple homotopy types of such $L$ are in one-to-one correspondence with the units in $R_G$. The correspondence is given by $u \in R_G$. |
− | + | ||
− | one-to-one correspondence with the units in $R_G$. The | + | {{endthm}} |
− | correspondence is given by $u \in R_G$. | + | |
− | + | \cite[Theorem 14E.3]{Wall(1999) | |
− | \ | + | |
− | + | The existence of a fake lens space in the homotopy type of such $L$ | |
− | such $L$ is addressed in \cite[Theorem 14E.4]{Wall(1999)}. Since the | + | is addressed in \cite[Theorem 14E.4]{Wall(1999)}. Since the units |
− | units $\varepsilon' (u) \in \ | + | $\varepsilon' (u) \in \Zz_N$ 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. | ||
\begin{cor} \label{lens-spaces-give-all-htpy-types} | \begin{cor} \label{lens-spaces-give-all-htpy-types} | ||
− | For any fake lens space $L^{2d-1}(\alpha)$ there exists $k \in \ | + | For any fake lens space $L^{2d-1}(\alpha)$ there exists $k \in \Nn$ |
and a homotopy equivalence | and a homotopy equivalence | ||
− | + | $$ | |
− | h \ | + | h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(\alpha_k). |
− | + | $$ | |
\end{cor} | \end{cor} | ||
</wikitex> | </wikitex> |
Revision as of 15:07, 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
- ...
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
- ...
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
- ...
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
- ...
6 Further discussion
- ...
7 References
This page has not been refereed. The information given here might be incomplete or provisional. |