Fake lens spaces

Revision as of 14:52, 7 June 2010

1 Introduction

A fake lens space is the orbit space of a free action of a finite cyclic group $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; A fake lens space is the orbit space of a free action of a finite cyclic group $G$ on a sphere $S^{2d-1}$. 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. </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; * $\pi_1 (L) = \Zz_m$, $\pi_i (L) = \pi_i (S^{2d-1})$ for $i \geq 2$ * $H_0 (L) = \Zz$, $H_{2d-1} (L) = \Zz$, $H_{2i-1} (L) = \Zz_m$ for G$ on a sphere $S^{2d-1}$. 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

• $\pi_1 (L) = \Zz_m$, $\pi_i (L) = \pi_i (S^{2d-1})$ for $i \geq 2$

• $H_0 (L) = \Zz$, $H_{2d-1} (L) = \Zz$, $H_{2i-1} (L) = \Zz_m$ for $1 \leq i \leq d-1$, $H_i (L) = 0$ for all other values of $i$.

• $\Delta$, $\rho$, ...

4 Homotopy Classification

We cite mainly from

[Wall(1999), chapter 14E].

We start by introducing some notation for {\it lens spaces} which

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$d$-dimensional complex vector space is isomorphic to some $\alpha_{\bar k}$. The representation $\alpha_{\bar k}$ induces a free action of $G$ on $S^{2d-1}$ which we still denote $\alpha_{\bar k}$.

\begin{defn} 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

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14E] we have $L(\alpha_{\bar k}) = L(N,k_1,\ldots,k_n)$.} \end{defn}

The lens space $L^{2d-1}(\alpha_{\bar k})$ is a $(2d-1)$-dimensional

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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 convenient choice of a CW-structure for $L^{2d-1}(\alpha_{\bar k})$ with one cell $e_i$ in each dimension $0 \leq i \leq 2d-1$.

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when $i \neq 0$ 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

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$\langle Z \rangle$ is the ideal generated by the norm element $Z$

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[Wall(1999), page 214]. The homotopy classification is stated

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for the homotopy and simple homotopy classification of finite

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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. \begin{defn} A {\it polarization} of a CW-complex $L$ as above is a pair $(T,e)$

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\end{defn} \noindent 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 $L^i(\alpha_1)$ is denoted the $i$-skeleton of the lens space $L^{2d-1}(\alpha_1)$. 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 $i-1$. \begin{prop}Wall(1999) Let $L$ be a finite CW-complex as above polarized by $(T,e)$. Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] preserving the polarization. It is unique up to homotopy and the 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

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complex $L$ is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such $L$ are in

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\item 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 $u \in R_G$. \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such $L$ is addressed in [Wall(1999), Theorem 14E.4]. Since the

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$L^{2d-1}(\alpha_k)$ we obtain the following corollary. \begin{cor}

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and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}

5 Homeomorphism classification

...

6 Further discussion

...

7 References

• [Milnor(1966)] Template:Milnor(1966)
• [Wall(1999)] Template:Wall(1999)

; We cite mainly from \cite[chapter 14E]{Wall(1999)}. We start by introducing some notation for {\it lens spaces} which are a special sort of fake lens spaces. Let $N \in \NN$, $\bar k = (k_1, \ldots k_d)$, where $k_i \in \ZZ$ are such that $(k_i,N)=1$. 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 $d$-dimensional complex vector space is isomorphic to some $\alpha_{\bar k}$. The representation $\alpha_{\bar k}$ induces a free action of $G$ on $S^{2d-1}$ which we still denote $\alpha_{\bar k}$. \begin{defn} \label{defn-lens-space} 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 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 14E]{Wall(1999)} we have $L(\alpha_{\bar k}) = L(N,k_1,\ldots,k_n)$.} \end{defn} The lens space $L^{2d-1}(\alpha_{\bar k})$ is a $(2d-1)$-dimensional manifold with $\pi_1 (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ_N$. Its 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 convenient choice of a CW-structure for $L^{2d-1}(\alpha_{\bar k})$ with one cell $e_i$ in each dimension $G$ on a sphere $S^{2d-1}$. 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

• $\pi_1 (L) = \Zz_m$, $\pi_i (L) = \pi_i (S^{2d-1})$ for $i \geq 2$

• $H_0 (L) = \Zz$, $H_{2d-1} (L) = \Zz$, $H_{2i-1} (L) = \Zz_m$ for $1 \leq i \leq d-1$, $H_i (L) = 0$ for all other values of $i$.

• $\Delta$, $\rho$, ...

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

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$N \in \NN$, $\bar k = (k_1, \ldots k_d)$, where

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$k_i \in \ZZ$ are such that $(k_i,N)=1$. When

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$G = \ZZ_N$ define a representation $\alpha_{\bar k}$ of $G$ on

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$\CC^d$ 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

$d$-dimensional complex vector space is isomorphic to some $\alpha_{\bar k}$. The representation $\alpha_{\bar k}$ induces a free action of $G$ on $S^{2d-1}$ which we still denote $\alpha_{\bar k}$.

\begin{defn} 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

group

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$G = \ZZ_N$ on $S^{2d-1}$ for some $\bar k = (k_1, \ldots k_d)$ as above.\footnote{In the notation of [Wall(1999), chapter

14E] we have $L(\alpha_{\bar k}) = L(N,k_1,\ldots,k_n)$.} \end{defn}

The lens space $L^{2d-1}(\alpha_{\bar k})$ is a $(2d-1)$-dimensional

manifold with

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$\pi_1 (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ_N$. Its

universal cover is $S^{2d-1}$, hence $\pi_i (L^{2d-1}(\alpha_{\bar k})) \cong \pi_i (S^{2d-1})$ for $i \geq 2$. There exists a convenient choice of a CW-structure for $L^{2d-1}(\alpha_{\bar k})$ with one cell $e_i$ in each dimension $0 \leq i \leq 2d-1$.

Moreover, we have

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$H_i (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ$ when $i = 0,2d-1$,

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$H_i (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ_N$ when $0 < i < 2d-1$ is odd and $H_i (L^{2d-1}(\alpha_{\bar k})) \cong 0$

when $i \neq 0$ 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

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$\QQ R_G$. This ring is defined as

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$\QQ R_G = \QQ \otimes R_G$ with

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$R_G = \ZZ G / \langle Z \rangle$ where

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$\ZZ G$ be the group ring of $G$ and

$\langle Z \rangle$ is the ideal generated by the norm element $Z$

of $G$. We also suppose that a generator

Tex syntax error

$T$ of $G$ is chosen. There is also an augmentation map

Tex syntax error

$\varepsilon' \co R_G \ra \ZZ_N$

[Wall(1999), page 214]. The homotopy classification is stated

in terms of a certain unit in

Tex syntax error

$\ZZ_N$. These invariants also suffice

for the homotopy and simple homotopy classification of finite

CW-complexes $L$ with

Tex syntax error

$\pi_1 (L) \cong \ZZ_N$ 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. \begin{defn} A {\it polarization} of a CW-complex $L$ as above is a pair $(T,e)$

where

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$T$ is a choice of a generator of $\pi_1 (L)$ and $e$ is a choice of a homotopy equivalence

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$e \co \wL \ra S^{2d-1}$.

\end{defn} \noindent 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 $L^i(\alpha_1)$ is denoted the $i$-skeleton of the lens space $L^{2d-1}(\alpha_1)$. 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 $i-1$. \begin{prop}Wall(1999) Let $L$ be a finite CW-complex as above polarized by $(T,e)$. Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] preserving the polarization. It is unique up to homotopy and the 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

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$U \in \ZZ G$ which maps to a unit $u \in R_G$. Furthermore, the

complex $L$ is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such $L$ are in

one-to-one correspondence with the units in

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$\ZZ_N$. The correspondence is given by

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$\varepsilon' (u) \in \ZZ_N$.

\item 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 $u \in R_G$. \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such $L$ is addressed in [Wall(1999), Theorem 14E.4]. Since the

units

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$\varepsilon' (u) \in \ZZ_N$ are exhausted by the lens spaces

$L^{2d-1}(\alpha_k)$ we obtain the following corollary. \begin{cor}

For any fake lens space $L^{2d-1}(\alpha)$ there exists

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$k \in \NN$

and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}

5 Homeomorphism classification

...

6 Further discussion

...

7 References

• [Milnor(1966)] Template:Milnor(1966)
• [Wall(1999)] Template:Wall(1999)

This page has not been refereed. The information given here might be incomplete or provisional.

\leq i \leq 2d-1$. Moreover, we have $H_i (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ$ when $i = 0,2d-1$, $H_i (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ_N$ when $G$ on a sphere $S^{2d-1}$. 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

• $\pi_1 (L) = \Zz_m$, $\pi_i (L) = \pi_i (S^{2d-1})$ for $i \geq 2$

• $H_0 (L) = \Zz$, $H_{2d-1} (L) = \Zz$, $H_{2i-1} (L) = \Zz_m$ for $1 \leq i \leq d-1$, $H_i (L) = 0$ for all other values of $i$.

• $\Delta$, $\rho$, ...

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

Tex syntax error

$N \in \NN$, $\bar k = (k_1, \ldots k_d)$, where

Tex syntax error

$k_i \in \ZZ$ are such that $(k_i,N)=1$. When

Tex syntax error

$G = \ZZ_N$ define a representation $\alpha_{\bar k}$ of $G$ on

Tex syntax error

$\CC^d$ 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

$d$-dimensional complex vector space is isomorphic to some $\alpha_{\bar k}$. The representation $\alpha_{\bar k}$ induces a free action of $G$ on $S^{2d-1}$ which we still denote $\alpha_{\bar k}$.

\begin{defn} 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

group

Tex syntax error

$G = \ZZ_N$ on $S^{2d-1}$ for some $\bar k = (k_1, \ldots k_d)$ as above.\footnote{In the notation of [Wall(1999), chapter

14E] we have $L(\alpha_{\bar k}) = L(N,k_1,\ldots,k_n)$.} \end{defn}

The lens space $L^{2d-1}(\alpha_{\bar k})$ is a $(2d-1)$-dimensional

manifold with

Tex syntax error

$\pi_1 (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ_N$. Its

universal cover is $S^{2d-1}$, hence $\pi_i (L^{2d-1}(\alpha_{\bar k})) \cong \pi_i (S^{2d-1})$ for $i \geq 2$. There exists a convenient choice of a CW-structure for $L^{2d-1}(\alpha_{\bar k})$ with one cell $e_i$ in each dimension $0 \leq i \leq 2d-1$.

Moreover, we have

Tex syntax error

$H_i (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ$ when $i = 0,2d-1$,

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$H_i (L^{2d-1}(\alpha_{\bar k})) \cong \ZZ_N$ when $0 < i < 2d-1$ is odd and $H_i (L^{2d-1}(\alpha_{\bar k})) \cong 0$

when $i \neq 0$ 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

Tex syntax error

$\QQ R_G$. This ring is defined as

Tex syntax error

$\QQ R_G = \QQ \otimes R_G$ with

Tex syntax error

$R_G = \ZZ G / \langle Z \rangle$ where

Tex syntax error

$\ZZ G$ be the group ring of $G$ and

$\langle Z \rangle$ is the ideal generated by the norm element $Z$

of $G$. We also suppose that a generator

Tex syntax error

$T$ of $G$ is chosen. There is also an augmentation map

Tex syntax error

$\varepsilon' \co R_G \ra \ZZ_N$

[Wall(1999), page 214]. The homotopy classification is stated

in terms of a certain unit in

Tex syntax error

$\ZZ_N$. These invariants also suffice

for the homotopy and simple homotopy classification of finite

CW-complexes $L$ with

Tex syntax error

$\pi_1 (L) \cong \ZZ_N$ 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. \begin{defn} A {\it polarization} of a CW-complex $L$ as above is a pair $(T,e)$

where

Tex syntax error

$T$ is a choice of a generator of $\pi_1 (L)$ and $e$ is a choice of a homotopy equivalence

Tex syntax error

$e \co \wL \ra S^{2d-1}$.

\end{defn} \noindent 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 $L^i(\alpha_1)$ is denoted the $i$-skeleton of the lens space $L^{2d-1}(\alpha_1)$. 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 $i-1$. \begin{prop}Wall(1999) Let $L$ be a finite CW-complex as above polarized by $(T,e)$. Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] preserving the polarization. It is unique up to homotopy and the 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

Tex syntax error

$U \in \ZZ G$ which maps to a unit $u \in R_G$. Furthermore, the

complex $L$ is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such $L$ are in

one-to-one correspondence with the units in

Tex syntax error

$\ZZ_N$. The correspondence is given by

Tex syntax error

$\varepsilon' (u) \in \ZZ_N$.

\item 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 $u \in R_G$. \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such $L$ is addressed in [Wall(1999), Theorem 14E.4]. Since the

units

Tex syntax error

$\varepsilon' (u) \in \ZZ_N$ are exhausted by the lens spaces

$L^{2d-1}(\alpha_k)$ we obtain the following corollary. \begin{cor}

For any fake lens space $L^{2d-1}(\alpha)$ there exists

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$k \in \NN$

and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}

5 Homeomorphism classification

...

6 Further discussion

...

7 References

• [Milnor(1966)] Template:Milnor(1966)
• [Wall(1999)] Template:Wall(1999)

This page has not been refereed. The information given here might be incomplete or provisional.

< i < 2d-1$ is odd and $H_i (L^{2d-1}(\alpha_{\bar k})) \cong 0$ when $i \neq 0$ is even. The classification of the lens spaces up to homotopy equivalence and simple homotopy equivalence is presented for example in \cite{Milnor(1966)}. The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in $\QQ R_G$. This ring is defined as $\QQ R_G = \QQ \otimes R_G$ with $R_G = \ZZ G / \langle Z \rangle$ where $\ZZ G$ be the group ring of $G$ and $\langle Z \rangle$ is the ideal generated by the norm element $Z$ of $G$. We also suppose that a generator $T$ of $G$ is chosen. There is also an augmentation map $\varepsilon' \co R_G \ra \ZZ_N$ \cite[page 214]{Wall(1999)}. The homotopy classification is stated in terms of a certain unit in $\ZZ_N$. These invariants also suffice for the homotopy and simple homotopy classification 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 following definition. \begin{defn} \label{def-pol-lens-spc} A {\it polarization} of a CW-complex $L$ as above 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 $e \co \wL \ra S^{2d-1}$. \end{defn} \noindent 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 $L^i(\alpha_1)$ is denoted the $i$-skeleton of the lens space $L^{2d-1}(\alpha_1)$. 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 $i-1$. \begin{prop}{\cite[Theorem 14E.3]{Wall(1999)}} \label{prop-simple-htpy-class} Let $L$ be a finite CW-complex as above polarized by $(T,e)$. Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] preserving the polarization. It is unique up to homotopy and the 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 $U \in \ZZ G$ which maps to a unit $u \in R_G$. Furthermore, the complex $L$ is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such $L$ are in one-to-one correspondence with the units in $\ZZ_N$. The correspondence is given by $\varepsilon' (u) \in \ZZ_N$. \item 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 $u \in R_G$. \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such $L$ is addressed in \cite[Theorem 14E.4]{Wall(1999)}. Since the units $\varepsilon' (u) \in \ZZ_N$ are exhausted by the lens spaces $L^{2d-1}(\alpha_k)$ we obtain the following corollary. \begin{cor} \label{lens-spaces-give-all-htpy-types} For any fake lens space $L^{2d-1}(\alpha)$ there exists $k \in \NN$ and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}

2 Construction and examples

...

3 Invariants

• $\pi_1 (L) = \Zz_m$, $\pi_i (L) = \pi_i (S^{2d-1})$ for $i \geq 2$

• $H_0 (L) = \Zz$, $H_{2d-1} (L) = \Zz$, $H_{2i-1} (L) = \Zz_m$ for $1 \leq i \leq d-1$, $H_i (L) = 0$ for all other values of $i$.

• $\Delta$, $\rho$, ...

4 Homotopy Classification

We cite mainly from

[Wall(1999), chapter 14E].

We start by introducing some notation for {\it lens spaces} which

Tex syntax error

Tex syntax error

Tex syntax error

Tex syntax error

$d$-dimensional complex vector space is isomorphic to some $\alpha_{\bar k}$. The representation $\alpha_{\bar k}$ induces a free action of $G$ on $S^{2d-1}$ which we still denote $\alpha_{\bar k}$.

\begin{defn} 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

Tex syntax error

14E] we have $L(\alpha_{\bar k}) = L(N,k_1,\ldots,k_n)$.} \end{defn}

The lens space $L^{2d-1}(\alpha_{\bar k})$ is a $(2d-1)$-dimensional

Tex syntax error

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 convenient choice of a CW-structure for $L^{2d-1}(\alpha_{\bar k})$ with one cell $e_i$ in each dimension $0 \leq i \leq 2d-1$.

Tex syntax error

Tex syntax error

when $i \neq 0$ 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

Tex syntax error

Tex syntax error

Tex syntax error

Tex syntax error

$\langle Z \rangle$ is the ideal generated by the norm element $Z$

Tex syntax error

Tex syntax error

[Wall(1999), page 214]. The homotopy classification is stated

Tex syntax error

for the homotopy and simple homotopy classification of finite

Tex syntax error

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. \begin{defn} A {\it polarization} of a CW-complex $L$ as above is a pair $(T,e)$

Tex syntax error

Tex syntax error

\end{defn} \noindent 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 $L^i(\alpha_1)$ is denoted the $i$-skeleton of the lens space $L^{2d-1}(\alpha_1)$. 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 $i-1$. \begin{prop}Wall(1999) Let $L$ be a finite CW-complex as above polarized by $(T,e)$. Then there exists a simple homotopy equivalence \[ h \co L \lra L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1} \] preserving the polarization. It is unique up to homotopy and the 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

Tex syntax error

complex $L$ is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such $L$ are in

Tex syntax error

Tex syntax error

\item 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 $u \in R_G$. \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such $L$ is addressed in [Wall(1999), Theorem 14E.4]. Since the

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$L^{2d-1}(\alpha_k)$ we obtain the following corollary. \begin{cor}

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and a homotopy equivalence \[ h \co L^{2d-1}(\alpha) \lra L^{2d-1}(\alpha_k). \] \end{cor}

5 Homeomorphism classification

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6 Further discussion

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7 References

• [Milnor(1966)] Template:Milnor(1966)
• [Wall(1999)] Template:Wall(1999)