Fake lens spaces

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1 Introduction

A fake lens space is the orbit space of a free action of a finite cyclic group $G$$\newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \DeclareMathOrd\GL{GL} % general linear group \DeclareMathOrd\SL{SL} % special linear group \DeclareMathOrd\SO{SO} % special orthogonal group \DeclareMathOrd\SU{SU} % special unitary group \DeclareMathOrd\Spin{Spin} % Spin group \DeclareMathOrd\RP{\Rr\mathrm P} % real projective space \DeclareMathOrd\CP{\Cc\mathrm P} % complex projective space \DeclareMathOrd\HP{\Hh\mathrm P} % quaternionic projective space \DeclareMathOrd\Top{\mathrm{Top}} % topological category \DeclareMathOrd\PL{\mathrm{PL}} % piecewise linear category \DeclareMathOrd\Cat{\mathrm{Cat}} % any category \DeclareMathOrd\KS{KS} % Kirby-Siebenmann class \DeclareMathOrd\Hud{Hud} % Hudson torusG$ on a sphere $S^{2d−1}$$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

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3 Invariants

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

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

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$G = \ZZ_N$ define a representation $\alpha_{\bar k}$$\alpha_{\bar k}$ of $G$$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})$$(z_1 \ldots , z_n) \mapsto (z_1 e^{2\pi i k_1/N}, \ldots, z_n e^{2\pi i k_d/N})$. Any free representation of $G$$G$ on a

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

\begin{defn} A {\it lens space} $L^{2d-1}(\alpha_{\bar k})$$L^{2d-1}(\alpha_{\bar k})$ is a manifold obtained as the orbit pace of a free action $\alpha_{\bar k}$$\alpha_{\bar k}$ of the

group

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$G = \ZZ_N$ on $S^{2d-1}$$S^{2d-1}$ for some $\bar k = (k_1, \ldots k_d)$$\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)$$L(\alpha_{\bar k}) = L(N,k_1,\ldots,k_n)$.} \end{defn}

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

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

when $i \neq 0$$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$$G$ and

$\langle Z \rangle$$\langle Z \rangle$ is the ideal generated by the norm element $Z$$Z$ of $G$$G$. We also suppose that a generator $T$$T$ of $G$$G$ is chosen. There

is also an augmentation map

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

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

in terms of a certain unit in

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$\ZZ_N$. These invariants also suffice

for the homotopy and simple homotopy classification of finite

CW-complexes $L$$L$ with

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$\pi_1 (L) \cong \ZZ_N$ and with the universal

cover homotopy equivalent to $S^{2d-1}$$S^{2d-1}$ of which fake lens spaces are obviously a special case. It is convenient to make the following definition. \begin{defn} A {\it polarization} of a CW-complex $L$$L$ as above is a pair $(T,e)$$(T,e)$ where $T$$T$ is a choice of a generator of $\pi_1 (L)$$\pi_1 (L)$ and $e$$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)$$L^{2d-1}(\alpha_k)$ the lens space $L^{2d-1}(\alpha_{\bar k})$$L^{2d-1}(\alpha_{\bar k})$ with $\bar k = (1,\ldots,1,k)$$\bar k = (1,\ldots,1,k)$. By $L^i(\alpha_1)$$L^i(\alpha_1)$ is denoted the $i$$i$-skeleton of the lens space $L^{2d-1}(\alpha_1)$$L^{2d-1}(\alpha_1)$. If $i$$i$ is odd this is a lens space, if $i$$i$ is even this is a CW-complex obtained by attaching an $i$$i$-cell to the lens space of dimension $i-1$$i-1$. \begin{prop}Wall(1999) Let $L$$L$ be a finite CW-complex as above polarized by $(T,e)$$(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$$G$. The chain complex differential on the right hand side is given by $\partial_{2d-1} e^{2d-1} = e_{2d-2} (T-1) U$$\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$$u \in R_G$. Furthermore, the

complex $L$$L$ is a Poincar\'e complex. \begin{enumerate} \item The polarized homotopy types of such $L$$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$$L$ are in one-to-one correspondence with the units in $R_G$$R_G$. The correspondence is given by $u \in R_G$$u \in R_G$. \end{enumerate} \end{prop} \noindent The existence of a fake lens space in the homotopy type of such $L$$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)$$L^{2d-1}(\alpha_k)$ we obtain the following corollary. \begin{cor}

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

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6 Further discussion

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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.