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

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

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

• $\Delta$$\Delta$, $\rho$$\rho$, ...

4 Homotopy Classification

All the results are taken from chapter 14E of [Wall1999].

Notation

Recall the arithmetic (Rim) square:

$\displaystyle \xymatrix{ \Zz G \ar[r]^{\eta} \ar[d]_{\varepsilon} & R_G \ar[d]^{\varepsilon'} \\ \Zz \ar[r]_{\eta'} & \Zz_N }$

where $R_G = \Zz G / \langle Z \rangle$$R_G = \Zz G / \langle Z \rangle$ with $\Zz G$$\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$. The maps $\varepsilon$$\varepsilon$, $\varepsilon'$$\varepsilon'$ are the augmentation maps.

Recall that the Reidemeister torsion is a unit in $\Qq R_G$$\Qq R_G$ where $\Qq R_G = \Qq \otimes R_G$$\Qq R_G = \Qq \otimes R_G$.

The homotopy classification is stated in the a priori broader context of finite CW-complexes $L$$L$ with $\pi_1 (L) \cong \Zz_N$$\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.

Definition 4.1.

A polarization of a CW-complex $L$$L$ with $\pi_1 (L) \cong \Zz_N$$\pi_1 (L) \cong \Zz_N$ and with the universal cover homotopy equivalent to $S^{2d-1}$$S^{2d-1}$ is a pair $(T,e)$$(T,e)$ where $T$$T$ is a choice of a generator of $\pi_1 (L)$$\pi_1 (L)$ and $e$$e$ is a choice of a homotopy equivalence $e \colon \widetilde L \rightarrow S^{2d-1}$$e \colon \widetilde L \rightarrow S^{2d-1}$.

Recall the classical lens space $L^{2d-1}(N,k,1,\ldots,1)$$L^{2d-1}(N,k,1,\ldots,1)$. By $L^i(N,k,1,\ldots,1)$$L^i(N,k,1,\ldots,1)$ is denoted its $i$$i$-skeleton with respect to the standard cell decomposition. If $i$$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$.

Proposition 4.2.

Let $L$$L$ be a finite CW-complex with $\pi_1 (L) \cong \Zz_N$$\pi_1 (L) \cong \Zz_N$ and universal cover $S^{2d-1}$$S^{2d-1}$ polarized by $(T,e)$$(T,e)$. Then there exists a simple homotopy equivalence

$\displaystyle h \colon L \rightarrow L^{2d-2}(N,1,\ldots,1) \cup_\phi e^{2d-1}$ $$\displaystyle h \colon L \rightarrow L^{2d-2}(N,1,\ldots,1) \cup_\phi e^{2d-1}$$

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 $U \in \Zz G$$U \in \Zz G$ which maps to a unit $u \in R_G$$u \in R_G$. Then $L$$L$ is a simple Poincare complex with Reidemeister torsion $\Delta (L) = (T-1)^d \cdot u$$\Delta (L) = (T-1)^d \cdot u$.

• The polarized homotopy types of such $L$$L$ are in one-to-one correspondence with the units in $\Zz_N$$\Zz_N$. The correspondence is given by $\varepsilon' (u) \in \Zz_N$$\varepsilon' (u) \in \Zz_N$. The invariant $\varepsilon' (u)$$\varepsilon' (u)$ can be identified with the first non-trivial $k$$k$-invariant of $L$$L$ (in the sense of homotopy theory) $k_{2d-1} (L) \in H^{2d} (B \Zz_N ; \Zz)$$k_{2d-1} (L) \in H^{2d} (B \Zz_N ; \Zz)$.

• 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$.

See Theorem 14E.3 in [Wall1999].

The existence of a fake lens space in the homotopy type of such $L$$L$ is addressed in [Theorem 14E.4] of [Wall1999].

Since the units $\varepsilon' (u) \in \Zz_N$$\varepsilon' (u) \in \Zz_N$ are exhausted by the lens spaces $L^{2d-1}(\alpha_k)$$L^{2d-1}(\alpha_k)$ we obtain the following corollary.

Corollary 4.3. For any fake lens space $L^{2d-1}(\alpha)$$L^{2d-1}(\alpha)$ there exists $k \in \Nn$$k \in \Nn$ and a homotopy equivalence

$\displaystyle h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(N,k,1,\ldots,1).$ $$\displaystyle h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(N,k,1,\ldots,1).$$

5 Homeomorphism classification

There is the following commutative diagram of abelian groups and homomorphisms with exact rows

$\displaystyle \xymatrix{ 0 \ar[r] & {\widetilde L}^s_{2d} (G) \ar[r]^(0.4){\partial} \ar[d]_{\cong}^{G-sign} & {\mathcal S}^s (L^{2d-1}(\alpha)) \ar[r]^{\eta} \ar[d]^{\widetilde \rho}& \widetilde {\mathcal N} (L^{2d-1}(\alpha)) \ar[r] \ar[d]^{[\widetilde \rho]}& 0 \\ 0 \ar[r] & 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G}/ 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & 0 }$

where $[\widetilde \rho]$$[\widetilde \rho]$ is the homomorphism induced by $\widetilde \rho$$\widetilde \rho$ (see ?).

The map ${\widetilde \rho} \colon {\mathcal S}^s (L^{2d-1}(\alpha)) \to {\mathbb Q} R^{(-1)^d}_{\widehat G}$${\widetilde \rho} \colon {\mathcal S}^s (L^{2d-1}(\alpha)) \to {\mathbb Q} R^{(-1)^d}_{\widehat G}$ is injective if $G = {\mathbb Z}_N$$G = {\mathbb Z}_N$ with $N$$N$ odd (compare [Wall1999, Corollary on page 222]?).

The following theorem is taken from [Wall1999, Theorem 14E.7].

Theorem 5.1. Let $L^{2d-1}$$L^{2d-1}$ and ${L'}^{2d-1}$${L'}^{2d-1}$ be oriented fake lens spaces with fundamental group $G$$G$ cyclic of odd order $N$$N$. Then there is an orientation preserving homeomorphism $L \to L'$$L \to L'$ inducing the identity on $G$$G$ if and only if $\Delta(L) = \Delta(L')$$\Delta(L) = \Delta(L')$ and $\rho(L) = \rho(L')$$\rho(L) = \rho(L')$.

Given $\Delta \in R_G$$\Delta \in R_G$ and $\rho \in {\mathbb Q}R_{\widehat G}$$\rho \in {\mathbb Q}R_{\widehat G}$, there exists a corresponding fake lens space $L^{2d-1}$$L^{2d-1}$ if and only if the following four statements hold:

• $\Delta$$\Delta$ and $\rho$$\rho$ are both real ($d$$d$ even) or imaginary ($d$$d$ odd).

• $\Delta$$\Delta$ generates ...

• ...

• ...

The following theorem is proved in [Macko&Wegner2010, Theorem 1.2]).

Theorem 5.2. Let $L^{2d-1}(\alpha)$$L^{2d-1}(\alpha)$ be a fake lens space with $\pi_1(L^{2d-1}(\alpha)) \cong {\mathbb Z}_N$$\pi_1(L^{2d-1}(\alpha)) \cong {\mathbb Z}_N$ where $N=2^K \cdot M$$N=2^K \cdot M$ with $K \geq 0$$K \geq 0$, $M$$M$ odd and $d \geq 3$$d \geq 3$. Then we have

$\displaystyle {\mathcal S}^s (L^{2d-1}(\alpha)) \cong \bar \Sigma_N (d) \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,1\}}} \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,2i\}}}$ $$\displaystyle {\mathcal S}^s (L^{2d-1}(\alpha)) \cong \bar \Sigma_N (d) \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,1\}}} \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,2i\}}}$$

where $\bar \Sigma_N (d)$$\bar \Sigma_N (d)$ is a free abelian group. If $N$$N$ is odd then its rank is $(N-1)/2$$(N-1)/2$. If $N$$N$ is even then its rank is $N/2-1$$N/2-1$ if $d=2e+1$$d=2e+1$ and $N/2$$N/2$ if $d=2e$$d=2e$. In the torsion summand we have $c = \lfloor (d-1)/2 \rfloor$$c = \lfloor (d-1)/2 \rfloor$.

6 Further discussion

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7 References

• [Macko&Wegner2010] Template:Macko&Wegner2010
• [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003

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