Fake lens spaces

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	The existence of a fake lens space in the homotopy type of such $L$		The existence of a fake lens space in the homotopy type of such $L$
−	is addressed in [Theorem 14E.4] of \cite{Wall1999}.	+	is addressed in {{cite|Wall1999|Theorem 14E.4}}.
	Since the units $\varepsilon' (u) \in \Zz_N$ are exhausted by the lens spaces		Since the units $\varepsilon' (u) \in \Zz_N$ are exhausted by the lens spaces

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Revision as of 17:31, 7 June 2010

Contents 1 Introduction 2 Construction and examples 3 Invariants 4 Homotopy Classification 5 Homeomorphism classification 6 Further discussion 7 References

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 $G \cong \Zz_N$ on a sphere $S^{2d-1}$. It is a generalization of the notion of a [[Lens spaces|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>; * $\pi_1 (L) = \Zz_N$, $\pi_i (L) = \pi_i (S^{2d-1})$ for $i \geq 2$ * $H_0 (L) = \Zz$, $H_{2d-1} (L) = \Zz$, $H_{2i-1} (L) = \Zz_N$ for G \cong \Zz_N$ 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

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

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

• $\Delta$, $\rho$, ...

4 Homotopy Classification

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

Notation

Recall the arithmetic (Rim) square:

where $R_G = \Zz G / \langle Z \rangle$ with $\Zz G$ be the group ring of $G$ and $\langle Z \rangle$ is the ideal generated by the norm element $Z$ of $G$. The maps $\varepsilon$, $\varepsilon'$ are the augmentation maps.

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

The homotopy classification is stated in the a priori broader context 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.

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

See Theorem 14E.3 in [Wall1999].

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

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

5 Homeomorphism classification

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

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

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

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

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

6 Further discussion

...

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

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

\leq i \leq d-1$, $H_i (L) = 0$ for all other values of $i$. * $\Delta$, $\rho$, ... == Homotopy Classification == ; All the results are taken from chapter 14E of \cite{Wall1999}. '''Notation''' Recall the arithmetic (Rim) square: $$ \xymatrix{ \Zz G \ar[r]^{\eta} \ar[d]_{\varepsilon} & R_G \ar[d]^{\varepsilon'} \ \Zz \ar[r]_{\eta'} & \Zz_N } $$ where $R_G = \Zz G / \langle Z \rangle$ with $\Zz G$ be the group ring of $G$ and $\langle Z \rangle$ is the ideal generated by the norm element $Z$ of $G$. The maps $\varepsilon$, $\varepsilon'$ are the augmentation maps. Recall that the Reidemeister torsion is a unit in $\Qq R_G$ where $\Qq R_G = \Qq \otimes R_G$. The homotopy classification is stated in the a priori broader context 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. {{beginthm|Definition}} \label{def-pol-lens-spc} Let $L$ be a CW-complex with $\pi_1 (L) \cong \Zz_N$ and with universal cover homotopy equivalent to $S^{2d-1}$. A '''polarization''' of $L$ 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 $e \colon \widetilde L \rightarrow S^{2d-1}$. {{endthm}} Recall the classical lens space $L^{2d-1}(N,k,1,\ldots,1)$. By $L^i(N,k,1,\ldots,1)$ is denoted its $i$-skeleton with respect to the standard cell decomposition. 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 $i-1$. {{beginthm|Proposition}} \label{prop-simple-htpy-class} Let $L$ be a finite CW-complex with $\pi_1 (L) \cong \Zz_N$ and universal cover $S^{2d-1}$ polarized by $(T,e)$. Then there exists a simple homotopy equivalence $$ h \colon L \rightarrow L^{2d-2}(N,1,\ldots,1) \cup_\phi e^{2d-1} $$ preserving the polarization. It is unique up to homotopy and the action of $G$. 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, $L$ is a simple Poincare complex with Reidemeister torsion $\Delta (L) = (T-1)^d \cdot u$. * The polarized homotopy types of such $L$ are in one-to-one correspondence with the units in $\Zz_N$. The correspondence is given by $\varepsilon' (u) \in \Zz_N$. The invariant $\varepsilon' (u)$ can be identified with the first non-trivial $k$-invariant of $L$ (in the sense of homotopy theory) $k_{2d-1} (L) \in H^{2d} (B \Zz_N ; \Zz)$. * 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 $u \in R_G$. {{endthm}} See Theorem 14E.3 in \cite{Wall1999}. The existence of a fake lens space in the homotopy type of such $L$ is addressed in {{cite|Wall1999|Theorem 14E.4}}. Since the units $\varepsilon' (u) \in \Zz_N$ are exhausted by the lens spaces $L^{2d-1}(\alpha_k)$ we obtain the following corollary. {{beginthm|Corollary}} \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 $$ h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(N,k,1,\ldots,1). $$ {{endthm}} == Homeomorphism classification == ; There is the following commutative diagram of abelian groups and homomorphisms with exact rows $$ \xymatrix{ 0 \ar[r] & {\widetilde L}^s_{2d} (G) \ar[r]^(0.4){\partial} \ar[d]_{\cong}^{G-sign} & {\mathcal S}^s (L^{2d-1}(\alpha)) \ar[r]^{\eta} \ar[d]^{\widetilde \rho}& \widetilde {\mathcal N} (L^{2d-1}(\alpha)) \ar[r] \ar[d]^{[\widetilde \rho]}& 0 \ 0 \ar[r] & 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G} \ar[r] & {\mathbb Q} R^{(-1)^d}_{\widehat G}/ 4 \cdot R^{(-1)^d}_{\widehat G} \ar[r] & 0 } $$ where $[\widetilde \rho]$ is the homomorphism induced by $\widetilde \rho$ (see ?). The map ${\widetilde \rho} \colon {\mathcal S}^s (L^{2d-1}(\alpha)) \to {\mathbb Q} R^{(-1)^d}_{\widehat G}$ is injective if $G = {\mathbb Z}_N$ with $N$ odd (compare {{cite|Wall1999|Corollary on page 222}}?). The following theorem is taken from {{cite|Wall1999|Theorem 14E.7}}. \begin{theorem} Let $L^{2d-1}$ and ${L'}^{2d-1}$ be oriented fake lens spaces with fundamental group $G$ cyclic of odd order $N$. Then there is an orientation preserving homeomorphism $L \to L'$ inducing the identity on $G$ if and only if $\Delta(L) = \Delta(L')$ and $\rho(L) = \rho(L')$. Given $\Delta \in R_G$ and $\rho \in {\mathbb Q}R_{\widehat G}$, there exists a corresponding fake lens space $L^{2d-1}$ if and only if the following four statements hold: * $\Delta$ and $\rho$ are both real ($d$ even) or imaginary ($d$ odd). * $\Delta$ generates ... * ... * ... \end{theorem} The following theorem is proved in {{cite|Macko&Wegner2010|Theorem 1.2}}). \begin{theorem} Let $L^{2d-1}(\alpha)$ be a fake lens space with $\pi_1(L^{2d-1}(\alpha)) \cong {\mathbb Z}_N$ where $N=2^K \cdot M$ with $K \geq 0$, $M$ odd and $d \geq 3$. Then we have $$ {\mathcal S}^s (L^{2d-1}(\alpha)) \cong \bar \Sigma_N (d) \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,1\}}} \oplus \bigoplus_{i=1}^{c} {\mathbb Z}_{2^{\min\{K,2i\}}} $$ where $\bar \Sigma_N (d)$ is a free abelian group. If $N$ is odd then its rank is $(N-1)/2$. If $N$ is even then its rank is $N/2-1$ if $d=2e+1$ and $N/2$ if $d=2e$. In the torsion summand we have $c = \lfloor (d-1)/2 \rfloor$. \end{theorem} == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]] {{Stub}}G \cong \Zz_N 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

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

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

• $\Delta$, $\rho$, ...

4 Homotopy Classification

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

Notation

Recall the arithmetic (Rim) square:

where $R_G = \Zz G / \langle Z \rangle$ with $\Zz G$ be the group ring of $G$ and $\langle Z \rangle$ is the ideal generated by the norm element $Z$ of $G$. The maps $\varepsilon$, $\varepsilon'$ are the augmentation maps.

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

The homotopy classification is stated in the a priori broader context 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.

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

See Theorem 14E.3 in [Wall1999].

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

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

5 Homeomorphism classification

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

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

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

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

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

6 Further discussion

...

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

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