Fake lens spaces

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	<wikitex>;		<wikitex>;
−	We cite mainly from \cite[chapter 14E]{Wall(1999)}.	+	All the results are taken from chapter 14E of \cite{Wall1999}.
−	We start by introducing some notation for {\it lens spaces} which	+	'''Notation'''
−	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}	+	Recall the arithmetic (Rim) square:
−	A {\it lens space} $L^{2d-1}(\alpha_{\bar k})$ is a manifold	+	$$
−	obtained as the orbit pace of a free action $\alpha_{\bar k}$ of the	+	\xymatrix{
−	group $G = \Zz_N$ on $S^{2d-1}$ for some $\bar k = (k_1, \ldots	+	\Zz G \ar[r]^{\eta} \ar[d]_{\varepsilon} & R_G \ar[d]^{\varepsilon'} \\
−	k_d)$ as above.\footnote{In the notation of \cite[chapter	+	\Zz \ar[r]_{\eta'} & \Zz_N
−	14E]{Wall(1999)} we have $L(\alpha_{\bar k}) =	+	}
−	L(N,k_1,\ldots,k_n)$.}	+	$$
−	\end{defn}	+	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. We also suppose that a generator $T$ of $G$ is chosen.
−	The lens space $L^{2d-1}(\alpha_{\bar k})$ is a $(2d-1)$-dimensional	+	Recall that the Reidemeister torsion is a unit in $\Qq R_G$ where $\Qq R_G = \Qq \otimes R_G$.
−	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 $0 \leq i \leq 2d-1$.	+
−	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 $0	+
−	< i < 2d-1$ 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	+	The homotopy classification is stated in terms of a certain unit in $\Zz_N$. These invariants also
−	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 $\varepsilon' \colon R_G \rightarrow	+
−	\Zz_N$ \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		suffice for the homotopy and simple homotopy classification of
	finite CW-complexes $L$ with $\pi_1 (L) \cong \Zz_N$ and with the		finite CW-complexes $L$ with $\pi_1 (L) \cong \Zz_N$ and with the
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	following definition.		following definition.
−	{{beginthm|Definition}} \label{def-pol-lens-spc} A polarization of a	+	The simple homotopy classification is stated in
−	CW-complex $L$ as above is a pair $(T,e)$ where $T$ is a choice of a	+	terms of Reidemeister torsion which is a unit in
−	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	+	{{beginthm|Definition}} \label{def-pol-lens-spc}
−	$L^{2d-1}(\alpha_{\bar k})$ with $\bar k = (1,\ldots,1,k)$. By	+
−	$L^i(\alpha_1)$ is denoted the $i$-skeleton of the lens space	+	A 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 $e \colon \widetilde L \rightarrow S^{2d-1}$.
−	$L^{2d-1}(\alpha_1)$. 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	+	{{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 $i-1$.		lens space of dimension $i-1$.
	{{beginthm|Proposition}} \label{prop-simple-htpy-class}		{{beginthm|Proposition}} \label{prop-simple-htpy-class}
−	Let $L$ be a finite CW-complex as above polarized by $(T,e)$. Then	+	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
−	there exists a simple homotopy equivalence	+
	$$		$$
−	h \colon L \rightarrow L^{2d-2}(\alpha_1) \cup_\phi e^{2d-1}	+	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		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
−	$U \in \Zz G$ which maps to a unit $u \in R_G$. Furthermore, the	+	$U \in \Zz G$ which maps to a unit $u \in R_G$. Then $L$ is a simple Poincare complex with Reidemeister torsion $\Delta (L) = (T/1)^d \cdot u$.
−	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 $\varepsilon' (u) \in \Zz_N$.	+	* 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 $u \in R_G$.		* 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 $u \in R_G$.
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	{{endthm}}		{{endthm}}
−	\cite[Theorem 14E.3]{Wall(1999)	+	See Theorem 14E.3 in \cite{Wall1999}
	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 \cite[Theorem 14E.4]{Wall(1999)}. Since the units	+	is addressed in [Theorem 14E.4] of \cite{Wall(1999)}.
−	$\varepsilon' (u) \in \Zz_N$ are exhausted by the lens spaces	+
		+	Since the units $\varepsilon' (u) \in \Zz_N$ are exhausted by the lens spaces
	$L^{2d-1}(\alpha_k)$ we obtain the following corollary.		$L^{2d-1}(\alpha_k)$ we obtain the following corollary.
−	\begin{cor} \label{lens-spaces-give-all-htpy-types}	+	{{beginthm|Corollary}} \label{lens-spaces-give-all-htpy-types}
	For any fake lens space $L^{2d-1}(\alpha)$ there exists $k \in \Nn$		For any fake lens space $L^{2d-1}(\alpha)$ there exists $k \in \Nn$
	and a homotopy equivalence		and a homotopy equivalence
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	h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(\alpha_k).		h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(\alpha_k).
	$$		$$
−	\end{cor}	+	{{endthm}}
	</wikitex>		</wikitex>

---

Revision as of 16:50, 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 $G$ 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>; * $\pi_1 (L) = \Zz_m$, $\pi_i (L) = \pi_i (S^{2d-1})$ 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

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. We also suppose that a generator $T$ of $G$ is chosen.

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 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.

The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in

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 [Theorem 14E.4] of [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.

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 [Macko&Wegner2010, Proposition 3.5]).

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].

6 Further discussion

...

7 References

• [Macko&Wegner2010] Template:Macko&Wegner2010
• [Wall(1999)] Template:Wall(1999)
• [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. We also suppose that a generator $T$ of $G$ is chosen. 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 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. The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in {{beginthm|Definition}} \label{def-pol-lens-spc} A 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 $e \colon \widetilde L \rightarrow S^{2d-1}$. {{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 $u \in R_G$. Then $L$ 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 [Theorem 14E.4] of \cite{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. {{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 $$ h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(\alpha_k). $$ {{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 {{cite|Macko&Wegner2010|Proposition 3.5}}). 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} == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]] {{Stub}}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

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. We also suppose that a generator $T$ of $G$ is chosen.

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 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.

The simple homotopy classification is stated in terms of Reidemeister torsion which is a unit in

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 [Theorem 14E.4] of [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.

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 [Macko&Wegner2010, Proposition 3.5]).

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].

6 Further discussion

...

7 References

• [Macko&Wegner2010] Template:Macko&Wegner2010
• [Wall(1999)] Template:Wall(1999)
• [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.