Fake lens spaces

Revision as of 10:24, 8 June 2010 (view source) Tibor Macko (Talk | contribs) (→Homeomorphism classification) ← Older edit		Revision as of 10:26, 8 June 2010 (view source) Christian Wegner (Talk | contribs) m (→Homotopy Classification and simple homotopy classification) Newer edit →
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	{{endthm}}		{{endthm}}
−	Recall the classical [[Lens spaces|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 $i-1$.	+	Recall the classical [[Lens spaces|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|Theorem|Wall}} \label{prop-simple-htpy-class}		{{beginthm|Theorem|Wall}} \label{prop-simple-htpy-class}
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	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		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}	+	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
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	and a homotopy equivalence		and a homotopy equivalence
	$$		$$
−	h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(N,k,1,\ldots,1).	+	h \colon L^{2d-1}(\alpha) \rightarrow L^{2d-1}(N;k,1,\ldots,1).
	$$		$$
	{{endthm}}		{{endthm}}

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Revision as of 10:26, 8 June 2010

Contents 1 Introduction 2 Definition 3 Invariants 4 Homotopy Classification and simple 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 on a sphere. It is a generalization of the notion of a lens space which is the orbit space of such a free action which comes from a unitary representation.

The classification of fake lens spaces can be seen as one of the basic questions in topology of manifolds. It is systematically obtained in three stages. First, homotopy classification using classical homotopy theory. Second, simple homotopy classification using Reidemeister torsion. Finally, surgery theory is employed to obtain a classification within the respective simple homotopy types. In fact, this classification was one of the early spectacular applications of surgery theory.

2 Definition

Throughout this page we use the following notation. By $<!-- 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 on a sphere. It is a generalization of the notion of a [[Lens spaces|lens space]] which is the orbit space of such a free action which comes from a unitary representation. The classification of fake lens spaces can be seen as one of the basic questions in topology of manifolds. It is systematically obtained in three stages. First, homotopy classification using classical homotopy theory. Second, simple homotopy classification using [[Wikipedia:Reidemeister torsion|Reidemeister torsion]]. Finally, [[Wikipedia:surgery theory|surgery theory]] is employed to obtain a classification within the respective simple homotopy types. In fact, this classification was one of the early spectacular applications of surgery theory. </wikitex> == Definition == <wikitex>; Throughout this page we use the following notation. By $G$ is denoted the finite cyclic group of order $N$. Let $\alpha$ be a free action of $G$ on the sphere $S^{2d-1}$. By $L^{2d-1} (\alpha)$ is denoted the orbit space of $\alpha$. Sometimes, when the dimension and the action are clear, we leave them from notation and simply write $L$. </wikitex> == Invariants == <wikitex>; For $L = L^{2d-1} (\alpha)$ we have * $\pi_1 (L) = \Zz_N$, $\pi_i (L) = \pi_i (S^{2d-1})$ for $i \geq 2$ * $H_0 (L) = \Zz$, $H_{2d-1} (L) = \Zz$, $H_{2i-1} (L) = \Zz_N$ for G$ is denoted the finite cyclic group of order $N$.

Let $\alpha$ be a free action of $G$ on the sphere $S^{2d-1}$. By $L^{2d-1} (\alpha)$ is denoted the orbit space of $\alpha$. Sometimes, when the dimension and the action are clear, we leave them from notation and simply write $L$.

3 Invariants

For $L = L^{2d-1} (\alpha)$ we have

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

Interesting invariants for fake lens spaces are

• the Reidemeister torsion $\Delta (L) \in \Qq [t] / (t^N - 1)$ and

• the $\rho$-invariant $\rho (L) \in \Qq R_{\widehat G}^{(-1)^d}$.

4 Homotopy Classification and simple 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 [Wall1999, Theorem 14E.3].

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

The homeomorphism classification, as already noted, is an excellent application of the non-simply connected surgery theory. Recall that for a topological manifold $X$ the surgery theoretic homeomorphism classification of manifolds wihin the homotopy type of $X$ is stated in terms of the surgery structure set $\mathcal{S} (X)$ and that the primary tool for its calculation is the surgery exact sequence.

For a fake lens space $L$ there is enough information about the normal invariants, the L-groups and the surgery obstruction in the surgery exact sequence so that one is left with just an extension problem. The strategy to proceed further is to relate the surgery exact sequence to representation theory of $G$. This is done via the following commutative diagram of abelian groups and homomorphisms with exact rows

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

See [Wall1999, Theorem 14E.7].

For $N$ odd Wall managed to obtain an even beter result, namely the complete classification of fake lens spaces of a given dimension $(2d-1) \geq 5$ with the fundamental group $G \cong \Zz_N$ which goes as follows:

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

For general $N$ the following theorem is proved in [Macko&Wegner2008, Theorem 1.2]).

6 Further discussion

...

7 References

• [Macko&Wegner2008] T. Macko and C. Wegner, On the classification of fake lens spaces, to appear in Forum. Math. Available at the arXiv:0810.1196.
• [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 $i$. Interesting invariants for fake lens spaces are * the [[Wikipedia:Reidemeister torsion|Reidemeister torsion]] $\Delta (L) \in \Qq [t] / (t^N - 1)$ and * the $\rho$-[[Rho-invariant|invariant]] $\rho (L) \in \Qq R_{\widehat G}^{(-1)^d}$. == Homotopy Classification and simple 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 spaces|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|Theorem|Wall}} \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 {{cite|Wall1999|Theorem 14E.3}}. 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 == ; The homeomorphism classification, as already noted, is an excellent application of the non-simply connected surgery theory. Recall that for a topological manifold $X$ the surgery theoretic homeomorphism classification of manifolds wihin the homotopy type of $X$ is stated in terms of the [[Wikipedia:Surgery structure set|surgery structure set]] $\mathcal{S} (X)$ and that the primary tool for its calculation is the [[Wikipedia:Surgery exact sequence|surgery exact sequence]]. For a fake lens space $L$ there is enough information about the [[Wikipedia:normal invariants|normal invariants]], the [[Wikipedia:L-theory|L-groups]] and the [[Wikipedia:surgery obstruction|surgery obstruction]] in the surgery exact sequence so that one is left with just an extension problem. The strategy to proceed further is to relate the surgery exact sequence to representation theory of $G$. This is done via 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 ?). {{beginthm|Theorem|Wall}} If $N$ is odd, then the map $$ {\widetilde \rho} \colon {\mathcal S}^s (L^{2d-1}(\alpha)) \to {\mathbb Q} R^{(-1)^d}_{\widehat G} $$ is injective. {{endthm}} See {{cite|Wall1999|Theorem 14E.7}}. For $N$ odd Wall managed to obtain an even beter result, namely the complete classification of fake lens spaces of a given dimension $(2d-1) \geq 5$ with the fundamental group $G \cong \Zz_N$ which goes as follows: {{beginthm|Theorem|Wall}} 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 ... * ... * ... {{endthm}} The following theorem is taken from {{cite|Wall1999|Theorem 14E.7}}. For general $N$ the following theorem is proved in {{cite|Macko&Wegner2008|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 is denoted the finite cyclic group of order $N$.

Let $\alpha$ be a free action of $G$ on the sphere $S^{2d-1}$. By $L^{2d-1} (\alpha)$ is denoted the orbit space of $\alpha$. Sometimes, when the dimension and the action are clear, we leave them from notation and simply write $L$.

3 Invariants

For $L = L^{2d-1} (\alpha)$ we have

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

Interesting invariants for fake lens spaces are

• the Reidemeister torsion $\Delta (L) \in \Qq [t] / (t^N - 1)$ and

• the $\rho$-invariant $\rho (L) \in \Qq R_{\widehat G}^{(-1)^d}$.

4 Homotopy Classification and simple 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 [Wall1999, Theorem 14E.3].

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

The homeomorphism classification, as already noted, is an excellent application of the non-simply connected surgery theory. Recall that for a topological manifold $X$ the surgery theoretic homeomorphism classification of manifolds wihin the homotopy type of $X$ is stated in terms of the surgery structure set $\mathcal{S} (X)$ and that the primary tool for its calculation is the surgery exact sequence.

For a fake lens space $L$ there is enough information about the normal invariants, the L-groups and the surgery obstruction in the surgery exact sequence so that one is left with just an extension problem. The strategy to proceed further is to relate the surgery exact sequence to representation theory of $G$. This is done via the following commutative diagram of abelian groups and homomorphisms with exact rows

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

See [Wall1999, Theorem 14E.7].

For $N$ odd Wall managed to obtain an even beter result, namely the complete classification of fake lens spaces of a given dimension $(2d-1) \geq 5$ with the fundamental group $G \cong \Zz_N$ which goes as follows:

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

For general $N$ the following theorem is proved in [Macko&Wegner2008, Theorem 1.2]).

6 Further discussion

...

7 References

• [Macko&Wegner2008] T. Macko and C. Wegner, On the classification of fake lens spaces, to appear in Forum. Math. Available at the arXiv:0810.1196.
• [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.