Fake lens spaces

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Classical [[lens spaces]] are of course examples of fake lens spaces. To get fake lens spaces which are not homeomorphic to classical ones one can employ the construction of [[fake complex projective spaces]]. Note that a fake complex projective space is an orbit space of a free tame action of $S^1$ on $S^{2d-1}$ and that we obviously have $\Zz_N \cong G < S^1$. Restricting the action to the subgroup we obtain a fake lens space. Its $\rho$-[[rho-invariant|invariant]] can be calculated by the naturality using the formula ...
Classical [[lens spaces]] are of course examples of fake lens spaces. To get fake lens spaces which are not homeomorphic to classical ones one can employ the construction of [[fake complex projective spaces]]. Note that a fake complex projective space is an orbit space of a free tame action of $S^1$ on $S^{2d-1}$ and that we obviously have $\Zz_N \cong G < S^1$. Restricting the action to the subgroup we obtain a fake lens space. Its $\rho$-[[rho-invariant|invariant]] can be calculated by the naturality using the formula ...
−
The above consruction does not exhaust all the fake lens spaces. To get all of them there is a construction which produces from a given fake lens space $L$ another fake lens space $L'$ such that the difference of their $\rho$-invariants is a prescribed element
+
The above construction does not exhaust all the fake lens spaces. To get all of them there is a construction which produces from a given fake lens space $L$ another fake lens space $L'$ such that the difference of their $\rho$-invariants is a prescribed element
$$
$$
\rho (L) - \rho(L') = x \in 4 \cdot R_{\widehat G}^{(-1)^d}.
\rho (L) - \rho(L') = x \in 4 \cdot R_{\widehat G}^{(-1)^d}.

Revision as of 11:27, 8 June 2010

An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print.

You may view the version used for publication as of 15:18, 25 April 2013 and the changes since publication.

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Contents

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 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 \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:

\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 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.

Definition 4.1.

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}.

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.

Theorem 4.2 Wall.

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

\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. 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.

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.

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

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

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

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

Theorem 5.1 Wall. If N is odd, then the map

\displaystyle {\widetilde \rho} \colon {\mathcal S}^s (L^{2d-1}(\alpha)) \to {\mathbb Q} R^{(-1)^d}_{\widehat G}

is injective.

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:

Theorem 5.2 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 I_G^n, \rho \in I_{\widehat G}^{-n}.
  • The classes of \rho \mod I_{\widehat G}^{-n+1} and (-2)^n \Delta \mod I_G^{n+1} correspond under
\displaystyle I_{\widehat G}^{-n} / I_{\widehat G}^{-n+1} \cong {\widehat H}^{2n}({\widehat G};\Zz) \cong {\widehat H}^{-2n}(G;\Zz) \cong I_G^n / I_G^{n+1}.
  • \rho \equiv - \displaystyle\sum_{\phi \in {\widehat G}, \phi \neq 1} \textup{sign}(i^n \phi(\Delta)) \phi \mod 4.

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

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

Theorem 5.3. 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

\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) 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.


6 Construction and examples

Classical lens spaces are of course examples of fake lens spaces. To get fake lens spaces which are not homeomorphic to classical ones one can employ the construction of fake complex projective spaces. Note that a fake complex projective space is an orbit space of a free tame action of S^1 on S^{2d-1} and that we obviously have \Zz_N \cong G < S^1. Restricting the action to the subgroup we obtain a fake lens space. Its \rho-invariant can be calculated by the naturality using the formula ...

The above construction does not exhaust all the fake lens spaces. To get all of them there is a construction which produces from a given fake lens space L another fake lens space L' such that the difference of their \rho-invariants is a prescribed element

\displaystyle \rho (L) - \rho(L') = x \in 4 \cdot R_{\widehat G}^{(-1)^d}.

The construction is just the Wall realization from surgery theory, alias a non-simply connected generalization of the plumbing construction.


7 Further discussion

...

8 References

\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 $I_G^n$, $\rho \in I_{\widehat G}^{-n}$. * The classes of $\rho \mod I_{\widehat G}^{-n+1}$ and $(-2)^n \Delta \mod I_G^{n+1}$ correspond under $$I_{\widehat G}^{-n} / I_{\widehat G}^{-n+1} \cong {\widehat H}^{2n}({\widehat G};\Zz) \cong {\widehat H}^{-2n}(G;\Zz) \cong I_G^n / I_G^{n+1}.$$ * $\rho \equiv - \displaystyle\sum_{\phi \in {\widehat G}, \phi \neq 1} \textup{sign}(i^n \phi(\Delta)) \phi \mod 4$. {{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} == Construction and examples == ; Classical [[lens spaces]] are of course examples of fake lens spaces. To get fake lens spaces which are not homeomorphic to classical ones one can employ the construction of [[fake complex projective spaces]]. Note that a fake complex projective space is an orbit space of a free tame action of $S^1$ on $S^{2d-1}$ and that we obviously have $\Zz_N \cong G < S^1$. Restricting the action to the subgroup we obtain a fake lens space. Its $\rho$-[[rho-invariant|invariant]] can be calculated by the naturality using the formula ... The above construction does not exhaust all the fake lens spaces. To get all of them there is a construction which produces from a given fake lens space $L$ another fake lens space $L'$ such that the difference of their $\rho$-invariants is a prescribed element $$ \rho (L) - \rho(L') = x \in 4 \cdot R_{\widehat G}^{(-1)^d}. $$ The construction is just the Wall realization from surgery theory, alias a non-simply connected generalization of the [[Wikipedia:plumbing|plumbing]] construction. == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]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 \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:

\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 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.

Definition 4.1.

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}.

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.

Theorem 4.2 Wall.

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

\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. 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.

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.

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

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

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

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

Theorem 5.1 Wall. If N is odd, then the map

\displaystyle {\widetilde \rho} \colon {\mathcal S}^s (L^{2d-1}(\alpha)) \to {\mathbb Q} R^{(-1)^d}_{\widehat G}

is injective.

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:

Theorem 5.2 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 I_G^n, \rho \in I_{\widehat G}^{-n}.
  • The classes of \rho \mod I_{\widehat G}^{-n+1} and (-2)^n \Delta \mod I_G^{n+1} correspond under
\displaystyle I_{\widehat G}^{-n} / I_{\widehat G}^{-n+1} \cong {\widehat H}^{2n}({\widehat G};\Zz) \cong {\widehat H}^{-2n}(G;\Zz) \cong I_G^n / I_G^{n+1}.
  • \rho \equiv - \displaystyle\sum_{\phi \in {\widehat G}, \phi \neq 1} \textup{sign}(i^n \phi(\Delta)) \phi \mod 4.

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

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

Theorem 5.3. 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

\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) 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.


6 Construction and examples

Classical lens spaces are of course examples of fake lens spaces. To get fake lens spaces which are not homeomorphic to classical ones one can employ the construction of fake complex projective spaces. Note that a fake complex projective space is an orbit space of a free tame action of S^1 on S^{2d-1} and that we obviously have \Zz_N \cong G < S^1. Restricting the action to the subgroup we obtain a fake lens space. Its \rho-invariant can be calculated by the naturality using the formula ...

The above construction does not exhaust all the fake lens spaces. To get all of them there is a construction which produces from a given fake lens space L another fake lens space L' such that the difference of their \rho-invariants is a prescribed element

\displaystyle \rho (L) - \rho(L') = x \in 4 \cdot R_{\widehat G}^{(-1)^d}.

The construction is just the Wall realization from surgery theory, alias a non-simply connected generalization of the plumbing construction.


7 Further discussion

...

8 References

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