Lens spaces

1 Introduction

A lens space is the orbit spaces of a free linear action of a finite cyclic group on a sphere. The importance of lens spaces stems from the fact that they provide examples of peculiar phenomena. For example, there are pairs of lens spaces where both lens spaces have the same homotopy and homology groups but are not homotopy equivalent, and there are also pairs where both lens spaces are homotopy equivalent, but not homeomorphic. The lens spaces also play a role in Milnor's disproof of Hauptvermutung for polyhedra, that means they were used to find two polyhedra which are homeomorphic but combinatorially distinct.

2 Construction and examples

Let $m$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; A lens space is the orbit spaces of a free linear action of a finite cyclic group on a sphere. The importance of lens spaces stems from the fact that they provide examples of peculiar phenomena. For example, there are pairs of lens spaces where both lens spaces have the same homotopy and homology groups but are not homotopy equivalent, and there are also pairs where both lens spaces are homotopy equivalent, but not homeomorphic. The lens spaces also play a role in Milnor's disproof of Hauptvermutung for polyhedra, that means they were used to find two polyhedra which are homeomorphic but combinatorially distinct. </wikitex> == Construction and examples == <wikitex>; Let $m$, $l_i$ for $i = 1, \ldots, d$ be natural numbers such that $(l_i,m) = 1$ for all $i$. The lens space $L(m;l_1,\ldots,l_d)$ is defined to be the orbit space of the free action of the cyclic group $\Zz_m$ on the sphere $S^{2d-1} = S (\Cc^d)$ given by the formula $$ (z_1,\ldots,z_d) \mapsto (z_1 \cdot e^{2\pi i l_1/m}, \ldots ,z_d \cdot e^{2\pi i l_d/m}). $$ </wikitex> == Invariants == <wikitex>; Abbreviate $L = L(m;l_1,\ldots,l_d)$. * $\pi_1 (L) = \Zz_m$, $\pi_i (L) = \pi_i (S^{2d-1})$ for $i \geq 2$ * $H_0 (L) = \Zz$, $H_{2d-1} (L) = \Zz$, $H_{2i-1} (L) = \Zz_m$ for m$, $l_i$$l_i$ for $i = 1, \ldots, d$$i = 1, \ldots, d$ be natural numbers such that $(l_i,m) = 1$$(l_i,m) = 1$ for all $i$$i$. The lens space $L(m;l_1,\ldots,l_d)$$L(m;l_1,\ldots,l_d)$ is defined to be the orbit space of the free action of the cyclic group $\Zz_m$$\Zz_m$ on the sphere $S^{2d-1} = S (\Cc^d)$$S^{2d-1} = S (\Cc^d)$ given by the formula

$\displaystyle (z_1,\ldots,z_d) \mapsto (z_1 \cdot e^{2\pi i l_1/m}, \ldots ,z_d \cdot e^{2\pi i l_d/m}).$ $$\displaystyle (z_1,\ldots,z_d) \mapsto (z_1 \cdot e^{2\pi i l_1/m}, \ldots ,z_d \cdot e^{2\pi i l_d/m}).$$

3 Invariants

Abbreviate $L = L(m;l_1,\ldots,l_d)$$L = L(m;l_1,\ldots,l_d)$.

• $\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$.

• Let $r_i$$r_i$ be natural numbers satisfying $r_i \cdot l_i \equiv 1$$r_i \cdot l_i \equiv 1$ mod $m$$m$ for all $i$$i$. Then the Reidemeister torsion is given by

$\displaystyle \Delta (L) = \prod_{i_1}^{d}(t^{r_i} -1) \in \Qq [t] / (t^m-1).$ $$\displaystyle \Delta (L) = \prod_{i_1}^{d}(t^{r_i} -1) \in \Qq [t] / (t^m-1).$$

• Let $r_i$$r_i$ be natural numbers satisfying $r_i \cdot l_i \equiv 1$$r_i \cdot l_i \equiv 1$ mod $m$$m$ for all $i$$i$. Then the Rho-invariant is defined by

$\displaystyle \rho (L) = \prod_{i_1}^{d}\frac{(t^{r_i} +1)}{(t^{r_i}-1)} \in \Qq [t] / (t^m-1).$ $$\displaystyle \rho (L) = \prod_{i_1}^{d}\frac{(t^{r_i} +1)}{(t^{r_i}-1)} \in \Qq [t] / (t^m-1).$$

4 Classification/Characterization

Abbreviate $L = L(m;l_1,\ldots,l_d)$$L = L(m;l_1,\ldots,l_d)$ and $L' = L(m;l'_1,\ldots,l'_d)$$L' = L(m;l'_1,\ldots,l'_d)$.

Homotopy classification:

Theorem 4.1 [Olum1953]. $L \simeq L'$$L \simeq L'$ if and only if $l_1 \cdots l_d \equiv \pm k^d l'_1 \cdots l'_d$$l_1 \cdots l_d \equiv \pm k^d l'_1 \cdots l'_d$ mod $m$$m$ for some $k \in \Zz_m$$k \in \Zz_m$.

See also [Cohen1973].

Homeomorphism classification:

Theorem 4.2 [Franz1935].$L \cong L'$$L \cong L'$ if and only if for some permutation $\sigma$$\sigma$ and some $k \in \Zz_m$$k \in \Zz_m$ we have $l_i \equiv \pm k l'_{\sigma(i)}$$l_i \equiv \pm k l'_{\sigma(i)}$ for all $i$$i$.

See also [Milnor1966]

5 Further discussion

More details and a discussion of fake lens spaces are planned. This includes the $\rho$$\rho$-invariant.

6 References

• [Cohen1973] M. M. Cohen, A course in simple-homotopy theory, Springer-Verlag, New York, 1973. MR0362320 (50 #14762) Zbl 0261.57009
• [Franz1935] W. Franz, Über die Torsion einer Überdeckung., Journ. f. Math. 173 (1935), 245-254. Zbl 61.1350.01
• [Milnor1966] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR0196736 (33 #4922) Zbl 0147.23104
• [Olum1953] P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458–480. MR0058212 (15,338a) Zbl 0052.19901

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 $i$. * Let $r_i$ be natural numbers satisfying $r_i \cdot l_i \equiv 1$ mod $m$ for all $i$. Then the [[Wikipedia:Reidemeister_torsion|Reidemeister torsion]] is given by $$ \Delta (L) = \prod_{i_1}^{d}(t^{r_i} -1) \in \Qq [t] / (t^m-1). $$ * Let $r_i$ be natural numbers satisfying $r_i \cdot l_i \equiv 1$ mod $m$ for all $i$. Then the [[Rho-invariant|Rho-invariant]] is defined by $$ \rho (L) = \prod_{i_1}^{d}\frac{(t^{r_i} +1)}{(t^{r_i}-1)} \in \Qq [t] / (t^m-1). $$ == Classification/Characterization == ; Abbreviate $L = L(m;l_1,\ldots,l_d)$ and $L' = L(m;l'_1,\ldots,l'_d)$. Homotopy classification: {{beginthm|Theorem|{{cite|Olum1953}}}} $L \simeq L'$ if and only if $l_1 \cdots l_d \equiv \pm k^d l'_1 \cdots l'_d$ mod $m$ for some $k \in \Zz_m$. {{endthm}} See also {{cite|Cohen1973}}. Homeomorphism classification: {{beginthm|Theorem|{{cite|Franz1935}}}}$L \cong L'$ if and only if for some permutation $\sigma$ and some $k \in \Zz_m$ we have $l_i \equiv \pm k l'_{\sigma(i)}$ for all $i$. {{endthm}} See also {{cite|Milnor1966}} == Further discussion == ; More details and a discussion of fake lens spaces are planned. This includes the $\rho$-invariant. == References == {{#RefList:}} [[Category:Manifolds]] {{Stub}}m, $l_i$$l_i$ for $i = 1, \ldots, d$$i = 1, \ldots, d$ be natural numbers such that $(l_i,m) = 1$$(l_i,m) = 1$ for all $i$$i$. The lens space $L(m;l_1,\ldots,l_d)$$L(m;l_1,\ldots,l_d)$ is defined to be the orbit space of the free action of the cyclic group $\Zz_m$$\Zz_m$ on the sphere $S^{2d-1} = S (\Cc^d)$$S^{2d-1} = S (\Cc^d)$ given by the formula

$\displaystyle (z_1,\ldots,z_d) \mapsto (z_1 \cdot e^{2\pi i l_1/m}, \ldots ,z_d \cdot e^{2\pi i l_d/m}).$ $$\displaystyle (z_1,\ldots,z_d) \mapsto (z_1 \cdot e^{2\pi i l_1/m}, \ldots ,z_d \cdot e^{2\pi i l_d/m}).$$

3 Invariants

Abbreviate $L = L(m;l_1,\ldots,l_d)$$L = L(m;l_1,\ldots,l_d)$.

• $\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$.

• Let $r_i$$r_i$ be natural numbers satisfying $r_i \cdot l_i \equiv 1$$r_i \cdot l_i \equiv 1$ mod $m$$m$ for all $i$$i$. Then the Reidemeister torsion is given by

$\displaystyle \Delta (L) = \prod_{i_1}^{d}(t^{r_i} -1) \in \Qq [t] / (t^m-1).$ $$\displaystyle \Delta (L) = \prod_{i_1}^{d}(t^{r_i} -1) \in \Qq [t] / (t^m-1).$$

• Let $r_i$$r_i$ be natural numbers satisfying $r_i \cdot l_i \equiv 1$$r_i \cdot l_i \equiv 1$ mod $m$$m$ for all $i$$i$. Then the Rho-invariant is defined by

$\displaystyle \rho (L) = \prod_{i_1}^{d}\frac{(t^{r_i} +1)}{(t^{r_i}-1)} \in \Qq [t] / (t^m-1).$ $$\displaystyle \rho (L) = \prod_{i_1}^{d}\frac{(t^{r_i} +1)}{(t^{r_i}-1)} \in \Qq [t] / (t^m-1).$$

4 Classification/Characterization

Abbreviate $L = L(m;l_1,\ldots,l_d)$$L = L(m;l_1,\ldots,l_d)$ and $L' = L(m;l'_1,\ldots,l'_d)$$L' = L(m;l'_1,\ldots,l'_d)$.

Homotopy classification:

Theorem 4.1 [Olum1953]. $L \simeq L'$$L \simeq L'$ if and only if $l_1 \cdots l_d \equiv \pm k^d l'_1 \cdots l'_d$$l_1 \cdots l_d \equiv \pm k^d l'_1 \cdots l'_d$ mod $m$$m$ for some $k \in \Zz_m$$k \in \Zz_m$.

See also [Cohen1973].

Homeomorphism classification:

Theorem 4.2 [Franz1935].$L \cong L'$$L \cong L'$ if and only if for some permutation $\sigma$$\sigma$ and some $k \in \Zz_m$$k \in \Zz_m$ we have $l_i \equiv \pm k l'_{\sigma(i)}$$l_i \equiv \pm k l'_{\sigma(i)}$ for all $i$$i$.

See also [Milnor1966]

5 Further discussion

More details and a discussion of fake lens spaces are planned. This includes the $\rho$$\rho$-invariant.

6 References

• [Cohen1973] M. M. Cohen, A course in simple-homotopy theory, Springer-Verlag, New York, 1973. MR0362320 (50 #14762) Zbl 0261.57009
• [Franz1935] W. Franz, Über die Torsion einer Überdeckung., Journ. f. Math. 173 (1935), 245-254. Zbl 61.1350.01
• [Milnor1966] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR0196736 (33 #4922) Zbl 0147.23104
• [Olum1953] P. Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458–480. MR0058212 (15,338a) Zbl 0052.19901

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