Lens spaces

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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.

For historical information about 3-dimensional lens spaces, see Lens spaces in dimension three: a history.

For information about manifold homotopy equivalent to lens spaces, see the page on fake lens spaces.

2 Construction and examples

Let $m$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}m$, $l_i$$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

$\displaystyle \Delta (L) = \prod_{i=1}^{d}(T^{r_i} -1) \in (\Qq R_G)^\times / \langle -1, T \rangle$ $$\displaystyle \Delta (L) = \prod_{i=1}^{d}(T^{r_i} -1) \in (\Qq R_G)^\times / \langle -1, T \rangle$$

where $\Qq R_G = \Qq [T] / (1+T+\cdots + T^{m-1})$$\Qq R_G = \Qq [T] / (1+T+\cdots + T^{m-1})$ (see p406 of [Milnor1966]).

• 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 (p187 of [Wall1999])

$\displaystyle \rho (L) = \prod_{i=1}^{d}\frac{(\chi^{r_i} +1)}{(\chi^{r_i}-1)} \in \Qq R_{\widehat G} = \Qq [\chi] / \langle 1 + \chi + \cdots + \chi^{N-1} \rangle.$ $$\displaystyle \rho (L) = \prod_{i=1}^{d}\frac{(\chi^{r_i} +1)}{(\chi^{r_i}-1)} \in \Qq R_{\widehat G} = \Qq [\chi] / \langle 1 + \chi + \cdots + \chi^{N-1} \rangle.$$

For the notation in the last two points click here

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)$.

1 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].

2 PL homeomorphism classification

Theorem 4.2 [Franz1935],[De Rham1936]. $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$.

3 Homeomorphism classification

Theorem 4.3 [Brody1960a]. $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]

4 $h$$h$-cobordism classification

Theorem 4.4 [Atiyah&Bott1968]. Two lens spaces $L$$L$, $L'$$L'$ are $h$$h$-cobordant if and only if they are homeomorphic.

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

• [Atiyah&Bott1968] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR0232406 (38 #731) Zbl 0167.21703
• [Brody1960a] E. J. Brody, The topological classification of the lens spaces, Ann. of Math. (2) 71 (1960), 163–184. MR0116336 (22 #7125) Zbl 0119.18901
• [Cohen1973] M. M. Cohen, A course in simple-homotopy theory, Springer-Verlag, New York, 1973. MR0362320 (50 #14762) Zbl 0261.57009
• [De Rham1936] G. de Rham, Sur les nouveaux invariants topologiques de M. Reidemeister, Rec. Math. [Mat. Sbornik], Moscou, Volume 1(43), (1936), 737-742. Zbl 0016.04501
• [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
• [Wall1999] C. T. C. Wall, Surgery on compact manifolds, American Mathematical Society, Providence, RI, 1999. MR1687388 (2000a:57089) Zbl 0935.57003

7 External links

• The Wikipedia page about lens spaces.