Lens spaces

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Contents

[edit] 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.

[edit] 2 Construction and examples

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

\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}).

[edit] 3 Invariants

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 1 \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 Reidemeister torsion is given by
\displaystyle  \Delta (L) = \prod_{i_1}^{d}(T^{r_i} -1) \in \Qq R_G = \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 is defined by
\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

[edit] 4 Classification/Characterization

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

[edit] 1 Homotopy classification

Theorem 4.1 [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.

See also [Cohen1973].

[edit] 2 PL homeomorphism classification

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

[edit] 3 Homeomorphism classification

Theorem 4.3 [Brody1960a]. 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.

See also [Milnor1966]

[edit] 4 h-cobordism classification

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

See also [Milnor1966]

[edit] 5 Further discussion

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

[edit] 6 References

[edit] 7 External links

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