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Categories: 3-manifolds | Differential topology | Topological spaces

Lens space

A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions.

Contents

1 Definition

2 In three dimensions

3 References

4 See also

Definition

Sit the $2 n - 1$ -sphere $S 2 n - 1$ inside $\mathbb C^n$ as the set of all $n$ -tuples of unit absolute value. Let $ω$ be a primitive $p$ th root of unity and let $q_1,\ldots,q_n$ be integers coprime to $p$ . Let the set of powers $\mathbb Z_p=\{1,\omega,\ldots,\omega^{p-1}\}$ act on the sphere by

$\omega\cdot(z_1,\ldots,z_n)=(\omega^{q_1}z_1,\ldots,\omega^{q_n}z_n).$

The resulting orbit space is a lens space, written as $L(p;q_1,\ldots,q_n)$ .

We can also define the infinite-dimensional lens spaces as follows. These are the spaces $L(p;q_1,q_2,\ldots)$ formed from the union of the increasing sequence of spaces $L(p;q_1,\ldots,q_n)$ for $n=1,2,\ldots$ . As before, the $q_1,q_2,\ldots$ must be coprime to $p$ .

In three dimensions

By specializing the above definition to $n = 2$ , we get 3-manifolds. In this case, a more picturesque description of a lens space is that of a space resulting from gluing two solid torii together by a homeomorphism of their boundaries. Of course, to be consistent, we should exclude the 3-sphere and $S^2 \times S^1$ , both of which can be obtained as just described; some mathematicians include these two manifolds in the class of lens spaces.

Three-dimensional lens spaces were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone. J.W. Alexander in 1919 showed that the lens spaces $L (1;5)$ and $L (2;5)$ were not homeomorphic even though they have isomorphic fundamental groups and the same homology.

There is a complete classification of three-dimensional lens spaces.

References

G. Bredon, Topology and Geometry, Springer Graduate Texts in Mathematics 139, 1993.
A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

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