Lens space

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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.

In the 3-manifold case, a picturesque description of a lens space is that of a space resulting from gluing two solid tori 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.

The three-dimensional lens spaces $L (p, q)$ were introduced by Tietze in 1908. They 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 $L (5;1)$ and $L (5;2)$ were not homeomorphic even though they have isomorphic fundamental groups and the same homology.

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

1 Definition
2 Three-dimensional lens spaces
3 See also
4 References

[edit] 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 $\zeta_p=\{1,\omega,\ldots,\omega^{p-1}\}$ 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 $p$ .

[edit] Three-dimensional lens spaces

Three-dimensional lens spaces arise as quotients of $S^3 \subset \mathbb{C}^2$ by the action of the group that is generated by elements of the form $\begin{pmatrix}\omega&0\\0&\omega^q\end{pmatrix}$ .

Such a lens space $L (p; q)$ has fundamental group $\mathbb{Z}/p\mathbb{Z}$ for all $q$ , so spaces with different $p$ are not homotopy equivalent. Moreover, classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces $L (p; q 1)$ and $L (p; q 2)$ are:

homotopy equivalent if and only if $q_1 q_2 \equiv \pm n^2 \pmod{p}$ for some $n \in \mathbb{N}$ ;
homeomorphic if and only if $q_1 \equiv \pm q_2^{\pm1} \pmod{p}$ .

[edit] See also

Spherical 3-manifold

[edit] References

G. Bredon, Topology and Geometry, Springer Graduate Texts in Mathematics 139, 1993.
A. Hatcher, Algebraic Topology available online, Cambridge University Press, 2002.
A. Hatcher, Notes on basic 3-manifold topology, available online (explains classification of L(p,q) up to homeomorphism)
H. Tietze, Ueber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten [1], Monatsh. fuer Math. und Phys. 19, 1-118 (1908) ( $\S$ 20)
M. Watkins, "A Short Survey of Lens Spaces" (1990 undergraduate dissertation)