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.

Definition

Sit the

2n-1

-sphere

S^{2n-1}

inside

\mathbb C^n

as the set of all

n

-tuples of unit absolute value. Let

\omega

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.

Lens Space

Definition

In three dimensions

References

See also