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Sit the 2n - 1-sphere S2n - 1 inside as the set of all n-tuples of unit absolute value. Let ω be a primitive pth root of unity and let be integers coprime to p. Let the set of powers act on the sphere by
The resulting orbit space is a lens space, written as .
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 , 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.
- G. Bredon, Topology and Geometry, Springer Graduate Texts in Mathematics 139, 1993.
- A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
- For more on 3-dimensional lens spaces, see spherical 3-manifold.
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