Geometric topology (object)

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For the mathematical subject area, see geometric topology.

In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume. Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds.

The following is a definition due to Troels Jorgensen:

A sequence

{M i}

in H converges to M in H if there are

a sequence of positive real numbers $ε i$ converging to 0, and
a sequence of $(1 + ε i)$ -bi-Lipschitz diffeomorphisms $\phi_i: M_{i, [\epsilon_i, \infty)} \rightarrow M_{[\epsilon_i, \infty)},$

where the domains and ranges of the maps are the

ε i

-thick parts of either the

M i

's or M.

There is an alternate definition due to Mikhail Gromov. Gromov's topology utilizes the Gromov-Hausdorff metric and is defined on pointed hyperbolic 3-manifolds. One essentially considers better and better bi-Lipschitz homeomorphisms on larger and larger balls. This results in the same notion of convergence as above as the thick part is always connected; thus, a large ball will eventually encompass all of the thick part.

As a further refinement, Gromov's metric can also be defined on framed hyperbolic 3-manifolds. This gives nothing new but this space can be explicitly identified with torsion-free Kleinian groups with the Chabauty topology.

[edit] See also

Algebraic topology (object)

[edit] References

William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978-1981).
Canary, R. D.; Epstein, D. B. A.; Green, P., Notes on notes of Thurston. Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), 3--92, London Math. Soc. Lecture Note Ser., 111, Cambridge Univ. Press, Cambridge, 1987.