Geometric topology (object)

In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume.

Use

Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds.

Definition

The following is a definition due to Troels Jorgensen:

A sequence ${\displaystyle \{M_{i}\}}$ in H converges to M in H if there are

• a sequence of positive real numbers ${\displaystyle \epsilon _{i}}$ converging to 0, and
• a sequence of ${\displaystyle (1+\epsilon _{i})}$-bi-Lipschitz diffeomorphisms ${\displaystyle \phi _{i}:M_{i,[\epsilon _{i},\infty )}\rightarrow M_{[\epsilon _{i},\infty )},}$

where the domains and ranges of the maps are the ${\displaystyle \epsilon _{i}}$-thick parts of either the ${\displaystyle M_{i}}$'s or M.

Alternate definition

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.

On framed manifolds

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.

See also

• Algebraic topology (object)

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.