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. 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 $epsilon\_i$ converging to 0, and :* a sequence of $(1+epsilon\_i)$-bi-Lipschitz diffeomorphisms $phi\_i:\; M\_\{i,\; [epsilon\_i,\; infty)\}\; ightarrow\; M\_\{\; [epsilon\_i,\; infty)\},$

:where the domains and ranges of the maps are the $epsilon\_i$-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.

ee also

*Algebraic topology (object)

References

* William Thurston, [http://www.msri.org/publications/books/gt3m/ "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.