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28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. 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 ${\varphi }_i:M_{i,[{ϵ}_i,\mathrm{\infty })}\to M_{[{ϵ}_i,\mathrm{\infty })},$

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.

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.

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