3-manifold A 3 dimensional manifoldMathworldPlanetmath is a topological spaceMathworldPlanetmath which is locally homeomorphic to the euclidean space 𝐑3.

One can see from simple constructions the great varietyMathworldPlanetmath of objects that indicate that they are worth to study.

First without boundary:

  1. 1.

    For example, with the cartesian product we can get:

    • S2×S1

    • 𝐑P2×S1

    • T×S1

  2. 2.

    Also by the generalizationPlanetmathPlanetmath of the cartesian product: fiber bundlesMathworldPlanetmath, one can build bundles E of the type


    where F is any closed surface.

  3. 3.

    Or interchanging the roles, bundles as:


    For the second type it is known that for each isotopyMathworldPlanetmath class [ϕ] of maps FF correspond to an unique bundle Eϕ. Any homeomorphism f:FF representing the isotopy class [ϕ] is called a monodromy for Eϕ.

    From the previuos paragraph we infer that the mapping class group play a important role inthe understanding at least for this subclass of objets.

    For the third class above one can use an orbifold instead of a simple surface to get a class of 3-manifolds called Seifert fiber spaces which are a large class of spaces needed to understand the modern classifications for 3-manifolds.


    • [G

      ] J.C. Gómez-Larrañaga. 3-manifolds which are unions of three solid tori, Manuscripta Math. 59 (1987), 325-330.

    • [GGH

      ] J.C. Gómez-Larrañaga, F.J. González-Acuña, J. Hoste. MinimalPlanetmathPlanetmath Atlases on 3-manifolds, Math. Proc. Camb. Phil. Soc. 109 (1991), 105-115.

    • [H

      ] J. Hempel. 3-manifolds, Princeton University Press 1976.

    • [O

      ] P. Orlik. Seifert Manifolds, Lecture Notes in Math. 291, 1972 Springer-Verlag.

    • [S

      ] P. Scott. The geometry of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401-487.

Title 3-manifold
Canonical name 3manifold
Date of creation 2013-03-11 19:23:28
Last modified on 2013-03-11 19:23:28
Owner juanman (12619)
Last modified by (0)
Numerical id 1
Author juanman (0)
Entry type Definition