In this brief note we define and give instances of the notion of a 3-manifold.

A 3-manifold is a Hausdorff topological space which is locally homeomorphic to the Euclidean spaceMathworldPlanetmath 3.

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

First examples without boundary:

  1. 1.

    For example, with the Cartesian product we can get:

    • S2×S1

    • P2×S1

    • T×S1

    • K×S1

    where S1 and S2 are the 1- and 2-dimensional spheres respectively, T is a torus, K a Klein bottle, and P2 is the 2-dimensional real projective space.

  2. 2.

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


    where F is any closed surface.

  3. 3.

    Or interchanging the roles, bundles as:

  4. 4.

    knots and links complements

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

From the previous paragraph we infer that the mapping class group play a important role in the 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.


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

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

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

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

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

Title 3-manifold
Canonical name 3manifold
Date of creation 2013-03-22 15:40:55
Last modified on 2013-03-22 15:40:55
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 17
Author juanman (12619)
Entry type Definition
Classification msc 57N10
Related topic manifoldMathworldPlanetmath
Related topic DehnsLemma
Related topic SphereTheorem
Related topic LoopTheorem
Related topic SeifertFiberSpace
Related topic Manifold