We will only deal here with the topological aspect of this notion.

A (topological) 3-dimensional manifold (or space), or 3-manifold, is a topological space locally homeomorphic to the 3-dimensional Euclidian space or to the half-space (i.e. for which every point has a neighborhood homeomorphic to or ); it is a 3-dimensional topological manifold.

The points for which a neighborhood is homeomorphic to the half-space constitute the "boundary" of the manifold; a compact manifold without boundary is said to be closed, a non compact manifold without boundary is said to be open.
 

Examples of simple Euclidian compact 3-manifolds (there are 10 kinds in total):
    - the hypertorus
    - the Klein space (that generalizes the Klein bottle)

See the description of the 18 Euclidian 3-manifolds, the classification of the spherical 3-manifolds, and examples of hyperbolic manifolds in "L'univers chiffonne" by Jean-Pierre Luminet page 409 to 417.

See also this article in Pour La Science and this popularization article.
Download this software that allows to visualize the various 3-dimensional spaces:
http://www.geometrygames.org/CurvedSpaces/index.html

See the 8 Thurston geometries that can be associated with a 3-variety in this popularization article and this Wikipedia article.
See beautiful visualizations of these geometries in www.3-dimensional.space.
 

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© Robert FERRÉOL  2017