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THREE-DIMENSIONAL MANIFOLD

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