Manifold

Classification of manifolds

It is known that every second-countable connected 1-manifold without boundary is homeomorphic either to R or the circle.

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(The unconnected ones are just disjoint unions of these.)

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For a classification of 2-manifolds, see Surface.

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The 3-dimensional case may be solved. Thurston's Geometrization Conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman may have proven this conjecture; his work is currently being evaluated, as of June 14, 2003.

Related Topics:
Thurston's Geometrization Conjecture - Grigori Perelman

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The classification of n-manifolds for n greater than three is known to be impossible; it is equivalent to the so-called word problem in group theory, which has been shown to be undecidable.

Related Topics:
Word problem - Group theory - Undecidable

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In other words, there is no algorithm for deciding whether a given manifold is simply connected. However, there is a classification of simply connected manifolds of dimension ≥ 5.

Related Topics:
Algorithm - Simply connected

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