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In [mathematics] , a 3-manifold is a 3-dimensional [manifold] . The topological, [piecewise-linear] , and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.

Phenomena in three dimensions can be strikingly different from that for other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. Perhaps surprisingly, this special role has led to the discovery of close connections to a diversity of other fields, such as [knot theory] , [geometric group theory] , [hyperbolic geometry] , [number theory] , [Teichmüller theory] , [topological quantum field theory] , [gauge theory] , [Floer homology] , and [partial differential equations] . 3-manifold theory is considered a part of [low-dimensional topology] or [geometric topology] .

An important scientific application of 3-manifolds is in [physical cosmology] , as models for the [Shape of the Universe] – the surface of the earth is locally approximately flat – it is roughly a 2-manifold, and globally the surface of the earth is a sphere. The universe, likewise, looks locally approximately like 3-dimensional Euclidean space, so the universe may be modeled as a 3-manifold, and one may ask which 3-manifold it is. In [physical cosmology] , [spacetime] is typically assumed to have a decomposition into a 3-dimensional spacial manifold and one dimension of time.

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