# Moise's theorem

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{{Short description|Any topological 3-manifold has unique PL and smooth structures}}
In [geometric topology](/source/geometric_topology), a branch of mathematics, '''Moise's theorem''', proved by [Edwin E. Moise](/source/Edwin_E._Moise) in {{harvtxt|Moise|1952}}, states that any topological [3-manifold](/source/3-manifold) has an essentially unique [piecewise-linear structure](/source/piecewise-linear_structure) and [smooth structure](/source/smooth_structure).

The analogue of Moise's theorem in dimension 4 (and above) is false: there are topological [4-manifold](/source/4-manifold)s with no piecewise linear structures, and others with an infinite number of inequivalent ones.

==See also==
* [Exotic sphere](/source/Exotic_sphere)

==References==
*{{Citation | last1=Moise | first1=Edwin E. | title=Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung | jstor=1969769 | mr=0048805  | year=1952 | journal=[Annals of Mathematics](/source/Annals_of_Mathematics) |series=Second Series | issn=0003-486X | volume=56 | pages=96–114 | doi=10.2307/1969769}}
*{{Citation | last1=Moise | first1=Edwin E. | title=Geometric topology in dimensions 2 and 3 | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | isbn=978-0-387-90220-3 | mr=0488059  | year=1977}}

Category:Geometric topology

{{topology-stub}}

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