Math 2.0

Manifolds and cell complexes Simplicial complexes and piecewise-linear topology Theorem

Moise's theorem

Builds on Topological manifoldTriangulationPiecewise linear manifold

Statement

Let MM be a topological manifold of dimension 33.

  1. MM has a triangulation.
  2. Any two triangulations h1 ⁣:∣K1∣→Mh_1 \colon \abs{K_1} \to M and h2 ⁣:∣K2∣→Mh_2 \colon \abs{K_2} \to M are related by a piecewise linear homeomorphism. That is, K1K_1 and K2K_2 have subdivisions that are isomorphic as simplicial complexes.

Remarks

Equivalently, every topological 33-manifold has a piecewise linear structure, unique up to piecewise linear homeomorphism. Combined with results comparing piecewise linear and smooth structures in low dimensions, it follows that every topological 33-manifold has a smooth structure, unique up to diffeomorphism. Edwin Moise proved the theorem in 1952; R. H. Bing later gave another proof. A key step approximates a homeomorphism of 33-manifolds by piecewise linear ones, which needs tameness results for surfaces in 33-manifolds. The statement fails in dimension 44, where the E8E_8 manifold has no triangulation, and part 2 fails for manifolds in higher dimensions.

Going deeper

For a student

The theorem lets 33-manifold topology use triangulations, normal surfaces and smooth tools freely, without losing generality. It is the dimension-three case of the Hauptvermutung and of the triangulation conjecture, both of which hold in dimensions up to three and fail in general.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Piecewise linear manifold, concept, no evidence levelPiecewise linear mani…conceptTopological manifold, concept, no evidence levelTopological manifoldconceptTriangulation, concept, no evidence levelTriangulationconceptMoise's theorem, statement, no evidence levelMoise's theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Piecewise linear manifold concept
  2. Topological manifold concept
  3. Triangulation concept
  4. Moise's theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Moise%27s_theorem
About this page
TagSV0816DJP9
URIm2://SV0816DJP9
Kindstatement, theorem
WritingW0 stub
MSC57Q15, 57K30
StatusThis copy of the site has no build data. Run m2 status SV0816DJP9.
AboutTriangulation
UsesTopological manifold, Triangulation, Piecewise linear manifold
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Changed2 November 2026

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