Math 2.0

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

Hauptvermutung

Builds on Simplicial complexHomeomorphismTriangulationRadó's theorem on triangulation of surfacesMoise's theoremKirby–Siebenmann class

Problem

Let KK and LL be simplicial complexes whose geometric realizations ∣K∣\abs{K} and ∣L∣\abs{L} are homeomorphic. Do KK and LL always have subdivisions K′K' and L′L' that are isomorphic as simplicial complexes?

Equivalently: are any two triangulations of the same space related by a piecewise linear homeomorphism?

Remarks

The name is German for "main conjecture". A positive answer would make the combinatorial invariants of a triangulation, such as simplicial homology, automatically topological invariants; homology was later shown to be a topological invariant by other means. The answer is yes for complexes of dimension at most 22, and for 33-manifolds by Moise's theorem; for surfaces it goes with Radó's theorem. The answer is no in general: John Milnor gave counterexamples in 1961 among finite simplicial complexes that are not manifolds. For manifolds the answer is also no. In dimensions at least 55, the theory behind the Kirby–Siebenmann class gives topological manifolds with several PL structures. The corresponding triangulations are not related by any piecewise linear homeomorphism.

Reading path

To understand this page from the ground up, read these 17 pages in order. The foundations come first.

Show the path as a graph
Circle, concept, no evidence levelCircleconceptClosed ball, concept, no evidence levelClosed ballconceptConnected space, concept, no evidence levelConnected spaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectHomeomorphism, concept, no evidence levelHomeomorphismconceptKirby–Siebenmann class, concept, no evidence levelKirby–Siebenmann classconceptPiecewise linear manifold, concept, no evidence levelPiecewise linear mani…conceptSimplicial complex, concept, no evidence levelSimplicial complexconceptTopological embedding, concept, no evidence levelTopological embeddingconceptTopological manifold, concept, no evidence levelTopological manifoldconceptTriangulation, concept, no evidence levelTriangulationconceptn-sphere, object, no evidence leveln-sphereobjectJordan curve theorem, statement, no evidence levelJordan curve theoremstatementMoise's theorem, statement, no evidence levelMoise's theoremstatementSchoenflies theorem, statement, no evidence levelSchoenflies theoremstatementRadó's theorem on triangulation of surfaces, statement, no evidence levelRadó's theorem on tri…statementHauptvermutung, problem, no evidence levelHauptvermutungproblem
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. Circle concept
  2. Closed ball concept
  3. Connected space concept
  4. Euclidean space object
  5. Homeomorphism concept
  6. Kirby–Siebenmann class concept
  7. Piecewise linear manifold concept
  8. Simplicial complex concept
  9. Topological embedding concept
  10. Topological manifold concept
  11. Triangulation concept
  12. n-sphere object
  13. Jordan curve theorem statement
  14. Moise's theorem statement
  15. Schoenflies theorem statement
  16. Radó's theorem on triangulation of surfaces statement
  17. Hauptvermutung problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hauptvermutung
About this page
Tag811NW2WEJH
URIm2://811NW2WEJH
Kindproblem
WritingW0 stub
MSC57Q25, 57Q15
StatusThis copy of the site has no build data. Run m2 status 811NW2WEJH.
AboutTriangulation
UsesSimplicial complex, Homeomorphism, Triangulation, Radó's theorem on triangulation of surfaces, Moise's theorem, Kirby–Siebenmann class
Same as
Changed2 November 2026

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