Hauptvermutung

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The Hauptvermutung (German for main conjecture) of geometric topology is the conjecture that every triangulable space has an essentially unique triangulation.

This is now known to be false. The non-manifold version was disproved by John Milnor in 1961. The manifold version was disproved by Robion C. Kirby and Larry Siebenmann in 1969, using the Kirby-Siebenmann class in the cohomology group H³(M,Z/2Z).

For manifolds of dimension at most 3 the Hauptvermutung is true. For manifolds of dimension 4 Simon Donaldson found compact simply connected examples with an infinite number of inequivalent PL structures. For compact manifolds of dimension at least 5 with at least 1 PL structure, the PL structures can be parametrized (up to isomorphism) by the cohomology group H⁴(M,Z/2Z), and in particular there are only a finite number of essentially distinct such structures.

[edit] External links

• http://www.maths.ed.ac.uk/~aar/haupt Additional material, including original sources
• Yuli B. Rudyak (2001). "Piecewise linear structures on topological manifolds." .
• Andrew Ranicki (ed.) The Hauptvermutung Book ISBN 0-7923-4174-0