Hauptvermutung

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The Hauptvermutung (German for main conjecture) of geometric topology is the conjecture that any two triangulations of a triangulable space have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by Steinitz and Tietze.

This conjecture is now known to be false. The non-manifold version was disproved by John Milnor^[1] in 1961 using Reidemeister torsion.

The manifold version is true in dimensions m ≤ 3. The cases m = 2 and 3 were proved by Tibor Radó and Edwin E. Moise^[2] in the 1920s and 1950s, respectively.

An obstruction to the manifold version was formulated by Andrew Casson and Dennis Sullivan in 1967–9 (originally in the simply-connected case), using the Rochlin invariant and the cohomology group H³(M;Z/2Z).

A homeomorphism ƒ : N → M of m-dimensional piecewise linear manifolds has an invariant κ(ƒ) ∈ H³(M;Z/2Z) such that for m ≥ 5, ƒ is isotopic to a piecewise linear (PL) homeomorphism if and only if κ(ƒ) = 0. In the simply-connected case and with m ≥ 5, ƒ is homotopic to a PL homeomorphism if and only if [κ(ƒ)] = 0 ∈ [M,G/PL]

The obstruction to the manifold Hauptvermutung is now seen as a relative version of the triangulation obstruction of Rob Kirby and Larry Siebenmann, obtained in 1970. The Kirby–Siebenmann obstruction is defined for any compact m-dimensional topological manifold M

$\kappa (M)\in H^{4}(M;{\mathbb {Z}}/2{\mathbb {Z}})$

again using the Rochlin invariant. For m ≥ 5, M has a PL structure (i.e. can be triangulated by a PL manifold) if and only if κ(ƒ) = 0, and if this obstruction is 0 the PL structures are parametrized by H³(M;Z/2Z). In particular there are only a finite number of essentially distinct PL structures on M.

For compact simply-connected manifolds of dimension 4 Simon Donaldson found examples with an infinite number of inequivalent PL structures, and Michael Freedman found the E8 manifold which not only has no PL structure, but is not even homeomorphic to a simplicial complex. In dimensions greater than 4 the question of whether all compact manifolds are homeomorphic to simplicial complexes is an important open question that may have recently been solved in the negative. On March 13, 2013, Ciprian Manolescu posted a preprint on the ArXiv claiming to show that there are 5-manifolds, and hence in every dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex, and thus that do not admit a simplicial triangulation.

References

↑ Milnor, John W. (1961). "Two complexes which are homeomorphic but combinatorially distinct". Annals of Mathematics 74 (2): 575–590. doi:10.2307/1970299. JSTOR 1970299. MR 133127
↑ Moise, Edwin E. (1977). Geometric Topology in Dimensions 2 and 3. New York: New York : Springer-Verlag. ISBN 978-0-387-90220-3.

External links

http://www.maths.ed.ac.uk/~aar/haupt Additional material, including original sources
Rudyak, Yuli B. (2001). "Piecewise linear structures on topological manifolds". arXiv:math.AT/0105047 [math.AT].
Andrew Ranicki (ed.) The Hauptvermutung Book ISBN 0-7923-4174-0
Andrew Ranicki High-dimensional manifolds then and now

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