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. 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 in 1961. The manifold version is true in dimensions $\leqslant$ 3. 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 f:N $\to$ M of m-dimensional piecewise linear manifolds has an invariant $κ$ (f) $\in$ H³(M;Z/2Z) such that for m $\geqslant$ 4 f is isotopic to a piecewise linear (PL) homeomorphism if and only if $κ$ (f)=0. In the simply-connected case and with m $\geqslant$ 4 f is homotopic to a PL homeomorphism if and only if [ $κ$ (f)]=0 $\in$ [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

κ

(M) $\in$ H⁴(M;Z/2Z)

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

For compact simply connected manifolds of dimension 4 Simon Donaldson found examples with an infinite number of inequivalent PL structures, and 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.