# Hauptvermutung

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 Ernst Steinitz and Heinrich Franz Friedrich Tietze.

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

## History

The manifold version is true in dimensions ${\displaystyle m\leq 3}$. The cases ${\displaystyle m=2}$ and 3 were proved by Tibor Radó and Edwin E. Moise in the 1920s and 1950s, respectively.[2][3][4]

An obstruction to the manifold version was formulated by Andrew Casson and Dennis Sullivan in 1967–69 (originally in the simply-connected case), using the Rochlin invariant and the cohomology group ${\displaystyle H^{3}(M;\mathbb {Z} /2\mathbb {Z} )}$.

A homeomorphism ${\displaystyle f\colon N\to M}$ of m-dimensional piecewise linear manifolds has an invariant ${\displaystyle \kappa (f)\in H^{3}(M;\mathbb {Z} /2\mathbb {Z} )}$ such that for ${\displaystyle m\geq 5}$, ${\displaystyle f}$ is isotopic to a piecewise linear (PL) homeomorphism if and only if ${\displaystyle \kappa (f)=0}$. In the simply-connected case and with ${\displaystyle m\geq 5}$, ${\displaystyle f}$ is homotopic to a PL homeomorphism if and only if ${\displaystyle [\kappa (f)]=0\in [M,G/{\rm {PL}}]}$.

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

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

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(M) = 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 (by work of Casson) is not even homeomorphic to a simplicial complex.[5]

In 2013, Ciprian Manolescu proved that there exist compact topological manifolds of dimension 5 (and hence of any dimension greater than 5) that are not even homeomorphic to a simplicial complex.[6] Thus Casson's example illustrates a more general phenomenon that is not merely limited to dimension 4.

## References

1. ^ 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 0133127.
2. ^ Radó, Tibor (1925). "Über den Begriff der Riemannschen Fläche". Acta Scientarum Mathematicarum Universitatis Szegediensis. 2 (1): 96–114. doi:10.2307/1969769. MR 0048805.
3. ^ Moise, Edwin E. (1952). "Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung". Annals of Mathematics. 56 (2): 101–121. doi:10.2307/1969769.
4. ^ Moise, Edwin E. (1977). Geometric Topology in Dimensions 2 and 3. New York: New York : Springer-Verlag. ISBN 978-0-387-90220-3.
5. ^ Akbulut, Selman; McCarthy, John D. (1990). Casson's invariant for oriented homology 3-spheres. Princeton, NJ: Princeton University Press. ISBN 0-691-08563-3. MR 1030042.
6. ^ Manolescu, Ciprian (2016) [2015]. "Pin(2)-equivariant Seiberg–Witten Floer homology and the Triangulation Conjecture". Journal of the American Mathematical Society. 29: 147–176. arXiv:1303.2354. doi:10.1090/jams829.