The main hypothesis of combinatorial topology (or Hauptvermutung ) is the hypothesis that any two triangulations of the same space admit isomorphic subdivisions.
It was formulated in 1908 by Ernst Steinitz and Heinrich Tietze .
This hypothesis has been refuted in general terms. Moreover, it turned out to be false for some varieties of dimension 4 and higher.
Solution History
A counterexample to the general case was constructed by John Milnor in 1961 using the torsion of Reidemeister . [one]
For manifolds, the hypothesis is valid in dimensions 2 and 3. These cases were proved by Tibor Rado And Edwin Moise In the 1920s and 1950s, respectively. [2]
An obstacle to the hypothesis for manifolds was found by Casson And Dennis Sullivan in 1967-1969 using the Rokhlin invariant .
A homeomorphism ƒ: N → M between m -dimensional piecewise linear manifolds has an invariant κ (ƒ) ∈ H 3 ( M ; Z / 2 Z ) such that for m ≥ 5, ƒ is isotopic to a piecewise linear homeomorphism if and only if κ (ƒ) = 0.
Obstruction to the hypothesis is a relative variant of the Kerby - Siebenmann class and is defined for any compact m -dimensional topological manifold
using the Rokhlin invariant. For m ≥ 5, M has a piecewise linear structure (that is, it can be triangulated by a piecewise linear manifold) if and only if κ (ƒ) = 0, and in this case the piecewise linear structures are determined by the element H 3 ( M ; Z / 2 Z ). In particular, there is only a finite number of different piecewise linear structures on M.
For compact simply connected manifolds of dimension 4, Simon Donaldson found examples with an infinite number of nonequivalent piecewise linear structures, and Mikhail Fridman found an E8-manifold that also does not allow triangulation.
In 2013, Cyprian Manolescu proved the existence of compact manifolds of dimension 5 (and, therefore, of any dimension greater than 5) that do not allow triangulation. [3]
Notes
- ↑ John W. Milnor. Two complexes which are homeomorphic but combinatorially distinct // Annals of Mathematics . - 1961. - Vol. 74. - P. 575-590. - DOI : 10.2307 / 1970299 . . MR : 133127 .
- ↑ Moise, Edwin E. Geometric Topology in Dimensions 2 and 3. - New York: Springer-Verlag, 1977 .-- ISBN 978-0-387-90220-3 .
- ↑ Ciprian Manolescu. Pin (2) -equivariant Seiberg –Witten Floer homology and the Triangulation Conjecture // J. Amer. Math. Soc .. - 2016. - Vol. 29. - P. 147-176. - DOI : 10.1090 / jams829 .