Triangulation (topology)

Triangulation of a subset of the plane.

In the topology , a branch of mathematics , a is triangulation or triangulation separation of a space in simplices ( triangles , tetrahedral or its higher dimensional generalizations).

definition

A triangulation of a topological space is given by an (abstract) simplicial complex and a homeomorphism ${\ displaystyle X}$ ${\ displaystyle K}$

{\ displaystyle h \ colon \ vert K \ vert \ rightarrow X}

the geometrical implementation on . ${\ displaystyle \ vert K \ vert}$ ${\ displaystyle X}$

Triangulability of manifolds

The triangulability conjecture says that every manifold is triangulable. It was set up in 1926 by Hellmuth Kneser . However, as will be shown below, there are counterexamples to the assumption of triangulability.

Manifolds up to the third dimension are always triangulable. This was proved by Tibor Radó in 1925 for surfaces and in 1952 by Edwin Moise for 3-manifolds. Even in higher dimensions, differentiable manifolds are always triangulable according to Whitehead's theorem . Hassler Whitney gave a simpler proof with the help of his embedding theorem .

All differentiable and all PL manifolds are triangulatable. Robion Kirby and Laurence Siebenmann showed that not all topological manifolds have a PL structure . But they also showed that there are triangulable manifolds without a PL structure.

Andrew Casson showed with the help of the Casson invariant named after him that 4-manifolds with straight cut form and signature 8 cannot be triangulated. We know from Freedman's work that there is such a 4-manifold. She is called. Michael Davis and Tadeusz Januszkiewicz proved that by hyperbolizing of one gets a non-triangulatable aspheric 4-manifold. ${\ displaystyle E_ {8}}$ ${\ displaystyle E_ {8}}$

In the late 1970s David Galewski and Ronald John Stern constructed a manifold that can be triangulated if and only if every manifold of the dimension can be triangulated. In 2013, Ciprian Manolescu proved that the Galewski-Stern manifold cannot be triangulated. The reason for this is that the Rochlin homomorphism does not split. By means of hyperbolization, Michael Davis, Jim Fowler, and Jean-François Lafont showed that there are aspherical manifolds that can not be triangulated in dimension . ${\ displaystyle \ geq 5}$ ${\ displaystyle \ geq 6}$

Main guess

The question of the uniqueness of triangulations became known as the so-called "main conjecture" ( Heinrich Tietze ): If the geometrical realizations and two simplicial complexes are homeomorphic, are there then combinatorially isomorphic subdivisions of the simplicial complexes and ? The main guess that there is such a subdivision is generally wrong. John Milnor first found evidence of this in 1961. Milnor's examples, however, were not manifolds. It was only from the work of R. Kirby and LC Siebenmann that manifolds emerged as counterexamples. ${\ displaystyle \ vert K \ vert}$ ${\ displaystyle \ vert L \ vert}$ ${\ displaystyle K ^ {\ prime}, L ^ {\ prime}}$ ${\ displaystyle K}$ ${\ displaystyle L}$

The original motivation for the main conjecture was the proof of the topological invariance of combinatorially defined invariants such as simplicial homology . Despite the failure of the main conjecture, questions of this kind can often be answered with the simplicial approximation theorem .

Number of triangulations

The number of triangulations of a manifold can grow exponentially with the number of vertices. For the 3-sphere, this was proven by Nevo and Wilson.

Any two different triangulations of the same manifold can be converted into one another by a sequence of Pachner trains .

literature

Tibor Radó: About the concept of the Riemann surface. Acta Sci. Math. (Szeged), 2 (1925), 101-121.
John Henry Constantine Whitehead : On C ¹ -complexes. Ann. of Math. (2) 41, (1940). 809-824.
Edwin Moise: Affine structures in 3 -manifolds. V. The triangulation theorem and main conjecture. Ann. of Math. (2) 56, (1952). 96-114.
Hassler Whitney: Geometric integration theory. Princeton University Press, Princeton, NJ, 1957. (With a proof of Whitehead's Theorem.)
David Galewski, Ronald Stern: Classification of simplicial triangulations of topological manifolds. Ann. of Math. (2) 111 (1980) no. 1, 1-34.
Michael Freedman: The topology of four-dimensional manifolds. J. Differential Geom. 17 (1982) no. 3, 357-453.
Michael Davis, Tadeusz Januszkiewicz: Hyperbolization of polyhedra. J. Differential Geom. 34 (1991) no. 2, 347-388.
Ciprian Manolescu: Pin (2) -equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture pdf
Michael Davis, Jim Fowler, Jean-François Lafont: Aspherical manifolds that cannot be triangulated . Alg. Geom. Top. 14 (2014), 795-803. pdf
John Milnor: Two complexes which are homeomorphic but combinatorially distinct. Ann. of Math. (2) 74, (1961). 575-590
Robion Kirby , Laurence Siebenmann : On the triangulation of manifolds and the main presumption. Bull. Amer. Math. Soc. 75 (1969). 742-749
Eran Nevo, Stedman Wilson: How many n-vertex triangulations does the 3-sphere have? pdf

Web links

Ciprian Manolescu: Lectures on the triangulation conjecture
Ciprian Manolescu: Triangulations of Manifolds
Andrew Ranicki: On the main guess

Individual evidence

↑ Kneser, The Topology of Manifolds, Annual Report DMV, Volume 34, 1926, pp. 1-14

[1] Kneser, The Topology of Manifolds, Annual Report DMV, Volume 34, 1926, pp. 1-14