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triangulation theorem

Contents

Context

Manifolds and cobordisms

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

For XX a manifold,

A triangulation conjecture conjectures and triangulation theorem proves that a given kind of triangulation exists for a given kind of manifold. Conversely, deep theorems assert that a given kind of triangulation does not generally exists for a given class of manifolds.

Statements

Existence of triangulations

It is easy to show that every piecewise-linear manifold admits a combinatorial triangulation, so that combinatorial triangulability is often understood to mean existence of a piecewise-linear structure.

A compatible piecewise-linear structure, hence a combinatorial triangulation, hence a simplicial triangulation, does exist for

Non-existence of triangulations

In each dimension \geq 4 there exist topological manifolds not admitting a combinatorial triangulation (Kirby & Siebenmann 1969).

In each dimension \geq 5 there exist topological manifolds not even admitting a simplicial triangulation (Manolescu 2016)

For topological manifolds XX of dimension dim(X)3dim(X) \leq 3 triangulations still exist in general, but for every dimension 4\geq 4 there exist topological manifolds which do not admit a triangulation. For dim(X)5dim(X) \geq 5 there is an obstruction class Δ(X)H 4(X;/2)\Delta(X) \in H^4(X; \mathbb{Z}/2) in the ordinary cohomology with coefficients in / 2 \mathbb{Z}/2 , in that XX admits a triangulation if and only if Δ(X)=0\Delta(X) = 0.

References

Triangulation theorems for manifolds

On triangulation conjectures and triangulation theorems on existence of triangulations of manifolds.

Review:

The question of triangulability of smooth manifolds was first raised in

and for general topological manifolds in

  • Hellmuth Kneser, Die Topologie der Mannigfaltigkeiten, Jahresbericht der Deutschen Mathematiker-Vereinigung (1926), Volume: 34, page 1-13 (eudml:145701)

Proof that every surface admits a combinatorial triangulation:

  • Tibor Radó, Über den Begriff der Riemannschen Fläche, Acta Litt. Sci. Szeged, 2 (101-121), 10 (pdf, pdf)

    Proof that every smooth manifold admits a combinatorial triangulation:

Proof that every smooth manifold admits a combinatorial triangulation is due to

with further accounts in:

A detailed exposition is available in Chapter II (see Thm. 10.6) of

Generalization to existence of equivariant triangulation for smooth G-manifolds (equivariant triangulation theorem):

  • Sören Illman, Equivariant algebraic topology, Princeton University 1972 (pdf)

  • Sören Illman, Smooth equivariant triangulations of GG-manifolds for GG a finite group, Math. Ann. (1978) 233: 199 (doi:10.1007/BF01405351)

  • Sören Illman, The Equivariant Triangulation Theorem for Actions of Compact Lie Groups, Mathematische Annalen (1983) Volume: 262, page 487-502 (dml:163720)

Proof that every 3-manifold admits the structure of a smooth manifold and hence of a combinatorial triangulation:

Proof that in every dimension dim4dim \geq 4 there exist topological manifolds without combinatorial triangulation:

Proof that in every dimension dim5dim \geq 5 there exist topological manifolds without simplicial triangulation:

Last revised on August 1, 2021 at 09:23:58. See the history of this page for a list of all contributions to it.