nLab triangulation theorem

Contents

Context

Manifolds and cobordisms

Definitions

Genera and invariants

Classification

Theorems

Contents

Idea

For $X$ a manifold,

• a simplicial triangulation is a simplicial complex $Tr(X)$ and a homeomorphism $\left\vert Tr(X)\right\vert \xrightarrow{homeo} X$ from its geometric realization to the underlying topological space of $X$.

• a combinatorial triangulation is a simplicial triangulation such that for each simplex $\sigma$ the link of $\sigma$ (the union of all simplices $\tau$ that intersect $\sigma$ such that both $\sigma$ and $\tau$ are faces of a simplex) is homeomorphic to a sphere.

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

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 $G$-manifolds for $G$ 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 $dim \geq 4$ there exist topological manifolds without combinatorial triangulation:

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

Last revised on August 1, 2021 at 13:23:58. See the history of this page for a list of all contributions to it.