nLab triangulation theorem

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

• every smooth manifold

  (Cairns 1935, Whitehead 1940)

• every topological manifold of dimension 2, hence every surface

  (Rado 1924)

• every topological manifold of dimension 3, hence every 3-manifold

  (Moise 1952)

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 $\Delta(X) = 0$.

Related concepts

• equivariant triangulation theorem

References

Triangulation theorems for manifolds

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

Review:

• Ciprian Manolescu, Triangulations of manifolds, Notices of the International Congress of Chinese Mathematicians 2 2 (2014) (pdf, pdf, doi:10.4310/ICCM.2014.v2.n2.a2)

The question of triangulability of smooth manifolds was first raised in

• Henri Poincaré, Complément à l’analysis situs, Rendiconti del Circolo Matematico di Palermo (1884-1940) volume 13, pages 285–343 (1899) (doi:10.1007/BF03024461)

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

• Stewart S. Cairns, Triangulation of the manifold of class one, Bull. Amer. Math. Soc. 41(8): 549-552 (euclid:1183498332)

• J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math. (2) 41 (1940), 809–824 (doi:10.2307/1968861, jstor:1968861)

with further accounts in:

• Hassler Whitney, Section IV.B of: Geometric Integration Theory (1957), Princeton Legacy Library (2016) (doi:10.1515/9781400877577)

• Stewart S. Cairns, A simple triangulation method for smooth manifolds, Bull. Amer. Math. Soc. 67 (1961), 389-390 (doi:10.1090/S0002-9904-1961-10631-9)

  (but see MO:a/177199, where this “simpler argument” is claimed to be wrong)

• Jacob Lurie, Existence of Triangulations (pdf), Lecture 4 in: Topics in Geometric Topology

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

• James R. Munkres, Elementary Differential Topology, Annals of Mathematics Studies 54 (1966), Princeton University Press (doi:10.1515/9781400882656).

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:

• Edwin E. Moise, Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung, Annals of Mathematics Second Series, Vol. 56, No. 1 (Jul., 1952), pp. 96-114 (doi:10.2307/1969769, jstor:1969769)

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

• Robion C. Kirby, Laurent C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75(4): 742-749 (July 1969) (euclid:1183530633)

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

• Ciprian Manolescu, $Pin(2)$-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture, J. Amer. Math. Soc. 29 (2016), 147-176 (arXiv:1303.2354, doi:10.1090/jams829)