# nLab triangulation theorems -- references

Triangulation theorems for manifolds

### 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:

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