Good evening,

When does a n-manifold have a triangulation?

Thank you in advance


A manifold with a triangulation is called a piecewise-linear manifold. To quote wikipedia's article on PL manifolds:

"The obstruction to placing a PL structure on a topological manifold is the Kirby-Siebenmann class. To be precise, the Kirby-Sibenmann class is the obstruction to placing a PL-structure on $M \times R$ and in dimensions $n$ > 4 this ensures that M has a PL-structure."

This means that a manifold has a triangulation if and only if a particular element of the fourth cohomology group with $\mathbb{Z}_2$ coefficients is zero. It looks like a full description is given in this paper on the arxiv.


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