3
$\begingroup$

Good evening,

When does a n-manifold have a triangulation?

Thank you in advance

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.