6
$\begingroup$

I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological and smooth manifolds are widely studied and there are tons of books about them, PL topology seems to be much less popular nowadays. Moreover, I saw in some place the assertion that PL topology is nowadays not nearly as useful as it used to be to study topological and smooth manifolds, due to new techniques developed in those categories, but I haven't seen it carefully explained.

My first question is: is this feeling about PL topology correct? If it is so, why is this? (If it is because of new techniques, I would like to know what these techniques are.)

My second question is: if I'm primarily interested in topological and smooth manifolds, is it worth to learn PL topology?

Also I would like to know some important open problems in the area, in what problems are working mathematicians in this field nowadays, and some recommended references (textbooks) for a begginer. I've seen that the most cited books on the area are from the '60's or 70's. Is there any more modern textbook on the subject?

Thanks in advance.

  • 1
    I think the main problem with PL topology is that it isn't very useful outside of geometric topology. Lots of different kinds of mathematicians need to know results about smooth manifolds and smooth topology (e.g. mathematical physicists, algebraic geometers, experts in differential equations, and so forth), so these subjects are much more popular. PL topology is only really useful as a technical tool for geometric topologists to prove theorems about geometric topology.2012-05-02

1 Answers 1