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.

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    $\begingroup$ 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. $\endgroup$ – Jim Belk May 2 '12 at 15:25

I think I've seen a similar question to this one on MO, but I'm not sure - I can't find it this quickly. It may also well be that my following answer will be mistaken, since I'm also just beginning to learn about PL topology, but what I've understood is the following:

PL manifolds lie somewhere between topological manifolds and smooth manifolds, so the hope was to say something about either of them by studying them (if something holds for PL manifolds, it may descend on smooth ones, and if something doesn't hold for topological ones, maybe it does for PL ones). This has been extensively studied during the sixties and seventies, with several main problems arising, mainly concerning triangulations and the "Hauptvermutung". However, the main problems have been solved (see http://www.maths.ed.ac.uk/~aar/books/haupt.pdf and the articles by Kirby-Siebenmann), which doesn't make research in this direction relatively uninteresting.

That doesn't, however, mean that one shouldn't study PL manifolds. I'm relatively new to topology myself, but I have the feeling that by understanding PL manifolds, I will be able to both understand more about the historical aspects of both topological and smooth manifolds, and just be able to understand them better. Once you know a bit about manifolds that fall in one category but not in the other, I think it is fair to say one knows a bit more about manifolds in general. It may take some time, though.

Concerning references for learning PL manifolds that are not from the '60s and '70s, I just posted this question (Standards in P.L. Topology) and hope some useful result will be found.

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