I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. I'd really want to know the state of the question, since I'm self-studying the material for pleasure and I don't have anyone to talk about it. Please feel free to close this post if you think the topic is not appropriate for this site.
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 mathematicians working in this field nowadays (if it is still an active field of research), 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.
Best Answer
According to a recent poll by the Central Planning Commitee for Universal Education Standards, some geometric topologists don't have a clue about regular neighborhoods, while others haven't heard of multijet transversality; but they all tend to be equally excited when it comes to Hilbert cube manifolds.
Rourke-Sanderson, Zeeman, Stallings, Hudson,
L. C. Glaser, Geometrical combinatorial topology (2 volumes)
Not really (as far as I know), but some more recent books related to PL topology include:
Turaev, Quantum invariants of knots and 3-manifolds (chapters on the shadow world)
Kozlov, Combinatorial algebraic topology (chapters on discrete Morse theory, lexicographic shellability, etc.)
Matveev, Algorithmic topology and classification of 3-manifolds
2D homotopy and combinatorial group theory
Daverman-Venema, Embeddings in manifolds (about a third of the book is on PL embedding theory)
Benedetti-Petronio, Branched standard spines of 3-manifolds
Buchstaber-Panov, Torus actions and their applications in topology and combinatorics
Buoncristiano, Rourke, and Sanderson, A geometric approach to homology theory (includes the PL transversality theorem)
The Hauptvermutung book
Buoncristiano, Fragments of geometric topology from the sixties
I'll mention two problems.
1) Alexander's 80-year old problem of whether any two triangulations of a polyhedron have a common iterated-stellar subdivision. They are known to be related by a sequence of stellar subdivisions and inverse operations (Alexander), and to have a common subdivision (Whitehead). However the notion of an arbitrary subdivision is an affine, and not a purely combinatorial notion. It would be great if one could show at least that for some family of subdivisions definable in purely combinatorial terms (e.g. replacing a simplex by a simplicially collapsible or constructible ball), common subdivisions exist. See also remarks on the Alexander problem by Lickorish and by Mnev, including the story of how this problem was thought to have been solved via algebraic geometry in the 90s.
2) MacPherson's program to develop a purely combinatorial approach to smooth manifold topology, as attempted by Biss and refuted by Mnev.