I see that we've recently just created the tag . Considering that the subject has its own arXiv subject code, I don't object to the tag's existence. But for me, geometric topology sort of lies in the fuzzy area between differential topology, differential geometry, and low dimensional topology.

Can someone explain what should be, and in particular, how one should differentiate it from the and tags?

(This question is asked with writing the TagWiki for in mind; especially to disambiguate it from the other tags covering similar topic areas.)

I think the short preface by Daverman and Sher to the Handbook of Geometric Topology and the Table of Contents gives a good idea what the subject is about. However, I find it hard to give a good criterion for distinguishing them other than on a "you know it when you see it" basis. –  t.b. Aug 25 '11 at 4:05
The first two sentences from the preface seem very usable for a blurb: Geometric Topology focuses on matters arising in special spaces such as manifolds, simplicial complexes, and absolute neighborhood retracts. Fundamental issues driving the subject involve the search for topological characterizations of the more important objects and for topological classification within key classes. –  t.b. Aug 25 '11 at 4:05
The Poincaré conjecture is then mentioned as the prototypical problem. Further excerpts (some key words): [...] matters closely studied in times past [...] PL topology, infinite-dimensional topology, and group actions on manifolds, [...] geometric group theory and 3-manifolds (knot theory included) and their invariants [...] cohomological dimension theory, fixed point theory, homology manifolds, invariants of high-dimensional manifolds, mapping class groups, structures on manifolds and topological dynamics. –  t.b. Aug 25 '11 at 4:11
For what it's worth (likely not much), Wikipedia considers low-dimensional topology to be a subset of geometric topology. –  Jesse Madnick Aug 30 '11 at 17:42
I created the tag. I certainly hope I haven't been indelicate in doing so. Although I consider myself a geometric topologist, I must confess I am not able to define the field precisely. (I find Ryan's answer and Theo's comments excellent). –  PseudoNeo Aug 31 '11 at 8:25
I personally think this is an excellent question and one I've had a lot of trouble getting a straight answer on from topologists.I haven't asked James Stasheff yet,though-and he's definitely someone I'd like to get the input of. But I'm interested in answers to the question as well,so upvoting the question. –  Mathemagician1234 Aug 31 '11 at 16:31

1 Answer 1

up vote 6 down vote accepted

The terminology "geometric topology" as far as I'm aware is a fairly recent historical phenomenon.

The words used by topologists to describe their areas has had a fair bit of flux over the years. Before the mid-40's, algebraic topology was called combinatorial topology. I think the urge to use the phrase geometric topology began sometime after the advent of the h-cobordism theorem, and the observation that high-dimensional manifold theory, via a rather elaborate formulation can be largely turned into elaborate algebraic problems.

So there was a desire to have a term that held-together all the aspects of topology where these techniques either don't apply, or were not used (or at least, not predominantly used). That's geometric topology. So 2, 3 and 4-dimensional manifold theory would be a big chunk of this area. But of course, even if high-dimensional manifold theory in principle reduces to algebra, that doesn't necessarily mean you want to use that reduction -- it may be too complicated to be useful. So there are higher-order type high-dimensional manifold theory problems that don't fit the traditional reductions. Like say Vassiliev's work on spaces of knots. So this would also be considered geometric topology.

Defining a subject by what it's not is kind of strange and artificial but all these subject-area definitions are kind of strange and artificial. I think the above-quoted blurbs also get at a key aspect of the area. Algebraic topology tends to be more focused on a broad set of tools. Geometric topology is focused more by the goals, things like the Poincare conjecture(s) and such. So the latter tends to have a more example-oriented culture.

Ryan: would you mind if I copy large chunks of your answer directly into the Tag Wiki for geometric-topology? –  Willie Wong Aug 31 '11 at 13:50
Sounds fine. You might want to paraphrase a bit. –  Ryan Budney Aug 31 '11 at 14:39
This has been referenced on the "Topology" page on Wikipedia. –  Brian Rushton Dec 30 '13 at 0:59

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