What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?

Question

Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) force us to work very hard to prove the main theorems setting up all of the big machinery to get the payoff we want (e.g. invariance of domain, fixed point theorems). But why should I care about these arbitrary and terrible spaces and functions in the first place when, as far as I can tell, any manifold which occurs in applications is at least piecewise-differentiable and any morphism which occurs in applications is at least homotopic to a piecewise-differentiable one?

In other words, do topological manifolds really naturally occur in the rest of mathematics (without some extra structure)?

Answer 1

As far as I know, historically smooth manifolds were the first manifolds studied by people like Poincare, Riemann, up to Whitney. There were a few major events that caused people to take things like topological and PL manifolds seriously, but originally people were not motivated to study these kinds of objects. Here are some of the big events/ideas that come to mind:

1) Poincare's original proof of Poincare duality was a proof for triangulated manifolds. That smooth manifolds had triangulations (and whether or not they were essentially unique) was a problem that took some time to solve. So the study of triangulations and PL manifolds picked up.

2) Smale's proof of the h-cobordism theorem, although written up for the smooth category when you look at it carefully there's a lot of "smoothing the corners" going on. You can think carefully about it and determine all the smoothing of the corners does not kill the proof but I know many strong mathematicians that were hesitant to accept Smale's proof, insisting that it was only a PL-category proof. FYI, the smoothing of the corners issue has been settled, there's a very nice write-up in Kosinski's manifolds text. But this was another issue that kept people thinking about the PL category.

3) If anything, topological manifolds play a role simply for comparison sake -- after all the forgetful functor from the smooth to the topological category is an interesting functor. Perhaps for different people in different ways. I've yet to be interested by a topological manifold that admits no smooth structure but I do find multiple smooth structures on the same topological manifold interesting. Is this purely psychological?

4) Topological and PL-manifold theory is where some "nasty" constructions work, like the Alexander trick. There are different versions of it, one being that the restriction map $Aut(D^n) \to Aut(S^{n-1})$ admits a section in the topological or PL categories. It does not in the smooth category. If anything, I find these kinds of facts informative on the smooth category. The smooth category is interesting largely because of facts like these. There's a similar Alexander trick for knots, for example, the space of topological or PL embeddings $\mathbb R^j \to \mathbb R^n$ which restrict to the standard inclusion $x \longmapsto (x,0)$ outside of the unit ball, this space is contractible, by "pulling the knot tight". But in the smooth category, this space isn't contractible.

I think one of the major events in the development of this subject is simply pragmatic. To get smooth manifold theory off the ground you need Sard's theorem and transversality. This requires analysis to the level of measure theory, and a solid multi-variable calculus background, which in many undergraduate educations is skimped on (especially since it comes early, and many curriculums are too service-based to teach calculus "well"). PL manifolds are inherently more combinatorial and so the learning curve for people with weak analysis backgrounds is easier to deal with. I think also some people really appreciate the combinatorial nature of the subject.

Anyhow, those are some thoughts off the top of my head.

Getting to your conversation with Mariano:

PL manifold theory by-and-large isn't terribly different from smooth manifold theory. So I think once you learn one, adapting to the other isn't so hard. But topological manifolds are really quite different. This may be my ignorance speaking to some extent, but I'm still working my way through Kirby-Siebenmann. I'm told various people are working on re-writing the main theorems of that text, to make it easier reading for people that are not named Larry Siebenmann. But we're probably still several years from that. I suspect in the next 10 years there should be several different accounts of most of that material. But I'm still some ways from understanding smoothing theory.

Manifolds when they come up "in nature" like in physics or engineering applications tend to always be smooth, and usually with plenty of extra structure. Sometimes the objects that come up aren't manifolds, but algebraic varieties, or even more degenerate (but smooth) stratified spaces.

Answer 2

Topological manifolds arise naturally as the background of other phenomena.

For example, nature throws at you examples of spaces which admit several smooth structures and when you try to describe that phenomenon you need to say something like «there are many different $Y$s one can put on an $X$». In the situation of exotic smooth structures, a natural class one can use for the role of $X$s is that of topological manifolds.

Answer 3

The topological category is inherently beautiful. Some very pretty and subtle phenomena happen. My favorite example of this is that some knots bound locally flat (that is they admit a locally trivial normal bundle) topological disks into the 4-ball, but they don't bound smooth disks. That is, there are topologically slice knots which are not smoothly slice. The topological disks themselves are practically impossible to visualize, coming from a limiting construction due to very deep work of Mike Freedman, and are a little bit fractal in nature. The world would be a poorer place if we never knew about this amazing structure.

Answer 4

A standard example of a non-smoothable 4-manifold is the E8-manifold $[E_8]$, that is, the manifold which has $E_8$ as its intersection form $H_2([E_8],\mathbb{Z}) \times H_2([E_8],\mathbb{Z}) \rightarrow \mathbb{Z}$. The manifold exists topologically (as the plumbing of disc-bundles, this thing has a homology 3-sphere as its boundary, but there is another construction (Freedman's fake 4-balls, see for example Alexandru Scorpan, The Wild World of 4-Manifolds) which always allows you to close this thing up), but isn't smoothable due to the theorem of Rohlin (see link for more and a few examples).

I think the main reason to study topological manifold is that they have a lot to say about the other structures (PL, Differential, Complex-Differential, Sympletic, etc).

Of course when you are strictly working with complex analytic things, I can understand (from a focus point of view) why you aren't bothered that much by underlying structures.

Secondly, the topological realm is a great source for counter-examples for things that seem obvious, but aren't true when you look more closely.

Answer 5

There is an aspect in which the topological category (for homotopy theorists) is quite simple and elegant. That is, the Quillen equivalence between $Top$ and $sSet$ shows that all of homotopy theory is combinatorially encoded in simplicial sets. Simplicial sets in turn, are completely determined just by the very simple class of polytopes: the $n$-simplices. So, homotopy theory boils down to understanding triangles. (It turns out triangles are quite complicated.) I find it quite miraculous that at the most general situation (from manifolds all the way to just topological spaces) all you basically have are the $n$-simplices. So, as motivation for the more practical cases of manifolds with extra structure, understanding the most basic polytopes is tantamount to understanding the homotopy category of topological spaces.

Another reason for general homotopy theory being relevant is that there are plenty of cases where there is no smooth structure around at all. For instance, the homotopy theory of finite topological spaces is very rich (e.g., non-trivial fundamental groups) and useful (e.g., computer graphics).