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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)?

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Why wouldn't you study them?! – Mariano Suárez-Alvarez Feb 4 '11 at 22:40
@Mariano: because I don't want to digest a lot of machinery if nobody actually uses it in anything like full generality! – Qiaochu Yuan Feb 4 '11 at 22:44
So you are asking why should you study them? – Mariano Suárez-Alvarez Feb 4 '11 at 22:56
@Mariano: all I am trying to do is eliminate a possibility which I find absurd: that the machinery is developed for arbitrary topological spaces / arbitrary manifolds but that it is only used on spaces that could be handled by simpler machinery (e.g. de Rham cohomology instead of singular cohomology). Note that I am assuming I am wrong about this. – Qiaochu Yuan Feb 4 '11 at 23:00
The following reason might be unsatisfactory to you but I think everything is worth studying whether or not it has applications. For example, abstract topological spaces are certainly studied in point-set/general/set-theoretic topology by professional mathematicians. Why not better understand abstract topological spaces? They may not occur in "nature" so to speak but their theory is certainly not trivial and any non-trivial theory is worth better understanding. Moreover, the tools of algebraic topology can be used to better understand abstract topological spaces as well as manifolds. – Amitesh Datta Jun 25 '11 at 3:00
up vote 30 down vote accepted

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.

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I am totally willing to think about the PL category, but I am still uncertain about the topological category. I get that it's a natural place to prove the easy stuff, but why do we also want to prove the hard stuff here? (Full disclosure: I have recently started reading Guillemin and Pollack and am being blown away at how nice and intuitive everything is once you take Sard's theorem and transversality for granted (both of which are themselves nice and intuitive unlike, say, the definition of singular cohomology).) – Qiaochu Yuan Feb 4 '11 at 23:26
@Qiaochu, but the PL category has its wierdnesses too... The fact that the Hauptvermutung is false, say. – Mariano Suárez-Alvarez Feb 4 '11 at 23:27
@Mariano: my naive opinion is that the failure of the Hauptvermutung has more to do with the fact that homeomorphisms are weird than that PL-manifolds are weird. I am not necessarily even convinced that the category of topological spaces is "right." – Qiaochu Yuan Feb 4 '11 at 23:30
That's great. IMO Guillemin and Pollack is one of the most enabling books for young topologists, as it gives you so many ready-made and intuitive constructions for things like homology, cohomology, Poincare duality, covering spaces, etc. – Ryan Budney Feb 4 '11 at 23:31

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.

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@Mariano: yes, certainly it's a good idea to talk about simple things in the topological category. Why should we also talk about complicated things in the topological category? For example, what's the point of proving the Jordan curve theorem for continuous curves? Does anyone actually care about arbitrary continuous curves? – Qiaochu Yuan Feb 4 '11 at 23:33
Layers of complexity are nice but complexity unfortunately isn't linearly ordered. For example, can you give me a 1-line construction of a topological manifold that can't be given a PL structure? Naturally occuring manifolds almost always are smooth. So in some rather formally non-intuitive way, they're the bottom-layer of complexity. Look at almost any intro textbook that defines topological and smooth manifolds and there isn't a single example of a genuine topological manifold that isn't naturally smooth. – Ryan Budney Feb 4 '11 at 23:33
@Qiaochu: Well, one point is that it is a difficult result :) – Mariano Suárez-Alvarez Feb 4 '11 at 23:46
Ultimately, I don't think one can argue about these things rationally. «Why should be also talk about complicated aspects of subject X?» is an essentially unanswerable question. Jacobi's pour l'honneur de l'esprit humain comes to mind. – Mariano Suárez-Alvarez Feb 4 '11 at 23:54
@Ryan: that is definitely a motivating factor in my question: I literally don't know any examples of topological manifolds that don't come with natural smooth structures, up to singularities and corners and boundaries and other things I am perfectly willing to accept. – Qiaochu Yuan Feb 5 '11 at 0:10

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.

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I like these results, too, for comparison sake. Similarly, if $M$ is Poincare dodecahedral space, there is a topological embedding of $M \times \mathbb R$ into $\mathbb R^4$, but no smooth embedding. But turning it on its head, this gives you reason to like the PL and smooth categories as they're more combinatorial. – Ryan Budney Feb 4 '11 at 23:25
That's just the problem: the topological category is too subtle. When I think about manifolds I would like to keep my geometric intuition as intact as possible, and I would like violations of that intuition to be justified by applications of the intuition-violating thing. – Qiaochu Yuan Feb 4 '11 at 23:28
@Qiaochu Yuan: why would you want to restrict to math that doesn't violate intuition? In my mind, when math violates intuition, that's when the most beautiful results occur. Why study something for which you already know all the answers? – Grumpy Parsnip Feb 4 '11 at 23:46
@Jim: I guess that requires elaboration. I have no qualms about the category of groups because groups perfectly capture any reasonable intuitive definition of symmetry (via Cayley's theorem). I have qualms about the category of manifolds because manifolds do not perfectly capture any reasonable intuitive definition of "space," even given the Whitney embedding theorem: the problem is that I have qualms about the category of topological spaces. In dimension 2, for compact manifolds, all reasonable definitions agree: that's great. In higher dimensions, for general manifolds, I have no such... – Qiaochu Yuan Feb 5 '11 at 0:02
I am with Jim on this one even though I don't study topological 4-manifolds explicitly and Jim does. The variety of different structures on 4-space makes it amazing. Furthermore, the original Kervaire and Milnor constructions of different smooth structures for higher dimensional manifolds are an inherent and big part of modern math not limited to but including Bott periodicity. Smoothness and PL-ness is nice, but to really understand what is under the hood, these other categories are amazing. – Scott Carter Feb 5 '11 at 0:52

I learned a lot from the answers and discussion above. Here is my personal view, not necessarily applicable to others.

I published my Lectures on Algebraic Topology (W. A. Benjamin) in 1967. The book is an extensive elaboration of notes from a course I taught. All the manifold theorems in my book refer to topological manifolds - I only make a few side remarks about differentiable manifolds, De Rham cohomology and PL-manifolds. My book was later expanded and slightly revised by John Harper, renamed as Algebraic Topology: A First Course (Perseus, 1981).

I was delighted, in the process of teaching that course, to learn that so much could be proved in this subject without bringing in analysis and with very little messy (imho) combinatorial argumentation. Purity of method is an essential part of mathematical aesthetics, as Hilbert emphasized. And to achieve that purity, one usually has to discover new techniques that apply in greater generality than the previously known methods.

I was glad to see that J. P. May, in his lovely 1999 treatise A Concise Course in Algebraic Topology (U. of Chicago Press), also focused on topological manifolds.

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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.

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That second reason is precisely why I think we shouldn't work in the topological category: when things seem obvious, it's because our intuition comes from nicer manifolds (say PL), and when that intuition breaks I just see it as evidence that the topological category is weird, not that it's interesting. – Qiaochu Yuan Jul 24 '11 at 15:17
@QiaochuYuan, that argument also applies to most of Banach space theory! :) – Mariano Suárez-Alvarez Jan 29 '12 at 8:58
@Qiaochu: That merely says that your notion of what is interesting differs from that of some other people $-$ like me, for instance. – Brian M. Scott Jan 29 '12 at 18:31

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).

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(1) The machinery mentioned is useful for much more than studying topological manifolds so one should learn homology and those techniques anyway. Persistant homology in the study of high dimensional data, configuration spaces in the study of robotic motion and so forth indicate that the machinery of algebraic topology is here to stay and rapidly spreading to other fields. Homology has permeated algebra (group cohomology, Lie algebra cohomology, Hochschild cohomology..), number theory (etale cohomology, p-adic cohomology,...) and many other fields. Algebraic Topology is useful for far more things than just worrying about topological manifolds. Honestly many facts about smooth manifolds have only been proven using algebraic topological techniques. Kervaire invariant problems or maybe just that every oriented 3-manifold is parallelizable.... so basically my viewpoint is get over the machinery EVEN IF you just want to stay in the smooth world.

(2) Many things in nature are fractals - continuity is fair for physical reasons (nature tends to dislike and rectify discontinuities/shockwaves - smoothness is mostly an abstraction or an approximation. In statics it can be reasonable to assume everything is smooth but not so much in dynamics. Objects like Koch snowflakes are the norm (nowhere differentiable embedding of the circle in this case). The boundary of the Mandelbrot set is not smooth - in dynamics both discrete and continuous, even if your functions or differential equations are smooth, useful invariant manifolds or boundaries of such need not be so.

(3) Even when working with smooth manifolds, it is important to know when things don't depend on the differential structure. THERE ARE MANY SMOOTH MANIFOLDS WHICH ARE HOMEOMORPHIC BUT NOT DIFFEOMORPHIC. One needs to decouple the smooth structure from its underlying topological manifold to understand what things differ for different smooth structures on the same topological manifold and what don't. For example in the uncountably many distinct smooth structures on R^4 or the finitely many distinct smooth structures on spheres in higher dimensions, a given differential equation can have different solution spaces. (Seiberg Witten invariants to distinguish them come from looking at solution spaces of certain DEQs) Thus your analysis depends on the differential structure for SOME differential equations. However for others it doesn't. For example the Hodge theorem shows that the dimension of the space of harmonic forms on a smooth closed Riemannian manifold in each degree is a topological invariant - in fact a homotopy invariant - it doesn't care about the smooth structure or the metric. In general one has to ponder what changing the differential structure might do to ones analysis - it is hard to do that systematically if you don't consider the basic underlying topological manifold. Sometimes it is better to even dump that and just consider the underlying homology-manifold or the underlying Poincare Duality space. (for example in Surgery theory).

(4) In many problems it is not sufficient to understand the problem in the case when everything is smooth but the continuous case can be reasonable (Lusin approximation in measure theory). In geometric analysis there is a wide difference between being able to prove something for C^1 or Lipschitz maps versus Holder continuous versus for continuous ones. It would be nice if everything in the world was smooth but it isn't. If I consider the function which keeps track of the mass of a set on one side of a hyperplane as a function of the normal of the hyperplane (a very basic ham-sandwich procedure necessary in many analytic combinatoric and measure theory constructions), it is easy to make reasonable conditions on the measure that guarantee continuity but less reasonable to guarantee smoothness. Thus many things have to be proved for continuous maps even if working in ambient smooth manifolds. You often end up having to use all the continuous machinery for this and so you might as well work on topological manifolds at that point.

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