13

In multiple sources (say here and here) I've see it asserted that a classification of topological spaces up to homeomorphism is either impossible, undesirable because homeomorphism is too strong, or both. (So, the next sentence will go, we should try classifying spaces up to homotopy equivalence instead).

Unfortunately, I've never seen someone elaborate on why classifying spaces by homeomorphism is either impossible or inconveniently strong. Regarding the former statement, what's the easiest way of demonstrating that classification up to homeomorphism is ridiculously hard or impossible, or a particularly good example of why that's the case?

But more importantly, the latter statement - that classification up to homeomorphism wouldn't even be a desirable thing anyway, because it's too fine an equivalence - is not obvious to me at all. In many realistic contexts, homeomorphism isn't even strong enough for what we want to achieve, and we instead need to talk about things like diffeomorphisms between smooth manifolds. Homotopy equivalence is often inconveniently weak, as it doesn't respect topological properties that are pretty important, like compactness in particular. So I don't get why classification up to homotopy equivalence would be a priori better, only why it would be simpler - but as I've said, those two sources seem to me like they are sort of making the claim that it would be better.

So there are some steps missing here for me. I'd appreciate either a quick explanation or, preferred, a reference for a longer one.

Billy Smith
  • 429
  • 1
    "homeomorphism isn't even strong enough for what we want to achieve" can you please elaborate on that? I get it if you're throwing metrics or differentiation into the mix, but then you're not strictly in the field of topology any more. Within topology proper, is there anywhere homeomorphism is too coarse an equivalence? – Arthur Dec 11 '18 at 11:55
  • I might not be right about that. Also, I don't know about the field demarcations exactly. First of all, in topology, do we ever classify smooth high-dimensional manifolds up to diffeomorphism, or is that not done? And second, is there actually an important difference between classification up to homeomorphism and classification up to diffeomorphism? If it's the case that homeomorphic differentiable manifolds are necessarily linked by some diffeomorphism, which I think on further reflection it might be, then what I said actually doesn't make sense. – Billy Smith Dec 11 '18 at 12:06
  • I'm certainly no expert. But after a little more looking, I think it's probably signficantly more complicated than I thought. From the paragraph linked below, I'm getting that diffeomorphism and homeomorphism are sometimes equivalent, and sometimes not, depending on the dimension of the manifold:

    https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture#Status

    Well, that's a shame. That part of my question still stands, then.

     – Billy Smith Dec 11 '18 at 13:33

1 Answers1

6

Often when people talk about "classifying" they think about having an algorithm (or a method) such that it takes a pair of spaces as an input and returns an answer to the question "are they homeomorphic/homotopic/diffeormorphic, etc?" as an output. All of that in a finite number of steps (see: decidability).

Such algorithm cannot exist because in particular we would be able to restrict the algorithm to manifolds. And in manifold case it is know that the problem is at least as hard as the word problem. And the word problem is known to have no solution.

Also IMO it would be very desirable if possible. I mean, what exactly would be a disadvantage of having such method?

freakish
  • 42,851
  • What, solving halting problem? Yeah, that would be interesting. But I'm not sure "desirable" is the right word. It would ruin a lot of good mathematics. – tomasz Dec 11 '18 at 13:28
  • Not taking into account the connection to the word problem, having such a method would be more desireable than not having it. However, i think homotopy equivalence may be much more useful, somewhat along the lines of how modular congruence often can give you the answers you're actually after much faster and much easier than equality does in number theory. – Arthur Dec 11 '18 at 13:36
  • @Arthur - could you expand on why that is? Because I don't really understand what you've just said, and that's actually kind of the primary thing I wanted to learn from this question. – Billy Smith Dec 11 '18 at 13:37
  • @freakish - Yeah, that makes sense. Pedagogically, though, this argument puts you in the position of having to assume the person you're talking to is well-acquainted with the idea of manifolds and has a great understanding of why classifying them is impossible, and yet knows nothing about homotopy theory (as motivating the development of the concept of homotopy equivalence was why people were making this argument to start with). That seems to me like a strange assumption to make. Is there an argument about why classification is impossible that can be made in terms of more point-set concepts? – Billy Smith Dec 11 '18 at 13:53
  • @BillySmith Not that I know of. There's a reason why books just mention it without actually explaining it. The stuff is far from trivial. Manifolds can be avoided. But at least the transition from topological spaces to group theory seems like the way to go. It is often done via fundamental group. The group theory is a simplier place to work with the word problem. – freakish Dec 11 '18 at 13:55
  • @BillySmith Also note that "are they homotopic?" question is also undecidable. Even the question "is it contractible?" is undecidable. So decidability was not the motivation behind those concepts (if I understand you correctly). – freakish Dec 11 '18 at 14:00
  • @freakish - You do understand me correctly. That's quite strange. If classifying up to homotopy equivalence, in full generality, can't be said to actually be any easier, then this argument for introducing homotopy theory is a bit suspect, right? I guess it may be easier for nice categories of spaces. Or somehow both classifications are uncomputable, but we can at least make some progress computing fundamental groups or homology groups, whereas homeomorphism is just too hard... but "less impossible" is a weird concept. I'm not sure how to rescue this argument given what you've just told me. – Billy Smith Dec 11 '18 at 14:09
  • @BillySmith The question "does $X$ have a trivial fundamental group?" is also undecidable. :D Even for simplicial complexes, can you find nicer category of spaces? I'm sorry, you're out of luck today. The point is that groups are hard (the question "does given presentation of a group yield a trivial group?" is undecidable). And if groups are hard then so are topological spaces due to algebraic topology (e.g. Eilenberg-MacLane construction). – freakish Dec 11 '18 at 14:12
  • @freakish Haha - but then how do you answer an undergraduate asking "Why do we need this idea of homotopy equivalence?" with anything other than "Because I say so, now roll with it"? You can hardly say "Well, classification up to homeomorphism is impossible, so we should classify up to homotopy equivalence instead. Which, yes, is also impossible. Just, like, slightly less impossible. You get me?"

    I guess what you're saying is, you need them to have significant intuition about why understanding groups is hard, and then once they've seen the definitions of homotopy groups, they'll get it.

     – Billy Smith Dec 11 '18 at 14:16
  • @BillySmith Forget about classification. That's not the point. There are many applications of the homotopy theory: https://math.stackexchange.com/questions/653/real-world-uses-of-homotopy-theory Some problems (e.g. Brouwer's fixed point theorem) are very hard to prove without it. – freakish Dec 11 '18 at 14:18
  • Yeah, focusing on fixed-point properties is a good way of introducing the concept, but I still feel it leaves something to be desired. Like, it works at first, but imagine being a student whose primary idea of the point of all algebraic topology is proving the contractibility or non-contractibility of the spheres. At some point, after having been introduced to the fundamental group and then homotopy groups and then homology groups and then CW-complexes and only then returning to the original question, you're gonna start losing the plot, you know? – Billy Smith Dec 15 '18 at 14:33