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Put another way, assuming it is somewhat fair to say that we (not I, but those who know better--part of my question is whether my stated assumption is in fact warranted) have in some sense a qualitatively better handle on manifolds up to and including dimension 4 than we do for the case of 5+ dimensions, is it just a coincidence (i.e., 'non-mathematical' factors) that it seems 'natural' for us to think of space-time in terms a four-dimensional manifold?

Edit: apologies for my naivety; I do not mean to imply that my assumption is solid. I probably overestimated the legitimacy of the question I was asking.

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    $\begingroup$ One reason comes from the fundamental group. For two-manifolds, they are all defined by one relation and much is known; in three-manifolds they form a rich landscape with much both in the ways of decidability and mystery. But in four-manifolds (and above) the fundamental group can be any finitely presented group, and all hope of fully general theorems disappears. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Apr 28 at 5:27
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    $\begingroup$ @Carl-FredrikNybergBrodda I know very little about this field, but that sounds like a good reason to have a cutoff at dimension 3 rather than 4 $\endgroup$
    – Alessandro Codenotti
    Commented Apr 28 at 6:56
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    $\begingroup$ A major point is certainly that the h-cobordism theorem (in the smooth category) is true in dimension at least 5, but not in dimension 4 and presumably also not in dimension 3 (and has a very different proof in dimension 2). $\endgroup$
    – ThiKu
    Commented Apr 28 at 10:57
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    $\begingroup$ Not everyone agrees that 5-manifolds (perhaps with boundary) are beyond the low-dimensional: eg see here $\endgroup$
    – mme
    Commented Apr 28 at 11:24
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    $\begingroup$ The term "low-dimensional topology" was created because of the Whitney trick and the h-cobordism theorem. Roughly speaking, these tools made manifold theory (for many/most purposes) much more uniform and algebraic in nature, in dimensions $n \geq 5$. $\endgroup$
    – Ryan Budney
    Commented Apr 28 at 18:58

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Contrary to the assumption in your question, understanding manifolds of dimension 5 and higher is considered to be easier than understanding 3- and 4-dimensional manifolds. Why? Because in $({\ge}5)$-manifolds there is more room to maneuver. Specifically, in a $({\ge}5)$-manifold any pair of embedded surfaces can be made disjoint by a small perturbation. This fact leads to the proof of the $h$-cobordism theorem and other tools for classifying $({\ge}5)$-manifolds.

I'm not aware of any convincing argument relating the above to the fact the space-time is 4-dimensional. But there is plenty of sub-convincing speculation on this topic.

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    $\begingroup$ @Neal: No, that’s not what the Whitney trick is. It’s a geometric move for cancelling intersections; see here: celebratio.org/Whitney_H/article/220 $\endgroup$
    – Misha
    Commented Apr 28 at 15:40
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    $\begingroup$ Making smooth submanifolds of dimensions K and L disjoint by a small perturbation in a smooth N manifold is an easy consequence of transversality when K + L < N. $\endgroup$
    – Daniel Asimov
    Commented Apr 28 at 20:39
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    $\begingroup$ Now I want to write a paper with a section titled "Sub-convincing speculations" $\endgroup$
    – Sam Hopkins
    Commented Apr 28 at 23:26
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    $\begingroup$ "Convincing sub-speculations", for vague philosophising below the level of actual concrete ideas, but that everything takes as a good motivating idea $\endgroup$
    – David Roberts ♦
    Commented Apr 29 at 2:22
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    $\begingroup$ @Municipal-Chinook-7: Don't you mean to say it's more difficult to study low-dimensional topology? Because you appear to be re-stating your previously-stated assumption. $\endgroup$
    – Ryan Budney
    Commented Apr 29 at 5:21
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There are of course ways to construct difficult problems in high-dimensional manifold theory. One of the ways high-dimensional manifold theory differs strongly from low-dimensional manifold theory is in the study of automorphism groups, like the diffeomorphism group of a manifold. In this case, dimensions $n \leq 3$ is very well understood, but dimensions $n \geq 4$ are considerably more difficult, and in many respects dimension $4$ more closely resembles high-dimensions than low dimensions, as far as we can tell at the moment.

But your question is specifically about what the term low-dimensional topology means. And this is tied to the story of the classification of manifolds, and the tools we developed for the task, like the Whitney trick and h-cobordism.

Here is maybe an analogy to help understand how h-cobordism makes life easier in high dimensions. If you are studying a rock pile and someone asks you to classify all the granite rocks in the pile. If you didn't know any better you would have to pick through all the rocks and investigate them one by one. Maybe you'd have to split some rocks if you were suspicious granite was fused into the centre of another rock. But if you have a magical rock sifter that could pick out all the granite for you, it would be quick and easy. h-cobordism is something like that magical rock sifter as it allows you to readily identify all the manifolds that satisfy certain homotopy properties. I like to think of it as something that makes the subject more uniform and homogenous -- it allows you to quickly find the things you are looking for. Low-dimensional topology is less uniform so you have a harder time finding out if certain types of manifolds exist. Like, say, a counter-example to the generalized Poincare conjecture.

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  • $\begingroup$ Ah, I recall seeing somewhere the notion that difficulties in high-dimensional topology are often explicitly homotopy-theoretic in nature. I'd assume that it would be a consequence of the h-cobordism theorem? $\endgroup$
    – xuq01
    Commented Apr 29 at 19:17
  • $\begingroup$ Not purely h-cobordism, but the h-cobordism theorem is the beginning of a body of work including what gets called "surgery theory", s-cobordism, k-theory, smoothing theory, and so on. With only a few exceptions this body of work is coherent in dimensions $n \geq 5$, like the Whitney trick. $\endgroup$
    – Ryan Budney
    Commented Apr 30 at 0:47

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