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Marcel Berger states Thom's Principle as:

"rich structures are more numerous in low dimension, and poor structures are more numerous in high dimension."

This is in Geometry II (Springer-Verlag, Berlin, 1987. Google books), pp.39-40, after a discussion of regular polytopes shows that there are only three convex regular polytopes in dimensions larger than 4, but six in dimension 4, five in dimension 3, and an infinite number in dimension 2. He then lists further examples of the principle: e.g., simple Lie groups illustrate rich structure in low dimensions, and topological vector spaces illustrate poor structure in high dimension (all homeomorphic in finite dimensions).

In so far as Thom's principle is true—or at least holds widely—my question is:

Why should low dimensions exhibit richer structure than high dimensions? Is there any generic reason to expect this?

It might also be interesting to track down Thom's own formulation of his principle, to understand the context in which he proposed it.

Edit. Here is a snapshot of Berger's examples (p.40):
   p.40

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Speculation: being low-dimensional beings, we care about things that are interesting/rich in low dimensions (but turn out to be poor/less interesting in high dimensions). That is, we are talking about a kind of selection bias here. –  Ketil Tveiten Sep 13 '11 at 12:06
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Sort of vaguely related: mathoverflow.net/questions/5372/dimension-leaps –  Andrew Stacey Sep 13 '11 at 12:17
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One way to say it is that in low dimensions there are richer structures. The other way is to say that in low dimensions there are more "exceptions" to the rules, as the "rules" start to get into their characteristic pattern only after some "low dimensional adjusting oscillations"... –  Qfwfq Sep 13 '11 at 12:51
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I kind of question the premise. "Rich" structures often arise when we classify an object and find that it comes in several infinite families with finitely many exceptional examples. The exceptional examples are by their very nature often the most interesting, and we try to understand them by fitting them to our low-dimensional tools. For example, the exceptional Lie group E8 has dimension 248 (is that low or high?) but thanks to the theory of root systems we can say a lot about it by doing algebra and geometry in 8 dimensional Euclidean space. –  Paul Siegel Sep 13 '11 at 12:54
    
Aside from that, there are plenty of examples which at the moment are not expressible in what could possibly be considered low dimensions. Last I checked the smallest known representation for the monster group is in dimension 196883, for example. –  Paul Siegel Sep 13 '11 at 12:59
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3 Answers

up vote 8 down vote accepted

One possible reason is that in higher dimensions there are more degrees of freedom that can be used to unravel and untangle things, which often leads to simpler structures. This reason has particularly been used as an explanation as to why the geometry/topology of high-dimensional manifolds is sometimes easier to deal with than that of lower dimensional ones. The canonical illustration that accompanies the previous sentence is the failure of the Whitney trick in dim < 5; the fact that the trick holds in higher dimensions was, for example, instrumental in Smale's proof of the $h$-cobordism theorem.

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I'm glad someone mentioned Whitney and cobordism! –  some guy on the street Sep 13 '11 at 15:46
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Pro

I think the examples given are instances of Guy's "strong law of small numbers". That seems at least poetic reason for low-dimensional specializations of your favorite theory to be different in character from high-dimensional specializations.

Con

An example of increasing richness indicating Thom was thinking about something else:

  • "The" connected 0-manifold is a point
  • "The" connected compact 1-manifold is a circle
  • Connected compact 2-manifolds are connect sums of tori or of projective planes; they are uniformizable.
  • Connected compact smooth 3-manifolds are piecewise geometrizable, where the joints are among spheres and tori.
  • Connected compact $(3+n)$-manifolds "solve" the word problem for groups; particularly weird: there are smooth examples that have contractible stable closed periodic geodesics in any smooth metric.

This is sort-of what I'd call rigidity in low dimensions, and richness in high dimensions. Perhaps the theories in low dimension are richer in the sense that there are more universal statements we can prove, but there seems to be a greater wealth of useful examples in higher dimension.

Whether this is an instance of Thom's principle as quoted or an exception, it is still an instance of Guy's law, in that the low-dimensional behaviour isn't representative of high-dimensional behaviour.

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I never heard of Guy's "law"! :-) Thanks! en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers –  Joseph O'Rourke Sep 13 '11 at 15:56
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I am reminded of the fact that for geometries of dimension strictly higher than 2, the Desarguesian and Pappian configurations are equivalent, so that (in my view) there is a greater degree of uniformity in higher dimensional geometric spaces. I think Thom's principle (as noted in another post) is a selection principle and tells the reader what Thom thinks of by rich structure. If structures are essentially (in mathematical logic, some notion of conservative) extensions of incidence relations, then I am not too surprised that something like this principle should hold, since a certain amount of uniqueness is lost in a structure that should be invariant under more geometric transformations. I am not a geometer, though, so go ask someone like Hrushovksii what they think (from a mathematical logic perspective) of Thom's principle.

Gerhard "Ask Me About System Design" Paseman, 2011.09.13

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