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In an appendix to his notes on intersection homology and perverse sheaves, MacPherson writes

Why do we want to consider only spaces $V$ that admit a decomposition into manifolds? The intuitive answer is found by considering the group of all self homeomorphisms of $V$. Certainly if $V$ is to be of "finite topological type", then this group should have finitely many orbits. It is these orbits that should be the natural strata of $V$. That the orbits of this group should be manifolds results from the meta-mathematical pricinple that a space of "finite topological type" whose group of self-homeomorphisms acts transitively must be a manifold. I don't know a precise mathematical statement that realizes this meta-mathematical principle, but I expect that there is one.

As these notes are dated from 1990, I was wondering if the past twenty years have seen any work done towards a precise formulation of this meta-mathematical principle.

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This is amazing timing! I was reading the same thing, and wondering the same thing, one week ago. –  Daniel Moskovich Jun 10 '11 at 19:29
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I think this business about "finite topological type" is intentionally left vague, and certainly any precise formulation of MacPherson's principle will have to explicitly explain what is meant by this notion. It's at least somewhat clear (in my opinion) that such spaces as uncountable totally disconnected groups shouldn't fit the bill; in fact, in MacPherson's notes, the Cantor set is given as an example of something that has "infinite topological type". –  Faisal Jun 10 '11 at 23:33
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MacPherson says "finite type" means "i.e. topological spaces that require finitely much data to specify them" (including singular spaces). He also says the cantor set and one point compactification of the surface of infinite genus are not, and In some sense, the quest for a definition of a stratified space is the same as the quest for the definition of a space of "finite topological type". "In fact, giving a workable definition [of stratified spaces] was historically a very difficult problem, and it is very possible that the definitions we have today will not turn out to be the best ones." –  Junkie Jun 11 '11 at 0:31
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@Junkie: MacPherson must mean something slightly more specific than this. For example a finite topological space requires finite data to specify, and some of them have transitive automorphism group but are not even Hausdorff (e.g. the indiscrete space on a finite set, the partition topology on a finite set where each block has the same size). –  Qiaochu Yuan Jun 11 '11 at 19:56
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@Qiaochu: purely as a guess, by "finitely much data", it's possible that MacPherson had in mind a subspace of some $\mathbb{R}^n$ which is definable by a predicate in finitary first-order logic applied to a suitably restricted language of real numbers. An example might be semi-algebraic sets, which are definable in some $\mathbb{R}^n$ using the language of the ordered fields. Such sets admit stratifications, and there are many extensions of this language (e.g., ordered fields with exponentiation) which admit similar structure theorems, known to model theorists who study o-minimal geometry. –  Todd Trimble Jun 11 '11 at 20:31
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3 Answers

up vote 18 down vote accepted

A precise version of this statement is the Bing-Borsuk conjecture that a homogeneous ENR is a manifold. Here is a recent survey, generally in the direction of my answer. There is a candidate counter-example, due to Bryant, Ferry, Mio, and Weinberger, but they can't show it is homogeneous. Some people think that these generalized manifolds are very nice (especially if it does turn out that they're homogeneous) and one should just weaken statements like MacPherson's to allow them. If you throw on enough hypotheses, the BFMW machinery applies. This uses surgery theory and so is limited to high dimensions. The question is wide open in dimension 3, where it implies the Poincare conjecture.

I think of the Euclidean neighborhood retract (ENR) condition as a finiteness hypothesis. It is that the space is an absolute neighborhood retract that embeds in Euclidean space; equivalently a retract of an open subset of Euclidean space. This rules out the Cantor set because a neighborhood can have only countably many components, while the Cantor set has uncountably many. This condition implies that the space is homotopy equivalent to a finite dimensional CW complex, but it imposes a lot of tameness on the topology as well. This is a local condition and does not rule out the integers or the infinite genus surface. If one wants to impose such a global finiteness, one can require that the one-point compactification also be an ENR.

Following things like Bing's proof of Kline's characterization of the 2-sphere, by being disconnected by all embedded circles, but no pairs of points, there were attempts to characterize $n$-manifolds by separation conditions, such as homology. A manifold is locally a disk, which is the cone on a sphere. Given a space and a point, one can define the homotopy link of that point, which would be the base of the cone, if the space were locally a cone. The local homology $H_k(X,X-\{x\})$ is the homology of the link. If these groups are the homology of spheres and they form a local system, the space is called a homology manifold. To be a manifold, the links must be simply connected. The disjoint disks property implies this and it was conjectured that a homology manifold with the disjoint disks property is a manifold. This implies the shocking Cannon-Edwards theorem that the double suspension of the Poincare homology three-sphere is a topological manifold, even though it is not homogeneous under piecewise linear maps (or bi-Lipschitz maps).

Bryant, Ferry, Mio, and Weinberger produced counterexamples that were not manifolds, but they showed that the obstruction was a local invariant and that these spaces were amenable to surgery theory and classified them up to s-cobordism. They conjectured that these generalized manifolds are homogeneous and that s-cobordism implies homeomorphism, as with manifolds.

Bredon and later Bryant showed that if the local homologies of a homogeneous ENR are finitely generated, the space is a homology manifold. This sounds like a pretty tame finiteness assumption. More recently, Bryant achieved the homology manifold conclusion by strengthening the hypothesis from homogeneity to arc-homogeneity. To get from homology manifold to the BFMW generalized manifolds, one needs a hypothesis like the disjoint disks property, but I don't think anyone knows how to get this from homogeneity.

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Lots of great stuff here; thanks! –  Faisal Jun 13 '11 at 13:48
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Let me point out that this appendix in MacPherson's notes is explicitly mentioned in a wonderful and highly accessible book by the model theorist Lou van den Dries, Tame Topology and O-minimal Structures (page 8). Indeed, the entire corpus of o-minimal geometry can be viewed as giving a precise response to the frequently expressed desire, perhaps most eloquently enunciated in Grothendieck's Esquisse d'un Programme, to put the sort of "tame topology" that MacPherson is pointing to on firm theoretical ground.

Where MacPherson says that only finitely many data are required to define a finite topological type (FTT), he says he means subsets of manifolds -- probably we can assume the manifolds are Euclidean spaces $\mathbb{R}^n$ without any real loss of generality -- and a reasonable guess is that he means the data are specified by finitely many conditions, for example a subset carved out by finitely many equalities and inequalities involving some basic staple functions like polynomials should qualify as an FTT. Which functions can be admitted is presumably open to discussion, so long as finite expressions involving them do not lead to things like the Cantor set being "finitely definable", which for the purposes of this discussion will be considered "pathological".

There are a number of formalisms which capture this intuition in one way or another; the best known or most investigated is probably that of o-minimal structures (there are also the $\mathcal{X}$-sets of Shiota, among others). Rather than spell out the precise definition, let me roughly describe an o-minimal structure as consisting of subsets of $\mathbb{R}^n$ (where $n = 0, 1, 2, \ldots$) which

  • Are closed under all first-order logical operations: unions, intersections, relative complements, cartesian products, and closed under taking direct images along coordinate projections $\mathbb{R}^{n+1} \to \mathbb{R}^n$ (in order to accommodate existential quantification);

  • Include the equality and inequality relations $\{(x, y) \in \mathbb{R}^2: x = y\}$ and $\{(x, y) \in \mathbb{R}^2: x < y\}$;

  • Do not include any subsets of $\mathbb{R}^1$ except for arbitrary finite unions of points and intervals $(a, b)$, allowing $a = -\infty$ and $b = \infty$. In other words, that contain only those subsets of $\mathbb{R}$ which have to be there by the preceding axioms, and no more. This is the famous order-minimality or o-minimality condition; it rules out for example the set of natural numbers as belonging to an o-minimal structure.

Regarding this last axiom: logicians know how to exploit the arithmetic of natural numbers to define all sorts of chaotic and pathological structures. For example, if graphs of polynomial functions are admitted, and if $\mathbb{N} \subseteq \mathbb{R}$ is also admitted, then chaos ensues: a logician can write down some complicated finite formula in these predicates to define any Borel set you jolly well please, or for that matter any set in the projective hierarchy. Thus, the o-minimality axiom is there to ensure a level of respectable tameness.

The archetypal example of an o-minimal structure is the class of semi-algebraic sets, but many others are known. In any o-minimal structure, the "definable sets" (i.e., the sets belonging to the structure) admit Whitney stratifications into definable manifolds, and quite a rich theory has been developed; I refer to the book by van den Dries for details.

It would be very interesting if model theorists had looked into this business of the Bing-Borsuk conjecture; it could be that adding in o-minimality hypotheses would help address technical difficulties people have experienced in trying to prove this (but I am hardly qualified to say anything about this). A quick Google search didn't turn up much that I could see, but a paper by Frank Quinn in these proceedings seemed to touch ever so briefly on such issues.

Come to think of it, there are some resident experts on o-minimal theory here at MO (Thierry Zell and Dave Marker come to mind), and it would be wonderful if they could weigh in here.

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Thanks, Todd. This was really informative. –  Faisal Jun 13 '11 at 13:47
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I suspect a rephrasing into something analogous would help. Knowing some metamathematics and effectively no homology or sheaf theory, I interpret his principle as: if everywhere I look at a structure locally, I see essentially the same view, then a number of statements I can make about a particular view should hold everywhere I will go in the structure. Being a conditional, it requires less faith than a statement like certain physical statements are universal and so must be laws, but to me it has much the same character.

Gerhard "Ask Me About System Design" Paseman, 2011.06.10

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