In an appendix to his [notes](http://faculty.tcu.edu/gfriedman/notes/ih.pdf) 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.



A "meta-mathematical principle" of MacPherson
A precise version of this statement is the Bing-Borsuk conjecture
that a homogeneous ENR is a manifold.
[Here](http://arxiv.org/abs/0811.0886) 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](http://en.wikipedia.org/wiki/Kline_sphere_characterization),
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](http://en.wikipedia.org/wiki/Double_suspension_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](http://www.calvin.edu/%7Evenema/workshop/proceedingspapers/bryant.pdf)
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](https://arxiv.org/abs/0903.3052), 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.


