When is a topological space a manifold?

Question

I'm looking for someone to point me in the direction of papers or books which discuss when a topological space (perhaps with the conditions locally compact, Hausdorff) is a topological manifold, differentiable manifold, etc. This is to say, what work work has been done on necessary conditions for these classifications, knowing very little about the original space?

Answer 1

If I understand your question, you are asking about the so-called "Recognition problem for (topological) manifolds". I am by no means an expert on this, but I recommend the following old survey of Cannon: http://www.maths.ed.ac.uk/~aar/papers/cannon2.pdf . (The survey is 35 years old, so surely there are more modern results.)

Anyways, this is a well-known problem in topology, which Cannon states in the following way:

Recognition problem: Find a short list of topological properties, reasonably easy to check, that characterize topologcal manifolds among topological spaces.

The point is that "being locally Euclidean" is not an easy-to-check "intrinsic" property.

One necessary condition is that the space should be a so-called "generalized manifold", meaning that it looks like a manifold from the point of view of local homology or cohomology. (These are also sometimes called "homology manifolds" or "cohomology manifolds".)

It is well-known however, (I think due to Cannon and Edwards) that this algebraic-topology condition alone is not enough to characterize honest topological manifolds, i.e. there are generalized manifolds in every dimension $\geq 3$ which are not topological manifolds.

I hope that points you in the right direction, or at least gives you some terms to google.