I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of spaces which are homotopy equivalent to finite CW complexes.
Now I would like to empathize the importance of the spaces which are homotopy equivalent to finite CW complexes, i.e, I try to show that there are a lot of spaces we care about that satisfy this property. Therefore i shoot my questions:
1) Are any sufficient conditions on Topological Manifolds to be homotopy equivalent to finite CW complexes.
2) Are any sufficient conditions on Smooth Manifolds to be homotopy equivalent to finite CW complexes.
3) Do you know sufficient conditions on other spaces to be homotopy equivalent to finite CW complexes.
Note that I only care about finite CW complexes.
Please I need references to papers or books since I will cite them. I won't prove any of the results since I will use this information only as an informal motivation to show that the results presented in that section of the essay are amazing and broadly useful.
Background: I'm an undergraduate, hence don't be very concise in your explanations and don't skip to many details please.
I have found some questions such as:
But they don't satisfy my curiosity.
And I also have seen the Corollary A.12 in Hatcher:
A compact manifold is homotopy equivalent to a CW complex.
But it doesn't say that the CW complex is finite so that doesn't work for me.
Maybe I should ask this at mathoverflow?
Thanks in advance and any help would be appreciated.
Best Answer
1) Every compact topological manifold is homotopy equivalent to a finite CW complex; see here. As a (very sketchy) sketch: You know via Hatcher that they're dominated by a finite CW complex, hence you can apply Wall's obstruction theory to being homotopy equivalent to a finite CW complex: see here. This immediately implies that every simply connected compact manifold is homotopy equivalent to a finite CW complex, and with more difficulty, that this is true for manifolds with eg fundamental group $\Bbb Z^n$. The general case is incredibly hard; the reference given in the MathOverflow post is as far as I know essentially the only proof of this fact. I would only suggest reading if you've got sufficient spunk. (For reference: I don't.)
2) Every compact smooth manifold is homeomorphic to a finite CW complex. This follows from Morse theory, which on a smooth manifold actually gives you a triangulation. This is much more elementary than (1).
3) That'd be Wall's finiteness obstruction, mentioned in the first paragraph. The linked notes of Lurie are accessible given a first course in algebraic topology, some experience with homological algebra, and some patience. To use it, you need your spaces to be finitely dominated; this is equivalent to satisfying Lurie's Lemma 6, which is not particularly helpful in practice. Usually you'll start with a space you actually know is finitely dominated and start using the finiteness obstruction then.
If you're willing to possibly care about countable CW complexes, this brief article of Milnor's implies every mapping space of finite CW complexes has the homotopy type of a countable CW complex.