Manifold as Homotopy Type with Extra Structure

cohomologydifferential-topologygt.geometric-topologyhomotopy-theorymanifolds

Motivation: Surfaces

Closed oriented 2-manifolds (surfaces) are "classified by their homotopy type". By this we mean that two closed oriented surfaces are diffeomorphic iff they're homotopy equivalent. This means that somehow the homotopy type of the surface contains essentially all information about the manifold.

Let's turn the question around. Let's choose a homotopy type $M$, and ask whether it specifies a manifold. I have to be more precise by what I mean here. First of all, let's fix a dimension, say 2 for now, although we can increase it later. Of course not every homotopy type will correspond to a surface. There a some restrictions on $M$, such as:

  1. It has to have cohomological dimension 2, i.e. isomorphisms $H^k(M) \cong 0 \quad \forall k > 2$.
  2. We have to specify an isomorphism $\phi\colon \mathbb{Z} \xrightarrow{\cong} H^2(M)$ which will in particular define the fundamental class $[M] := \phi(1)$, corresponding to orientation.
  3. Cohomology and homology have to exhibit Poincaré duality, which is to say that the cap product with the fundamental class is an isomorphism: $[M] \cap – \colon H_k(M) \xrightarrow{\cong} H^{2-k}(M)$

Now we're in better shape. Although I don't know a proof and haven't seen this statement anywhere, I'd venture the following conjecture, which should be easy to prove or disprove by anyone who knows more homotopy theory than me:

Conjecture Each homotopy type with the extra structure outlined in 1. – 3. corresponds to a closed, oriented surface. In particular, there is an equivalence between the category of homotopy types with extra structure and the category of closed, oriented surfaces.

Note also that the cap product is functorial, so a map of surfaces should be a map of the homotopy types preserving all of the structure.

The takeaway is this: I've come to believe that surfaces are essentially homotopy types with extra structure on cohomology and homology that comes from the manifold structure. Possibly I haven't captured all structure that is needed. But I guess one could amend the list in that case.

Question 1: Am I right so far?

Higher dimensions: Topological, PL and smooth structures

It gets hairier when we go up dimensions. There are closed 3-manifolds that are homotopy equivalent, but not homeomorphic. On the other hand, simply connected topological 4-manifolds are classified by their intersection form, so they can be completely recovered by the information in 1. – 3. ! For smooth structures, there is of course less luck, although the Kirby-Siebenmann class in 4th cohomology tells you whether there is a PL structure or not, so that sounds like a promising candidate for more extra structure along the lines of what we had so far.

Question 2: How far can we carry on the idea and classify higher dimensional (topological, PL, or smooth manifolds) by extra structure on the homotopy type, or its homology and cohomology?

Boundaries, noncompact manifolds

We could wonder whether it's possible to generalise the story to manifolds with boundaries, or noncompact manifolds. Then the homotopy type will certainly not be sufficient.

Already surfaces with boundary are not classified by their homotopy type. (Typical counterexample: The direct sum of two annuli, and the minimal 1-handlebody of a torus.) What is really relevant here is the homotopy type of the boundary inclusion $\partial M \hookrightarrow M$, and the corresponding relative cohomology.

(Similarly, for noncompact manifolds what seems to be relevant is compactly supported cohomology, which is related to the compactification of the manifold.)

How far can the idea be generalised here?

Best Answer

You are talking about the (much studied) Poincare duality spaces. For a survey, see the very nice one by John Klein: (seems to be unpublished, but dates to April 2010).

Related Question