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What do the differential topology theories on arbitrary fields have in common?

Different differential topology theories

  • There is "ordinary" differential topology on real manifolds, with its rich relations to homotopy theory (de-Rham cohomology, Morse (co)homology, surgery theory), and its unique new features (exotic smooth structures).
  • Arguably, there is "complex differential topology", although there are good reasons that complex manifolds are "more geometric" than real manifolds. Still, there's Dolbeault cohomology which is sometimes related to (topological) cohomology.
  • There is $p$-adic analysis, and although I know little about it, I'm guessing a theory of $p$-adic manifolds has been developed and can be developed further. For example, there are Berkovich spaces.

For the sake of the question, let's ask of a "good" theory of differential topology to have some kind of Morse theory, handle decompositions or surgery theory. I want to know in which fields one can develop such a theory, and how they are related.

Common constructions, relations and differences

There are quite a few of common constructions, like unit balls and spheres, and projective spaces, which work for different fields and have similar properties. (E.g. as real manifolds, $\mathbb{R}P^n \cong h_0 \cup h_1 \cup \dots \cup h_n$ and $\mathbb{C}P^n \cong h_0 \cup h_2 \cup \dots \cup h_{2n}$.)

A lot of the basic questions in complex geometry vs. real differential topology come from looking at $\mathbb{R}$ as a subfield of $\mathbb{C}$, observing the forgetful functor from complex manifolds and real manifolds and asking which constructions and theorems can be lifted the other way. For example, we can ask which real manifolds have an almost complex structure (algebraic topology question), and which have almost complex structures that are integrable. The second question is hard to answer in general, but the questions themselves are straightforward to ask, so I'm wondering whether there is a general theory for asking such questions for arbitrary "differentiable/topological" fields.

Complex manifolds have a lot of differences to real manifolds as well, they are much closer to algebraic geometry. (I guess this is similar to the $p$-adic case.) That's interesting, and a big reason why people study them, but not what I'm after. I want to know what they have in common, like in this question.

It's not straightforward to study Morse theory on complex manifolds, since a holomorphic function from a complex manifold to $\mathbb{C}$ is constant. But I heard there is an approach to study Morse theory to $\mathbb{C}P^1$, so maybe there's still a common theme here?

Questions

  • What's a "differentiable field"? What basic properties must a field have so we can define smooth manifolds over it? Ideally, this is a contravariant functor from the category of differentiable fields to the category of categories.
  • What basic constructions and theorems from real differential topology are applicable/generalisable over any field? Morse theory? De-Rham cohomology? Surgery theory?
  • In what generality can we transport results from manifolds over one field to another? The inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$ gives us a forgetful functor from complex to real manifolds, so it allows us to ask great questions like "Does $S^6$ have a complex structure?". What are the natural questions to ask for an arbitrary field morphism $f\colon k_1 \to k_2$, and what's the generic answer in terms of integrable "almost $k_2$-structures"?

Maybe I'm ultimately after something like cohesive $\infty$-topoi, but the theory seem so abstract that I fail to understand whether simple things like handle decompositions work there.

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    $\begingroup$ related M.SE question math.stackexchange.com/questions/581078/… $\endgroup$ – Harry Gindi May 18 '18 at 10:55
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    $\begingroup$ Two words: Berkovich spaces. $\endgroup$ – Alex M. May 18 '18 at 11:43
  • $\begingroup$ @AlexM., thanks, added a note to the question. What's the differential topology of Berkovich spaces, though? $\endgroup$ – Manuel Bärenz May 18 '18 at 13:02
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    $\begingroup$ @Manuel I think that all of the non-real versions of differential topology are analytic in nature rather than smooth. The smooth real case is unique due to the existence of partitions of unity. Without them, you really need the additional rigidity of analytic functions (and so things tend to resemble the algebraic and complex analytic situation). I believe this means you can do sheafy DeRham cohomology, but not really Morse theory or Surgery, though I could be wrong. $\endgroup$ – Harry Gindi May 18 '18 at 16:39
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    $\begingroup$ I just wanted to mention, I looked into it, and it looks like there is a version of Morse theory for complex manifolds called Picard-Lefschetz theory that was generalized by Deligne and Katz to varieties over more general fields, but I'm just going by Wikipedia here. $\endgroup$ – Harry Gindi May 18 '18 at 19:18

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