Just a trivial observation: There's a sense in which the Cobordism Hypothesis is a (far reaching) generalization of the Pontryagin Thom theorem about bordism groups. However, the latter relies very heavily on transverality (and is in fact false for topological manifolds). So a topological analog would have to somehow exclude this part of the story.

It seems like this could work. Although i'm not so familiar with SG symbols. I should brush up on that first (regarding the completness as you say).

hmmm, i'm not sure I understand your point about the boundness, could you say a few words about the boundness? This is the part i'm least sure about. Also the completeness isn't at all obvious to me for the second class of symbols you propose.

Could you elaborate on why the second algebra you give satisfies 3? it doesn't seem obvious but maybe im missing something.

@mcd The constancy in $p$ condition (3) might be a problem with nonzero $\delta$ as well. As for "SG" or "Shubin" i'm not familiar with these terms

@mcd In my symbol classes $\delta$ is always $0$, I got the impression that for arbitrary $\delta$ the completeness condition In my question may fail. In any case if i'm wrong about that I was hoping that perhaps the extra minimality comdition will take care of this.

Thanks, although I was aware of this fact I didn't think about this when forming the question. In any case is there an apriori reason why someone should expect the rotated heat equation to be ill posed from just looking at the derivatives? (without knowing beforehand the solution of the usual heat equation)

Thank you for this! I understand the example of the rotated heat equation but is there a way to make precise the general principle you alluded to "Cauchy problem is usually ill-posed, because the order of space derivatives is less than that of the time derivative"?