Not thinking about $h$-cobordism, one usually defines a *cobordism* between manifolds, realizes it is an equivalence relation, chooses an appropriate class of *structured manifolds* (framed, unoriented, spin, ...), and dreams about the group of cobordism classes of these manifolds.


Thinking about $h$-cobordism, whose definition I learned today, one seems to not think this way. At least, I've glanced through the Wikipedia page and some notes/articles about $h$-cobordism and there is no explicit mention of 'the set of $h$-cobordism classes of $n$-manifolds (possibly with structure)' or its (group) structure.


(Taking after Will Sawin and Connor Malin's comments)


There is not an interesting group of $h$-cobordism classes of manifolds in general (see comments.) But there is, for instance, interesting groups of $h$-cobordism classes of homotopy spheres.



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> Is there any interpretation of these groups of a similar spirit (e.g., using spectra) to the relationship between cobordism and Thom spectra?
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Do $h$-cobordism groups arise from a 'Thom-like' spectrum?
This topic has a very different flavor from what is usually meant by cobordism. No, the correspondence between cobordism classes and homotopy classes (of a Thom space or Thom spectrum) has no analogue here; when manifolds, or cobordisms between manifolds, are created by transversality, we don't have much control over their homotopy types.


I would say that the main point of considering h-cobordisms is that surgery theory (in the sense of Wall, Browder, Novikov, ...) is pretty good at answering questions like "How many manifolds, if any, have a given homotopy type?" but where "how many" means up to the equivalence relation of h-cobordism. So, to complete the picture we need to know something about the distinction between this equivalence relation and the one that we originally cared about.


