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I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta_n$ of h-cobordism classes of homotopy $n$-spheres (with group operation coming from the connected sum). It occurred to me that there ought to be a whole $(\infty,0)$-category of homotopy $n$-spheres with h-cobordisms between them, diffeomorphisms between the cobordisms, and so forth.

Then we might hope that connected sum gives our h-cobordism category a symmetric monoidal / $E^\infty$-space structure, which is in fact grouplike (for the same reason that $\Theta_n$ has inverses: $M\#-M=S^n$). If I am not mistaken, then that would suggest that this h-cobordism category naturally comes equipped with the structure of a connective spectrum whose $\pi_0$ is exactly $\Theta_n$.

First of all, I want to know whether this procedure actually works, since I am a bit new to this and there are details that I am not sure about. And if it does work, I am curious if anything more is known about these spectra. Are they familiar objects? Do the cohomology theories have nice geometric descriptions?

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  • $\begingroup$ $E^n$ seems to act reasonably naturally, but where would you get an action of $E^\infty$? $\endgroup$
    – S. Carnahan ♦
    Commented Nov 11, 2014 at 4:16
  • $\begingroup$ In cobordism categories, the natural monoidal structure is disjoint union. This is cobordant to a (not "the," since taking connected sums requires making a choice) connected sum, but in cobordism categories, cobordisms are not required to be invertible. So I think the natural thing to do in this situation is to start with a category that has all the cobordisms you want and then invert the cobordisms. I don't know if a construction of this form can give the space in my answer though. $\endgroup$
    – Qiaochu Yuan
    Commented Nov 11, 2014 at 5:02
  • $\begingroup$ (If you don't do something like the above, then I don't see how you can hope to get a monoidal category: how do you take the tensor product of two h-cobordisms with respect to connected sum?) $\endgroup$
    – Qiaochu Yuan
    Commented Nov 11, 2014 at 5:07

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I don't know if this is the sort of thing you have in mind, but using surgery theory you can in fact write down a single connective spectrum / infinite loop space whose $n^{th}$ homotopy group is $\Theta_n$. This is the space $PL/O$, one definition of which is that it is the homotopy fiber of the natural map $BO \to BPL$, where $BO$ is the classifying space of stable vector bundles and $BPL$ is the classifying space of stable PL-microbundles (or something like that).

The relationship to exotic spheres (same as homotopy spheres by the generalized Poincaré conjecture) comes from looking at the set of homotopy classes of lifts of the unique PL-structure on the $n$-sphere to a smooth structure. Some details can be found in this article by Davis and Petrosyan.

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