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?