# Who defined the Inertia Group $I(M^n)\subset\Theta_n$ of a smooth manifold?

If you're unfamiliar with the definition, for an oriented smooth manifold $M^n$ we define the inertia group $I(M)$ to be the set of (h-cobordism classes of) homotopy spheres $\Sigma^n$ such that $M\#\Sigma$ is orientation-preserving diffeomorphic to $M$.

I'm trying to compile results into an expository Master's thesis on the subject, and it seems silly to not know the origin. Digging through old papers about the Inertia Group, I'm having a hard time finding the start of the trail. Many early papers refer to Tamura's "Sur les sommes connexes de certaines variétés différentiables" so I expect it to be close to the beginning, but I have been unable to find a copy of this paper.

I am aware of a few members here who are familiar with this theory, and maybe were even around when it started. Does anyone happen to know in which paper/book the Inertia Group originated?

• I think a natural first guess would be Milnor. – Ryan Budney Feb 26 '12 at 4:28
• Update: My supervisor found me a copy of Tamura's paper! In one corollary at the end he has a diffeomorphism between a $7$-manifold and its connected sum with a non-trivial Milnor sphere, but nowhere in the paper does he use "inertial" or "$I(M)$" (or French equivalents). I haven't managed to find the definition in anything by Milnor yet either (he does use "$I(M)$" in "Differentiable Manifolds which are homotopy spheres," but here it refers to the index aka signature). – William Mar 5 '12 at 1:20
• Did you find an answer to this? Also, does your reference to the occurrence of certain symbols mean that you have some way of doing bulk full-text searching, or was it a manual search? – LSpice Mar 26 '18 at 17:05
• An interesting question; I'd wondered the same myself at some point. Did you try writing to one of the original participants or who used this terminology early on? Eg Milnor, Browder, or perhaps A. Kosinski or T. Lawson? A possible explanation is that someone used the term in conversation or lectures and didn't bother with a definition in print. – Danny Ruberman Mar 27 '18 at 3:49