Aside from basic techniques of algebraic and differential topology and the

MR0148075 (26 #5584) 57.10 Kervaire, Michel A.; Milnor, John W. Groups of homotopy spheres. I. Annals of Mathematics. Second Series 77 (1963), 504–537. The authors aim to study the set of h-cobordism classes of smooth homotopy n-spheres; they call this set Θn. They remark that for n = 3, 4 the set Θn can also be described as the set of diffeomorphism classes of differentiable structures on S; but this observation rests on the “higher-dimensional Poincaré conjecture” plus work of Smale [Amer. J. Math. 84 (1962), 387–399], and it does not really form part of the logical structure of the paper. The authors show (Theorem 1.1) that Θn is an abelian group under the connected sum operation. (In § 2, the authors give a careful treatment of the connected sum and of the lemmas necessary to prove Theorem 1.1.) The main task of the present paper, Part I, is to set up methods for use in Part II, and to prove that for n = 3 the group Θn is finite (Theorem 1.2). (For n = 3 the authors’ methods break down; but the Poincaré conjecture for n = 3 would imply that Θ3 = 0.) We are promised more detailed information about the groups Θn in Part II. The authors’ method depends on introducing a subgroup bPn+1 ⊂ Θn; a smooth homotopy n-sphere qualifies for bPn+1 if it is the boundary of a parallelizable manifold. The authors prove in § 4 that the quotient group Θn/bPn+1 is finite (Theorem 4.1). More precisely, they prove that bPn+1 is the kernel of a homomorphism p′ : Θn → Πn/ImJ , where Πn is the stable group πn+k(S) and Im J is the image of the classical J-homomorphism. § 4 ends by giving (explicitly) the groups Θn/bPn+1 for n ≤ 8 and the groups bPn+1 for n ≤ 19, referring the reader to Part II for details. The proof given in § 4 depends on results in § 3. In this section, Theorem 3.1 states that every homotopy sphere is S-parallelizable, that is, its tangent bundle is stably trivial. The proof uses previous work of the same authors, and involves quoting information about the J-homomorphism. The remaining lemmas in § 3 concern the stability of bundles. It remains to prove that the groups bPn+1 are finite. The authors divide two cases. If n is even they prove that the groups bPn+1 are zero. That is, in §§ 5, 6 they prove (Theorem 5.1): If a smooth homotopy sphere of dimension 2k bounds an S-parallelizable manifold M , then it bounds a contractible manifold. The proof consists of simplifying M by surgery [J. Milnor, Proc. Sympos. Pure Math., Vol. III, pp. 39–55, Amer. Math. Soc., Providence, R.I., 1961; MR0130696 (24 #A556)]. The details are technical, and appear to be comparable with work of C. T. C. Wall, which also results in a proof of the same theorem [Trans. Amer. Math. Soc. 103 (1962), 421–433; MR0139185 (25 #2621)]. § 5 completes the proof for k even; the case in which k is odd is treated in § 6. Here the authors introduce the notion of a “framed manifold”, that is, a smooth manifold M plus a given trivialisation of