KERVAIRE INVARIANT ONE [ after

Around the year 1960 the theory of surgery was developed as part of a program to classify manifolds of dimension greater than 4. Among the questions it addresses is this: Does every framed cobordism class contain a homotopy sphere? Recall that a framing of a closed smooth manifold is an embedding into Euclidean space together with a trivialization of the normal bundle. A good example is given by the circle embedded in R, with a normal framing t such that the framing normal vector fields have linking number ±1 with the circle itself. A framed cobordism between framed n-manifolds M and N is an embedded manifold W ⊆ R × R+ such that W meets R ×{0} transversely in ∂W ; a diffeomorphism ∂W ∼= M ∐ N ; and a framing of the normal bundle of W that restricts to stabilizations of the framings of M and N . Cobordism classes of framed n-manifolds form an abelian group Ω n . The class of (S, t) in Ω 1 is written η. Any closed manifold of the homotopy type of S admits a framing [17], and the seemingly absurdly ambitious question arises of whether every class contains a manifold of the homotopy type (and hence, by Smale, of the homeomorphism type, for n > 4) of a sphere. (Classes represented by some framing of the standard n-sphere form a cyclic subgroup of order given by the resolution of the Adams conjecture in the late 1960’s.) A result of Pontryagin from 1950 implied that the answer had to be “No” in general: η 6= 0 in Ω 2 , while S is null-bordant with any framing. Kervaire and Milnor [17] showed that the answer is “Yes” unless n = 4k+2; but that there is an obstruction, the Kervaire invariant