The Kervaire Invariant of Framed Manifolds and its Generalization 论文

摘要

In 1960, Kervaire [11] introduced an invariant for almost framed (4k + 2)manifolds, (k # 0, 1, 3), and proved that it was zero for framed 10-manifolds, which was a key step in his construction of a piecewise linear 10-manifold which was not the homotopy type of a differential manifold. Haefliger [9] showed that Kervaire's invariant and the invariant of Pontrjagin [17] for 2, 6, and 14 dimensional framed manifolds, could be defined in a common fashion, and this invariant is the surgery obstruction in dimensions 4k + 2 (see [12], [15], [6]). A central question has remained, for which dimensions can a framed manifold have a non-zero Kervaire invariant. Pontrjagin's invariant is nonzero for certain framings on S' x S', S3 x S3 and S7 x S7, but until now all the results for the Kervaire invariant have been in the negative; Kervaire [11] showed it was zero in dimensions 10 and 18, and Brown and Peterson [8] showed it zero in dimensions 8k + 2. In this paper we will show that the Kervaire invariant is zero for dimensions # 2k2. For dimension 2k 2 we show that there is a framed manifold of Kervaire invariant 1 if and only if in the Adams spectral sequence for the stable homotopy groups of spheres the element h2 in E2 persists to Ed, (see [1], [2]). But it is a fact due to Mahowald and Tangora, (Topology 6 (1967) 349-370, ? 8) that hl in dimension 30 persists to Ed. Hence there is a framed 30-manifold of Kervaire invariant 1. (We are informed that recently Barratt and Mahowald have shown h' persists to Ed, so there is a framed 62-manifold of Kervaire invariant 1.)' Now we list some of the geometric corollaries which follow from our result (see [12]).