Relations Amongst Toda Brackets and the Kervaire Invariant in Dimension 62

Abstract

In this paper we use relations amongst Toda brackets and a lot of detailed information about the homotopy groups of spheres to show that there exists a 62-dimensional framed manifold with Kervaire invariant one. This paper, together with [4, 5], represents an effort to supply full details for a number of the results announced in [11], and to explore further some of the ideas behind that paper. The general problem of whether or not there are elements 6n e 71 +1-2 w i t n n o n ' zero Kervaire invariant will not be solved by the methods of this paper. The first three cases of this problem are trivial: 6X — rj , 62 = v , 03 = a , where rj, v and a are the three Hopf maps. It is known that #4 exists and indeed 04 is reasonably well understood [13, 16, 8]. The point of this paper is to construct another non-obvious example: 65 e n%2. However not even the most optimistic of us would claim that 65 is well understood. We begin this paper with some preliminaries on cell diagrams and Toda brackets. In §2 we outline the proof of the existence of 95, and in §§3 and 4 we give the detailed calculations needed to complete this proof. These calculations require constant references to [3, 6,13,14, 20, 21] for information about rc^ and the E2 and E^ terms of the mod 2 Adams spectral sequence. Some of this information is also contained in [10,16] and the tables at the end of [22]. There is an appendix with a proof of a general result on the vanishing of certain Toda brackets; a special case of this result is used in the main body of this paper.