K-THEORY AND THE HOPF INVARIANT

Abstract

[Received 29 December 1964] Introduction THE non-existence of elements of Hopf invariant one in ir%n-^(S ), for n ^ 1, 2, 4, or 8, was established in (1) by the use of secondary cohomology operations. The main purpose of this paper is to show how the use of primary operations in .K-theory provides an extremely simple alternative proof of this result. In fact ^-theory proofs have already been given in (8) and (4) but neither of these proofs is elementary: (8) uses results on complex cobordism, while (4) uses the connexion between the Chern character and the Steenrod squares established in (3) [see however (6) for a more elementary treatment of the results of (3)]. The simplicity and novelty of our present approach is that, unlike all previous attacks on the Hopf invariant problem, we consider not the stable but the unstable version of the problem: that is to say we shall prove directly