Two Proofs of the Stable Adams Conjecture

(P) where / is the complex /-homomorphism and , y denotes localization at p. Both J and ** are infinite loop maps, and it is natural to ask whether this result is infinitely deloopable; that is, whether J<ff = / as infinite loop maps. This is the Stable Adams Conjecture. We announce here two independent proofs of this conjecture. Details will appear in [2] and [6]. METHOD 1. Our proof is based upon a "geometric" criterion for pairs of maps into the spectrum (BsG) s (BsG)(p) to be homotopic, where ( ) A denotes the Bousfield-Kan Z/p-completion functor. We exploit the "galois symmetry" of (kU) [8] to show that J, J ° (*^) satisfy this criterion. We impose a Quillen closed model category structure on Segal's T-spaces [3], whose weak equivalences are level-wise weak equivalences of spaces. For any "suitably oriented, pointed C. W.-like space" X (e.g., X any pointed C. W. complex with no orientation specified), we obtain a T-space BsGx arising from distinguished homotopy equivalences of iterated smash products of X with itself. There is a natural functor $: Ho T-spaces —• HoSpectra sending BsGs2 to QÇ$sGs2) = BsG. Our basic representability theorem is a description of the functor