Quillen’s Work on the Adams Conjecture

In the 1960’s and 1970’s, the Adams Conjecture figured prominently both in homotopy theory and in geometric topology. Quillen sketched one way to attack the conjecture and then proved it with an entirely different line of argument. Both of his approaches led to spectacular and beautiful new mathematics. 1. Background on the Adams Conjecture For a finite CW -complex X, let KO(X) be the Grothendieck group of finite-dimensional real vector bundles over X, and J(X) the quotient of KO(X) by the subgroup generated by differences ξ− η, where ξ and η are vector bundles whose associated sphere bundles are fibrehomotopy equivalent. For any integer k, let Ψ be the Adams operation on KO(X) constructed in [1]. In [2, 1.2] J. F. Adams made the following proposal. 1.1. Conjecture. If k is an integer, X is a finite CW -complex and y ∈ KO(X), then there exists a non-negative integer e = e(k, y) such that k(y −Ψy) maps to zero in J(X). This quickly became known as the Adams Conjecture. Let O be the stable orthogonal group and G the monoid of stable self-homotopy equivalences of the sphere, so that the spaces BO and BG classify respectively stable real vector bundles and stable spherical fibrations. There is a natural map σ : BO → BG which assigns to each vector bundle its associated sphere bundle. Conjecture 1.1 is equivalent to the statement that for each k the composite map (1.2) BO 1−Ψk // BO σ // BG // BG[1/k] is null homotopic on finite skeleta. 1.3. Remark. There is also a complex form of Conjecture 1.1. Let K(X) be the Grothendieck group of finite-dimensional complex vector bundles over X, and J ′(X) the quotient of K(X) obtained as above by identifying two complex vector bundles if their associated sphere Date: October 20, 2011. 1