The purpose of this post is to describe Sullivan’s proof of the Adams conjecture via algebraic geometry; the conjecture and its motivation were described in the previous post (from where the notation is taken). The classical reference is Sullivan’s paper “Genetics of homotopy theory and the Adams conjecture,” and the MIT notes on “Geometric topology.”

1. First step: completion at a prime

Sullivan’s proof of the Adams conjecture  is based on interpreting the Adams operations via a surprising Galois symmetry in the (profinitely completed) homotopy types of classifying spaces. Let’s work in the complex case for simplicity. Our goal is to show that the composite

$\displaystyle BU(n) \stackrel{\psi^k - 1}{\rightarrow } BU \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic. (The map $J$ was defined in the previous post.)

Since the homotopy groups of ${B \mathrm{gl}_1(S)[1/k]}$ are finite, it will follow (by the Milnor exact sequence) that we can let ${n \rightarrow \infty}$ and conclude that the map

$\displaystyle BU \stackrel{\psi^k - 1}{\rightarrow } BU \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic (i.e., there are no phantom maps into a spectrum with finite homotopy groups).

Using again the finiteness of the homotopy groups of ${B \mathrm{gl}_1(S)[1/k]}$, we can get a splitting

$\displaystyle B \mathrm{gl}_1(S)[1/k] = \prod_{p \nmid k} \widehat{ B \mathrm{gl}_1(S)}_p$

into the respective profinite completions. There is a well-behaved theory of profinite completions for connective spectra, or for sufficiently nice (e.g. simply connected with finitely generated homology) spaces, which will be the subject of a different post. (more…)