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…)

The Sullivan conjecture on maps from classifying spaces originated in Sullivan’s work on localizations and completions in topology, which together with étale homotopy theory led him to a proof of the classical Adams conjecture. The purpose of this post is to briefly explain the conjecture; in the next post I’ll discuss Sullivan’s proof.

1. Vector bundles and spherical fibrations

Let ${X}$ be a finite CW complex. Given a real ${n}$-dimensional vector bundle ${V \rightarrow X}$, one can form the associated spherical fibration ${S(V) \rightarrow X}$ with fiber ${S^{n-1}}$ by endowing ${V}$ with a euclidean metric and taking the vectors of length one in each fiber. The Adams conjecture is a criterion on when the sphere bundles associated to vector bundles are fiber homotopy trivial. It will be stated in terms of the following definition:

Definition 1 Let ${J(X)}$ be the quotient of the Grothendieck group ${KO(X)}$ of vector bundles on ${X}$ by the relation that ${V \sim W}$ if ${V , W}$ have fiber homotopy equivalent sphere bundles (or rather, the group generated by it).

Observe that if ${V, W}$ and ${V', W'}$ have fiber homotopy equivalent sphere bundles, then so do ${V \oplus V', W \oplus W'}$; for example, this is because the sphere bundle of ${V \oplus V'}$ is the fiberwise join of that of ${V}$ and ${V'}$. It is sometimes more convenient to work with pointed spherical fibrations instead: that is, to take the fiberwise one-point compactification ${S^V}$ of a vector bundle ${V \rightarrow X}$ rather than the sphere bundle. In this case, the fiberwise join is replaced with the fiberwise smash product; we have

$\displaystyle S^{V \oplus W} \simeq S^V \wedge_X S^W,$

where ${\wedge_X}$ denotes a fiberwise smash product.

One reason is that this is of interest is that the group ${KO(X)}$ of vector bundles on a space ${X}$ is often very computable, thanks to Bott periodicity which identifies the ${KO}$-groups of a point. The set of spherical fibrations is much harder to describe: to describe the spherical fibrations over ${S^n}$ essentially amounts to computing a bunch of homotopy groups of spheres.