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.