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 be a finite CW complex. Given a real -dimensional vector bundle , one can form the associated *spherical fibration* with fiber by endowing 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 1Let be the quotient of the Grothendieck group of vector bundles on by the relation that if have fiber homotopy equivalent sphere bundles (or rather, the group generated by it).

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

where denotes a fiberwise smash product.

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