The Adams conjecture I

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 1 Let 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.

 

Rohlin’s theorem on the signature of four-manifolds

The signature of a -dimensional compact, oriented manifold is a classical cobordism invariant of ; the so-called Hirzebruch signature formula states that can be computed as a complicated polynomial in the Pontryagin classes of the tangent bundle (evaluated on the fundamental class of ). When is four-dimensional, for instance, we have

This implies that the Pontryagin number must be divisible by three.

There are various further divisibility conditions that hold in special cases. Here is an important early example:

Theorem 1 (Rohlin) If is a four-dimensional spin-manifold, then is divisible by (and so by ).

I’d like to describe the original proof of Rohlin’s theorem, which relies on a number of tools from the 1950s era of topology. At least, I have not gotten a copy of Rohlin’s paper; the proof is sketched, though, in Kervaire-Milnor’s ICM address, which I’ll follow.