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.

The signature ${\sigma(M)}$ of a ${4k}$-dimensional compact, oriented manifold ${M}$ is a classical cobordism invariant of ${M}$; the so-called Hirzebruch signature formula states that ${\sigma(M)}$ can be computed as a complicated polynomial in the Pontryagin classes of the tangent bundle ${TM}$ (evaluated on the fundamental class of ${M}$). When ${M}$ is four-dimensional, for instance, we have

$\displaystyle \sigma(M) = \frac{p_1}{3}.$

This implies that the Pontryagin number ${p_1}$ 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 ${M}$ is a four-dimensional spin-manifold, then ${\sigma(M)}$ is divisible by ${16}$ (and so ${p_1}$ by ${48}$).

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.  (more…)