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…)