In the previous post, I described the Sullivan conjecture and gave a vague outline of its proof by Miller. Shortly after Miller’s paper, various applications of the theorem to other problems in homotopy theory were discovered.

The intuition here is that there are two ways a space might appear to be finite: one is that its homotopy groups might be bounded and the other is that its homology groups might be bounded. It’s generally very hard for the two to happen at once, at least in the simply connected case. For instance, Eilenberg-MacLane spaces — the basic examples of spaces with bounded homotopy groups — have very messy cohomology. Similarly, finite complexes — spaces with bounded homology — generally have very complicated homotopy groups.

The purpose of this post is to explain a proof of the following result, conjectured by Serre:

Theorem 5 (McGibbon, Neisendorfer) Let ${X}$ be a simply connected finite complex such that ${\widetilde{H}_*(X; \mathbb{Z}/p) \neq 0}$. Then ${\pi_i X}$ contains ${p}$-torsion for infinitely many ${i}$.

As we’ll see, this result can be proved using the Sullivan conjecture. (more…)