I’ve been reading lately about the Sullivan conjecture and its proof (which is the subject of a course that Kirsten Wickelgren is teaching next semester). The resolution of this conjecture and work related to it led to a great deal of interesting algebra in the 1980s and 1990s, which I’ve been trying to understand a little about. Some useful references here are Haynes Miller’s 1984 paper, Lionel Schwartz’s book, and Jacob Lurie’s course notes.

1. Motivation

Let ${X}$ be a variety over the complex numbers. The complex points ${X(\mathbb{C})}$ are a topological space that has a homotopy type, which is often of interest. Étale homotopy theory (a refinement of étale cohomology) allows one to give a purely algebraic description of the profinite completion ${\widehat{X(\mathbb{C})}}$ of the homotopy type of ${X(\mathbb{C})}$. If ${X}$ is defined over the real numbers, though, then one can also study the topological space ${X(\mathbb{R})}$ of real points of ${X}$; one has

$\displaystyle X(\mathbb{R}) = X(\mathbb{C})^{\mathbb{Z}/2}$

for the conjugation action on ${X(\mathbb{C})}$. (more…)