Abstract

A surprising theorem of Wilton-Zalesski says that the geometric structure of a geometric 3-manifold can be determined purely from the set of finite quotients of its fundamental group, or equivalently the profinite completion. This raises the question of what other geometric and topological invariants can be seen in the profinite completion. Recall that the Stiefel-Whitney classes of a manifold are characteristic classes in mod 2 cohomology that detect important properties, such as orientability, the unoriented bordism class, and admitting a spin structure. In joint work with Sam Hughes, we show that for compact aspherical manifolds with fundamental group that is good in the sense of Serre, these characteristic classes are invariants of the profinite completion. This raises the possibility of applications to questions of profinite rigidity, and as a sample application we are able to show the profinite rigidity of the fundamental group of a flat 6-manifold.