The real section conjecture and Smith's fixed point theorem for pro-spaces

TL;DR

This paper proves a topological version of the section conjecture for profinite fundamental groups with group actions, leading to new results on real points of varieties and fixed points on complex curves.

Contribution

It introduces a topological approach to the section conjecture for pro-spaces with group actions, extending its applicability to real and complex algebraic varieties.

Findings

Section conjecture proven for real points of certain varieties

Derived fixed point results for finite group actions on complex curves

Established a topological analogue of the section conjecture for pro-spaces

Abstract

We prove a topological version of the section conjecture for the profinite completion of the fundamental group of finite CW-complexes equipped with the action of a group of prime order $p$ whose $p$-torsion cohomology can be killed by finite covers. As an application we derive the section conjecture for the real points of a large class of varieties defined over the field of real numbers and the natural analogue of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers.