A pro-algebraic fundamental group for topological spaces

Abstract

Consider a connected topological space $X$ with a point $x\in X$ and let $K$ be a field with the discrete topology. We study the Tannakian category of finite dimensional (flat) vector bundles on $X$ and its Tannakian dual ${\pi }_K(X,x)$ with respect to the fibre functor in $x$. The maximal pro-étale quotient of ${\pi }_K(X,x)$ is the étale fundamental group of $X$ studied by Kucharczyk and Scholze. For well behaved topological spaces, ${\pi }_K(X,x)$ is the pro-algebraic completion of the ordinary fundamental group ${\pi }_1(X,x)$. We obtain some structural results on ${\pi }_K(X,x)$ by studying (pseudo-)torsors attached to its quotients. This approach uses ideas of Nori in algebraic geometry and a result of Deligne on Tannakian categories. We also calculate ${\pi }_K(X,x)$ for some generalized solenoids.

The article was submitted in happier times as a contribution to a special volume celebrating the 80th birthday of Professor Alexei Nikolaevich Parshin. Sadly this volume has now become his orbituary

References in corpus (1)

• On Nori's Fundamental Group Scheme