A homotopical consequence of branched covers

TL;DR

The paper establishes a connection between the profinite completion of a pseudomanifold and its étale homotopy type derived from branched covers, confirming a conjecture by Sullivan from 1970.

Contribution

It proves that the profinite completion equals the étale homotopy type of branched covers, based on the existence of sufficient $K(,1)$ subspaces in pseudomanifolds.

Findings

Proves the equivalence between profinite completion and étale homotopy type.

Confirms Sullivan's conjecture from 1970.

Shows the existence of enough $K(,1)$ subspaces in pseudomanifolds.

Abstract

We prove that the profinite completion of a pseudomanifold is the Artin-Mazur's etale homotopy type construction on its branched covers, which was implicitly conjectured by Sullivan in his MIT note (page 247) around 1970. This is a consequence of the existence of enough $K(\pi,1)$ open dense subspaces in a pseudomanifold.