Genus Integration, Abelianization and Extended Monodromy

TL;DR

This paper explores the conditions under which the abelianization of a Lie algebroid admits a smooth integration, linking obstructions to extended monodromy groups and generalizing classical topological results.

Contribution

It introduces a path-space construction for the abelianization groupoid, relates obstructions to extended monodromy, and connects prequantization with smooth abelian integrations.

Findings

Obstructions to smooth abelian integration are related to extended monodromy groups.

A path-space construction for the abelianization groupoid is developed.

Prequantization conditions are equivalent to the smoothness of an abelian integration.

Abstract

Given a Lie algebroid we discuss the existence of a smooth abelian integration of its abelianization. We show that the obstructions are related to the extended monodromy groups introduced recently in \cite{CFMb}. We also show that this groupoid can be obtained by a path-space construction, similar to the Weinstein groupoid of \cite{CF1}, but where the underlying homotopies are now supported in surfaces with arbitrary genus. As an application, we show that the prequantization condition for a (possibly non-simply connected) manifold is equivalent to the smoothness of an abelian integration. Our results can be interpreted as a generalization of the classical Hurewicz theorem.