Submersions by Lie algebroids

TL;DR

This paper explores submersions by Lie algebroids, providing new proofs of local normal forms, establishing equivalences with Ehresmann connections, and extending cohomology localization results in the Lie algebroid setting.

Contribution

It introduces the notion of submersions by Lie algebroids, proves their equivalence to Ehresmann connections, and extends cohomology localization theorems within this framework.

Findings

New proof of local normal form for Lie algebroid transversals.

Equivalence between locally trivial submersions and Ehresmann connections.

Extension of de Rham cohomology localization theorem using local coefficients.

Abstract

In this note, we examine the bundle picture of the pullback construction of Lie algebroids. The notion of submersions by Lie algebroids is introduced, which leads to a new proof of the local normal form for lie algebroid transversals of [Bursztyn et al., Crelle, 2017], and which we use to deduce that Lie algebroids transversals concentrate all local cohomology. The locally trivial version of submersions by Lie algebroids $\mathfrak{S}$ is then discussed, and we show that this notion is equivalent to the existence of a complete Ehresmann connection for $\mathfrak{S}$, extending the main result in [del Hoyo, Indag. Math. 2016]. Finally, we show that locally trivial version of submersions by Lie algebroids gives rise to a system of local coefficients, which is an integral part of a version of the homotopy invariance of de Rham cohomology in the context of Lie algebroids, and we apply such local systems to extend the localization theorem of [Chen et al. 2006].