Lie algebroid Fibrations

TL;DR

This paper introduces a notion of fibration for Lie algebroids using super manifold language, establishes homotopy groups, and relates integrability obstructions to a transgression map within a long exact sequence.

Contribution

It defines a new concept of fibration for Lie algebroids, develops associated homotopy theory, and links integrability obstructions to this framework.

Findings

Defined Lie algebroid fibrations via Ehresmann connections.

Established homotopy groups and a long exact sequence for these fibrations.

Connected integrability obstructions to a transgression map in the sequence.

Abstract

A degree 1 non-negative graded super manifold equipped with a degree 1 vector field Q satisfying [Q, Q]=1, namely a so-called NQ-1 manifold is, in plain differential geometry language, a Lie algebroid. We introduce a notion of fibration for such super manifols, that essentially involves a complete Ehresmann connection. As it is the case for Lie algebras, such fibrations turn out not to be just locally trivial products. We also define homotopy groups and prove the expected long exact sequence associated to a fibration. In particular, Crainic and Fernandes's obstruction to the integrability of Lie algebroids is interpreted as the image of a transgression map in this long exact sequence.