On Relative Cohomology in Lie Theory

TL;DR

This paper develops the theory of relative cohomology for Lie groupoids and algebroids, clarifying their relationship and applications to characteristic classes in geometric structures.

Contribution

It introduces the notion of relative cohomology in Lie groupoid and algebroid theory, filling a gap in the literature and linking it via van Est maps.

Findings

Established the structural framework for relative cohomology in Lie groupoids and algebroids.

Demonstrated the relation between groupoid and algebroid cohomology through van Est maps.

Provided an intrinsic approach to defining characteristic classes using the developed theory.

Abstract

Motivated by our attempt to understand characteristic classes of Lie groupoids and geometric structures, we are brought back to the fundamentals of the cohomology theories of Lie groupoids and algebroids. One element that was missing in the literature was the notion of relative cohomology in this setting. The main aim of this paper is to develop the structural theory of this notion, the relation between the relative cohomology of groupoids and that of algebroids via van Est maps, and to indicate how it can be used to provide an intrinsic definition of characteristic classes.