[Paper Review] A fast convergence theorem for nearly multiplicative connections on proper Lie groupoids

Giorgio Trentinaglia|arXiv (Cornell University)|Mar 9, 2014

Homotopy and Cohomology in Algebraic Topology25 references3 citations

TL;DR

This paper establishes a fast convergence theorem for deforming nearly multiplicative connections into genuinely multiplicative connections on proper Lie groupoids using a recursive averaging technique. The key contribution is a constructive method ensuring convergence in the $C^k$-topology, reducing the study of multiplicative connections to longitudinal representations on regular groupoids and paving the way for an obstruction theory.

ABSTRACT

Motivated by the study of a certain family of classical geometric problems we investigate the existence of multiplicative connections on proper Lie groupoids. We show that one can always deform a given connection which is only approximately multiplicative into a genuinely multiplicative connection. The proof of this fact that we present here relies on a recursive averaging technique. As an application we point out that the study of multiplicative connections on general proper Lie groupoids reduces to the study of longitudinal representations of regular groupoids. We regard our results as a preliminary step towards the elaboration of an obstruction theory for multiplicative connections.

Motivation & Objective

• To address the global problem of existence and deformation of multiplicative (Cartan) connections on Lie groupoids, especially when source fibers are not simply connected.
• To generalize classical obstructions to $G$-structures by studying multiplicative connections on proper Lie groupoids.
• To develop a systematic method for deforming approximately multiplicative connections into genuine multiplicative ones, applicable even in non-simply connected settings.
• To reduce the analysis of multiplicative connections on general proper Lie groupoids to the study of longitudinal representations on regular groupoids.
• To lay foundational tools for an eventual obstruction theory for the existence and deformation of multiplicative connections.

Proposed method

• A recursive averaging procedure is applied to a nearly multiplicative connection to iteratively correct its failure to be multiplicative.
• The method relies on a normalized Haar system and integration over the groupoid's structure, ensuring compatibility with the groupoid multiplication.
• The convergence is proven in the $C^k$-topology using iterative estimates on the error between successive approximations.
• A key technical step involves extending compactly supported sections and using Haar integration to define a continuous averaging operator on sections of associated vector bundles.
• The construction is shown to be $C^k$-continuous via local coordinate reductions and partitions of unity.
• The method is applied to connections and generalized to pseudo-representations, establishing a unified framework for both cases.

Experimental results

Research questions

• RQ1 Under what conditions can a nearly multiplicative connection on a proper Lie groupoid be deformed into a genuinely multiplicative connection?
• RQ2 How can the existence of multiplicative connections on general proper Lie groupoids be reduced to the study of longitudinal representations on regular groupoids?
• RQ3 What is the rate of convergence of iterative averaging procedures in deforming approximate multiplicative structures?
• RQ4 Can the theory of multiplicative connections be extended to non-simply connected source fibers, where classical path-lifting methods fail?
• RQ5 What are the obstructions to the existence and deformation of multiplicative connections on proper Lie groupoids?

Key findings

• A recursive averaging technique ensures fast convergence of nearly multiplicative connections to genuine multiplicative connections in the $C^k$-topology.
• The convergence is guaranteed for any proper Lie groupoid, even when source fibers are not simply connected.
• The method constructs a multiplicative connection from any given connection that is only approximately multiplicative, via a sequence of corrections.
• The study of multiplicative connections on general proper Lie groupoids reduces to analyzing longitudinal representations on regular groupoids.
• The averaging operator is shown to be $C^k$-continuous, ensuring the method’s robustness in the smooth category.
• The results provide a foundational framework for developing an obstruction theory for multiplicative connections on proper Lie groupoids.