Convexity of Momentum Maps: A Topological Analysis

TL;DR

This paper extends the topological principles underlying convexity theorems for momentum maps to more general, non-closed maps with non-metric convexity structures, broadening the scope of convexity analysis.

Contribution

It introduces a new factorization technique and generalizes convexity results to non-closed maps and arbitrary connected spaces, removing previous restrictions.

Findings

Convexity of the image is established without local fiber connectedness.

The results apply to maps with target spaces lacking metric-based convexity.

A new topological framework for momentum map convexity is developed.

Abstract

The Local-to-Global-Principle used in the proof of convexity theorems for momentum maps has been extracted as a statement of pure topology enriched with a structure of convexity. We extend this principle to not necessarily closed maps $f\colon X\ra Y$ where the convexity structure of the target space $Y$ need not be based on a metric. Using a new factorization of $f$, convexity of the image is proved without local fiber connectedness, and for arbitrary connected spaces $X$.