Convexity of Momentum Maps: A Topological Analysis

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 $X$.

💡 Research Summary

The paper revisits the classical convexity theorems for momentum maps—most notably the Atiyah–Guillemin–Sternberg convexity result—and isolates the “local‑to‑global principle” that underlies their proofs. Rather than treating this principle as a technical lemma tied to symplectic geometry, the author reformulates it as a purely topological statement enriched by an abstract convexity structure on the target space. This abstraction removes the dependence on any metric, allowing the theory to be applied to spaces where convexity is defined algebraically rather than geometrically.

The central construction is a new factorization of an arbitrary continuous map (f\colon X\to Y). The map is decomposed as
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