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Connectedness of fibers beyond semitoric systems I: the non-degenerate case

Daniele Sepe, Susan Tolman

TL;DR

The paper addresses when fibers of an integrable system $(M,\omega,V,f=(\Phi,g))$ are connected, focusing on non-degenerate singularities that extend complexity-one $T$-spaces with proper moment maps. By constructing tall local models and applying a Morse-theoretic reduction on the reduced spaces $\Phi^{-1}(\beta)/T$, it proves a key proposition: if each critical point of $g$ modulo $\Phi$ admits local Morse charts on the reduced space, then $\Phi^{-1}(\beta)/T$ becomes a smooth oriented surface with $\overline{g}$ Morse, with index-1 points encoding hyperbolic blocks with connected stabilizers. The main result asserts a practical sufficient condition for fiber connectedness (no tall hyperbolic-block points with connected $T$-stabilizer) and demonstrates necessity under certain topological or genericity assumptions; it also extends semitoric fiber-connectedness to higher dimensions via a Morse-theoretic framework. Collectively, the work advances the topology of moment-map fibers in non-toric integrable systems and informs classification and quantization in higher-dimensional settings.

Abstract

In this paper we study the connectedness of the fibers of integrable systems that have only non-degenerate singular points and extend complexity one $T$-spaces with proper moment maps. Our main result states that if there are no tall singular points with a hyperbolic block and connected $T$-stabilizer, then each fiber is connected. Moreover, we prove that the above condition is necessary if either some reduced space is simply connected or the moment map for the integrable system is generic in a natural sense.

Connectedness of fibers beyond semitoric systems I: the non-degenerate case

TL;DR

The paper addresses when fibers of an integrable system are connected, focusing on non-degenerate singularities that extend complexity-one -spaces with proper moment maps. By constructing tall local models and applying a Morse-theoretic reduction on the reduced spaces , it proves a key proposition: if each critical point of modulo admits local Morse charts on the reduced space, then becomes a smooth oriented surface with Morse, with index-1 points encoding hyperbolic blocks with connected stabilizers. The main result asserts a practical sufficient condition for fiber connectedness (no tall hyperbolic-block points with connected -stabilizer) and demonstrates necessity under certain topological or genericity assumptions; it also extends semitoric fiber-connectedness to higher dimensions via a Morse-theoretic framework. Collectively, the work advances the topology of moment-map fibers in non-toric integrable systems and informs classification and quantization in higher-dimensional settings.

Abstract

In this paper we study the connectedness of the fibers of integrable systems that have only non-degenerate singular points and extend complexity one -spaces with proper moment maps. Our main result states that if there are no tall singular points with a hyperbolic block and connected -stabilizer, then each fiber is connected. Moreover, we prove that the above condition is necessary if either some reduced space is simply connected or the moment map for the integrable system is generic in a natural sense.
Paper Structure (7 sections, 24 theorems, 25 equations)

This paper contains 7 sections, 24 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Tall local models
  3. The key proposition
  4. Purely elliptic type
  5. The semitoric case
  6. Non-degenerate singular points
  7. Proof of the main result

Key Result

Theorem 1.6

Let $\left(M,\omega, {\mathfrak t} \times {\mathbb R}, f:=(\Phi,g)\right)$ be an integrable system such that $(M,\omega,\Phi)$ is a complexity one $T$-space with a proper moment map. Assume that each tall singular point in $\left(M,\omega, {\mathfrak t} \times {\mathbb R}, f\right)$ is non-degenerat

Theorems & Definitions (73)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Example 1.9
  • Theorem 1.10
  • ...and 63 more