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Homotopy type of stabilizers of circle-valued functions with non-isolated singularities on surfaces

Bohdan Feshchenko

TL;DR

The paper extends Maksymenko’s framework for stabilizers of circle-valued functions to a broad class $\mathcal{F}(M,P)$ of functions on oriented surfaces that may have non-isolated singularities, including Morse–Bott-type features. It introduces $H$-like vector fields $F$ associated to $f$ so that $f$ is constant along trajectories, and proves the stabilizer $\mathcal{S}_{\mathrm{id}}(f)$ is either contractible or homotopy equivalent to $S^1$; this dichotomy is governed by the presence of saddles, degenerate extrema, and the extremal circles $E_f$. The core approach combines shift-function analysis along $F$-trajectories, a robust fibration $\rho: \mathcal{S}_{\mathrm{id}}(f)\to \mathcal{D}_{\mathrm{id}}(E_f)$ with fiber $ \mathcal{G}(f,E_f)$, and a computation of $\pi_0 \mathcal{G}(f,E_f)$ which yields a free abelian structure of rank $|E_f|$ or $|E_f|-1$ depending on (T). The results generalize known classifications for isolated singularities to Morse–Bott-like settings, with implications for the homotopy types of stabilizers and related diffeomorphism groups on surfaces.

Abstract

The paper is devoted to the study of homotopy properties of stabilizers of smooth functions on oriented surfaces, i.e., groups of diffeomorphisms of surfaces preserving a given function. For some class of smooth functions which is a generalization of the class of Morse-Bott functions on oriented surfaces, the homotopy type of the connected component of the identity map of the stabilizer is completely described.

Homotopy type of stabilizers of circle-valued functions with non-isolated singularities on surfaces

TL;DR

The paper extends Maksymenko’s framework for stabilizers of circle-valued functions to a broad class of functions on oriented surfaces that may have non-isolated singularities, including Morse–Bott-type features. It introduces -like vector fields associated to so that is constant along trajectories, and proves the stabilizer is either contractible or homotopy equivalent to ; this dichotomy is governed by the presence of saddles, degenerate extrema, and the extremal circles . The core approach combines shift-function analysis along -trajectories, a robust fibration with fiber , and a computation of which yields a free abelian structure of rank or depending on (T). The results generalize known classifications for isolated singularities to Morse–Bott-like settings, with implications for the homotopy types of stabilizers and related diffeomorphism groups on surfaces.

Abstract

The paper is devoted to the study of homotopy properties of stabilizers of smooth functions on oriented surfaces, i.e., groups of diffeomorphisms of surfaces preserving a given function. For some class of smooth functions which is a generalization of the class of Morse-Bott functions on oriented surfaces, the homotopy type of the connected component of the identity map of the stabilizer is completely described.
Paper Structure (27 sections, 18 theorems, 45 equations, 3 figures)

This paper contains 27 sections, 18 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Layout of the paper
  3. Discussions of the proof of Theorem \ref{['thm:main-theorem']}
  4. Generalities on class $\mathcal{F}$
  5. Homogeneous polynomials
  6. Critical circles of $f$
  7. Foliated neighborhoods
  8. Smooth shifts along trajectories of flows
  9. Generalities
  10. Shift-map
  11. Singular foliations on the plane on horizontal lines
  12. Shifts along $F_{\mu,n}$
  13. $H$-fields for functions from $\mathcal{F}(M,P)$
  14. Hamiltonian vector fields
  15. Hamiltonian vector fields for functions from $\mathcal{F}(M,P)$
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $M$ be a smooth, compact, connected, and oriented surface, and let $f$ be a function from $\mathcal{F}(M,P)$. Then $\mathcal{S}_{\mathrm{id}}(f)$ is either contractible or homotopy equivalent to a circle $S^1$. To be more precise, $\mathcal{S}_{\mathrm{id}}(f)$ is contractible if either $f$ has

Figures (3)

  • Figure 1: Foliations by homogeneous polynomials: a degenerated extreme (left) and a $3$-saddle (right)
  • Figure 2: Neighborhood of a critical circle $C$ on an oriented surface: (a) $n_C$ is even, (b) $n_C$ is odd
  • Figure 3: Trajectories of $F_{\mu,n}$, $n\geq 1$ when $n$ is even (left) and $n$ is odd (right)

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Lemma 3.3: Lemma 2, Proposition 3, Theorem 19 in Maksymenko:TA:2003; Lemma 2.3 Maksymenko:OsakaJM:2011 or Lemma 6.1 Maksymenko:DefFuncI:2014
  • Lemma 3.4: Theorem 12, Proposition 14 in Maksymenko:TA:2003, Theorem 1.1 Maks:reparam-sh-map
  • Lemma 4.2: cf. p.24 case (3) in Maksymenko:OsakaJM:2011
  • proof
  • Proposition 5.4
  • Definition 5.5
  • proof : Proof of Proposition \ref{['prop:vector-field-f']}
  • ...and 21 more