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First order rigidity of homeomorphism groups of manifolds

Sang-hyun Kim, Thomas Koberda, J. de la Nuez González

TL;DR

This work proves a strong form of reconstruction: for every compact, connected manifold M there exists a single group-theoretic sentence φ_M such that another pair (N,H) in the vol- or non-vol class satisfies φ_M exactly when N is homeomorphic to M, thereby establishing first-order rigidity of homeomorphism groups. The authors build an expressive multi-sorted language AGAPE that uniformly interprets action structures, regular open sets, and arithmetical data inside the homeomorphism group G, enabling them to encode points, dimensions, and collar neighborhoods within G. They then interpret second-order arithmetic and analysis inside this framework, construct definable parametrizations of balls and collars, and finally complete the proof by aligning these definable topological features with Cheeger–Kister-type stability results to recover the homeomorphism type. The results generalize classical reconstruction theorems (e.g., Whittaker) and show that the full homeomorphism group’s first-order theory encodes rich geometric information, with potential implications for understanding rigidity phenomena across topological categories. The paper also outlines several intriguing open questions about extensions, weaker hypotheses, and higher-regularity analogs.

Abstract

For every compact, connected manifold $M$, we prove the existence of a sentence $φ_M$ in the language of groups such that the homeomorphism group of another compact manifold $N$ satisfies $φ_M$ if and only if $N$ is homeomorphic to $M$. We prove the analogous statement for groups of homeomorphisms preserving an Oxtoby--Ulam probability measure.

First order rigidity of homeomorphism groups of manifolds

TL;DR

This work proves a strong form of reconstruction: for every compact, connected manifold M there exists a single group-theoretic sentence φ_M such that another pair (N,H) in the vol- or non-vol class satisfies φ_M exactly when N is homeomorphic to M, thereby establishing first-order rigidity of homeomorphism groups. The authors build an expressive multi-sorted language AGAPE that uniformly interprets action structures, regular open sets, and arithmetical data inside the homeomorphism group G, enabling them to encode points, dimensions, and collar neighborhoods within G. They then interpret second-order arithmetic and analysis inside this framework, construct definable parametrizations of balls and collars, and finally complete the proof by aligning these definable topological features with Cheeger–Kister-type stability results to recover the homeomorphism type. The results generalize classical reconstruction theorems (e.g., Whittaker) and show that the full homeomorphism group’s first-order theory encodes rich geometric information, with potential implications for understanding rigidity phenomena across topological categories. The paper also outlines several intriguing open questions about extensions, weaker hypotheses, and higher-regularity analogs.

Abstract

For every compact, connected manifold , we prove the existence of a sentence in the language of groups such that the homeomorphism group of another compact manifold satisfies if and only if is homeomorphic to . We prove the analogous statement for groups of homeomorphisms preserving an Oxtoby--Ulam probability measure.
Paper Structure (33 sections, 38 theorems, 247 equations, 3 figures)

This paper contains 33 sections, 38 theorems, 247 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Elementary equivalence implies homeomorphism
  3. Outline of the paper
  4. Preliminaries
  5. Transitivity of balls in manifolds
  6. Regular open sets and homeomorphism groups
  7. First order logic
  8. The AGAPE structure and basic observations
  9. The langauge L-AGAPE and the structure AGAPE(M,G)
  10. Interpreting action structures in homeomorphism groups
  11. First order descriptions of basic topological properties
  12. Further topological properties
  13. Relative-compactness regarding good balls
  14. Detecting finiteness of components
  15. Touching and containing the boundary
  16. ...and 18 more sections

Key Result

Theorem 1.2

Let $M$ and $N$ be compact manifolds, and suppose is an isomorphism of groups. Then there exists a homeomorphism such that for all $f\in\mathop{\mathrm{Homeo}}\nolimits(M)$, we have $\phi(f)=\psi\circ f\circ\psi^{-1}$.

Figures (3)

  • Figure 1: The proof of Theorem \ref{['thm:output-balls']} (2) and a potential issue when $U$ is not "spaciously collared".
  • Figure 2: The condition $\operatorname{COL}(M,G;\underline{K})$.
  • Figure 3: Parts (c) and (d) of the condition $\operatorname{EMB}(M,G;f)$.

Theorems & Definitions (71)

  • Remark 1.1
  • Theorem 1.2: See whittaker
  • Theorem 1.3: Rubin1989
  • Theorem 1.4
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • Theorem 2.1: von Neumann vonNeumann1961, Oxtoby--Ulam oxtoby-ulam
  • ...and 61 more