Set theory, logic, and homeomorphism groups of manifolds

TL;DR

The paper analyzes how set-theoretic assumptions shape the logical classification of manifolds via their homeomorphism groups. It proves a sharp contrast: under V = L the homeomorphism groups of connected manifolds are first-order rigid, while under PD non-classification phenomena appear across dimensions, including non-homeomorphic manifolds with elementarily equivalent homeomorphism groups and non-conjugate yet type-equivalent homeomorphisms. It further shows that infinitary logics (L_{ω1ω}) can single-handedly encode homeomorphism types and conjugacy classes, and that these rigidity properties can be extended via product constructions and dimension-augmented frameworks. Finally, the authors establish that non-classification phenomena are consistent with ZFC (via forcing), underscoring the independence of such rigidity results from standard axioms. The work highlights deep connections between descriptive set theory, infinitary logic, and geometric topology in understanding the limits of classifying manifolds by group-theoretic or model-theoretic data.

Abstract

We investigate the relationship between axiomatic set theory and the first-order theory of homeomorphism groups of manifolds in the language of group theory, concentrating on first-order rigidity and type versus conjugacy. We prove that under the axiom of constructibility (i.e. {V=L}), homeomorphism groups of arbitrary connected manifolds are first-order rigid, and that the conjugacy class of a homeomorphism of a manifold is determined by its type. In contradistinction, under projective determinacy (PD), we show that in all dimensions greater than one, there exist pairs of noncompact, connected manifolds whose homeomorphism groups are elementarily equivalent but which are not homeomorphic. We also show that under PD, every manifold of positive dimension admits pairs of homeomorphisms with the same type which are not conjugate to each other. Finally, we show that infinitary sentences do determine conjugacy classes of homeomorphisms and homeomorphism types of manifolds; specifically, the conjugacy class of a homeomorphism of an arbitrary manifold is determined by a single $L_{ω_1ω}$ sentence. Similarly, the homeomorphism type of an arbitrary connected manifold is determined by a single $L_{ω_1ω}$ sentence.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1