Set theory, logic, and homeomorphism groups of manifolds

TL;DR

This paper explores the connections between set theory and the structure of homeomorphism groups of manifolds, revealing how different axioms influence their logical properties and classification.

Contribution

It establishes the first-order rigidity of homeomorphism groups under V=L and demonstrates how projective determinacy affects their elementary equivalence and conjugacy classes.

Findings

Homeomorphism groups are first-order rigid under V=L.

Conjugacy classes are determined by types in the first-order theory.

Under PD, non-homeomorphic manifolds can have elementarily equivalent homeomorphism groups.

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 $L_{\omega_1\omega}$ sentence. Similarly, the homeomorphism type of an arbitrary connected manifold is determined by a single $L_{\omega_1\omega}$ sentence.