Set theory, logic, and homeomorphism groups of manifolds

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 the regularity hypothesis that every projective set of reals has the Baire property, 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, under the same Baire-property hypothesis, that every manifold of positive dimension admits pairs of homeomorphisms with the same type which are not conjugate to each other. Projective determinacy implies the Baire-property hypothesis, so the corresponding consequences under PD follow immediately. Finally, we show that infinitary formulas 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 $L_{ω_1ω}$ formula. Similarly, the homeomorphism type of an arbitrary connected manifold is determined by a single $L_{ω_1ω}$ sentence.