Set theory, logic, and homeomorphism groups of manifolds

Summary · qwen2.5:32b

The article explores the interplay between set theory axioms and the first-order properties of homeomorphism groups of manifolds, showing that under V=L, these groups are first-order rigid with conjugacy classes determined by type, while under projective sets having the Baire property, noncompact connected manifolds can have elementarily equivalent but non-homeomorphic groups. Notably, infinitary $L_{\omega_1\omega}$ formulas determine both conjugacy classes of homeomorphisms and homeomorphism types of manifolds.

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Exploring the interplay between set theory and group theory in the context of homeomorphism groups of manifolds.

Excerpt

arXiv:2512.05206v2 Announce Type: replace 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 $L_{\omega_1\omega}$ formula. Similarly, the homeomorphism type of an arbitrary connected manifold is determined by a single $L_{\omega_1\omega}$ sentence.