Uniform first order interpretation of the second order theory of countable groups of homeomorphisms
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915850667163648 |
|---|---|
| author | Koberda, Thomas González, J. de la Nuez |
| author_facet | Koberda, Thomas González, J. de la Nuez |
| contents | We show that the first order theory of the homeomorphism group of a compact manifold interprets the full second order theory of countable groups of homeomorphisms of the manifold. The interpretation is uniform across manifolds of bounded dimension. As a consequence, many classical problems in group theory and geometry (e.g.~the linearity of mapping classes of compact $2$--manifolds) are encoded as elementary properties of homeomorphism groups of manifolds. Furthermore, the homeomorphism group uniformly interprets the Borel and projective hierarchies of the homeomorphism group, which gives a characterization of definable subsets of the homeomorphism group. Finally, we prove analogues of Rice's Theorem from computability theory for homeomorphism groups of manifolds. As a consequence, it follows that the collection of sentences that isolate the homeomorphism group of a particular manifold, or that isolate the homeomorphism groups of manifolds in general, is not definable in second order arithmetic, and that membership of particular sentences in these collections cannot be proved in ZFC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16334 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform first order interpretation of the second order theory of countable groups of homeomorphisms Koberda, Thomas González, J. de la Nuez Group Theory Geometric Topology Logic We show that the first order theory of the homeomorphism group of a compact manifold interprets the full second order theory of countable groups of homeomorphisms of the manifold. The interpretation is uniform across manifolds of bounded dimension. As a consequence, many classical problems in group theory and geometry (e.g.~the linearity of mapping classes of compact $2$--manifolds) are encoded as elementary properties of homeomorphism groups of manifolds. Furthermore, the homeomorphism group uniformly interprets the Borel and projective hierarchies of the homeomorphism group, which gives a characterization of definable subsets of the homeomorphism group. Finally, we prove analogues of Rice's Theorem from computability theory for homeomorphism groups of manifolds. As a consequence, it follows that the collection of sentences that isolate the homeomorphism group of a particular manifold, or that isolate the homeomorphism groups of manifolds in general, is not definable in second order arithmetic, and that membership of particular sentences in these collections cannot be proved in ZFC. |
| title | Uniform first order interpretation of the second order theory of countable groups of homeomorphisms |
| topic | Group Theory Geometric Topology Logic |
| url | https://arxiv.org/abs/2312.16334 |