{"title":"Generic dichotomy for homomorphisms for $E_0^\\mathbb{N}$","authors":"Assaf Shani","doi":"arxiv-2408.01261","DOIUrl":null,"url":null,"abstract":"We prove the following dichotomy. Given an analytic equivalence relation $E$,\neither ${E_0^{\\mathbb{N}}}\\leq_B{E}$ or else any Borel homomorphism from\n$E_0^{\\mathbb{N}}$ to $E$ is \"very far from a reduction\", specifically, it\nfactors, on a comeager set, through the projection map\n$(2^{\\mathbb{N}})^{\\mathbb{N}}\\to (2^{\\mathbb{N}})^k$ for some\n$k\\in\\mathbb{N}$. As a corollary, we prove that $E_0^{\\mathbb{N}}$ is a prime\nequivalence relation, answering a question on Clemens.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.01261","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove the following dichotomy. Given an analytic equivalence relation $E$,
either ${E_0^{\mathbb{N}}}\leq_B{E}$ or else any Borel homomorphism from
$E_0^{\mathbb{N}}$ to $E$ is "very far from a reduction", specifically, it
factors, on a comeager set, through the projection map
$(2^{\mathbb{N}})^{\mathbb{N}}\to (2^{\mathbb{N}})^k$ for some
$k\in\mathbb{N}$. As a corollary, we prove that $E_0^{\mathbb{N}}$ is a prime
equivalence relation, answering a question on Clemens.