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arXiv · 2408.01261

Generic dichotomy for homomorphisms for $E_0^\mathbb{N}$

Assaf Shani
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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.

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Assaf Shani. 2024-08-02. Generic dichotomy for homomorphisms for $E_0^\mathbb{N}$. https://arxiv.org/abs/2408.01261

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