The finite Friedman-Stanley jumps: generic dichotomies for Borel homomorphisms

Fix n=1,2,3, or n=ω. We prove a dichotomy for Borel homomorphisms from the n-th Friedman-Stanley jump =⁺ⁿ to an equivalence relation E which is classifiable by countable structures: if there is no reduction from =⁺ⁿ to E, then in fact all Borel homomorphisms are very far from a reduction. For this we use a different presentation of =⁺ⁿ, equivalent up to Borel bi-reducibility, which is susceptible to Baire-category techniques. This dichotomy is seen as a method for proving positive Borel reducibility results from =⁺ⁿ. As corollaries we prove: (1) for n≤ω, =⁺ⁿ is in the spectrum of the meager ideal. This extends a result of Kanovei, Sabok, and Zapletal for $n=1$; (2) =+ω is a regular equivalence relation. This answers positively a question of Clemens; (3) for n<ω, the equivalence relations, classifiable by countable structures, which do not Borel reduce =⁺ⁿ are closed under countable products. This extends a result of Kanovei, Sabok, and Zapletal for $n=1$.