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

Fix n=1, 2, 3, or n=. 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, =^+n is in the spectrum of the meager ideal. This extends a result of Kanovei, Sabok, and Zapletal for n=1; (2) =^+ is a regular equivalence relation. This answers positively a question of Clemens; (3) for n<, 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.