Key points are not available for this paper at this time.
We prove the following dichotomy. Given an analytic equivalence relation E, either E₀^{N}BE or else any Borel homomorphism from E₀^N to E is "very far from a reduction", specifically, it factors, on a comeager set, through the projection map (2^N) ^N (2^N) ᵏ for some k. As a corollary, we prove that E₀^N is a prime equivalence relation, answering a question on Clemens.
Assaf Shani (Fri,) studied this question.