No Borel upper bound for the $F_σ$ equivalence relations

Abstract

We prove that the family of coset relations of $F_σ$ real linear subspaces of Banach space $\ell_1$ has no common upper bound among Borel equivalence relations in the sense of Borel reducibility. These relations are indexed by a well-founded tree. Consequently, on a Polish space, a universal $Σ^0_α$ equivalence relation exists exactly when $α=1$, and a universal $Π^0_α$ equivalence relation exists exactly when $α=1$ or $2$. This strengthens the earlier nonexistence theorem for a universal $F_σ$ equivalence relation. This also answers a question of Kanovei and Lyubetsky about universal equivalence relations in Borel classes.