The purpose of this paper is to prove the upper bound in Malle's conjecture on the distribution of finite extensions of F q (t) with specified Galois group. As in [EVW16], our result is based upon computations of the homology of braid groups with certain (exponential) coefficients. However, the approach in this paper is new, relying on a connection between the cohomology of Hurwitz spaces and the cohomology of quantum shuffle algebras.Theorem 1.2. For each integer m and each transitive G ≤ S m , there are constants C(G), Q(G), and e(G) such that, for all q > Q(G) and all X > 0,More generally, we prove in Theorem 7.8 an upper bound for the number of extensions of F q (t) with a given Galois group and conditions on the local monodromy at the ramified places of F q (t). We note that any nontrivial lower bound for N G (F q (t), X) for general G would give a positive answer to the inverse Galois problem for F q (t), in a strong quantitative sense; this result is beyond the reach of our techniques for now.The exponent e(G) in Theorem 1.2 may be taken to be d + 1, where d is the Gelfand-Kirillov dimension of a certain graded ring R: this ring may be presented either as a ring of components of the family of Hurwitz spaces described below, or as a free braided commutative algebra on a braided vector space V determined by G. An upper bound of d ≤ #G − 2 is given by Proposition 5.3; this is almost certainly not optimal. We note that the theorem would hold with Q(G) = 1 (that is, without any condition on q) if Hypothesis 5.9 below holds for the braided vector space V ǫ appearing in the proof; this seems a very interesting direction for future work.

Abstract. Conjecture F from [VW12] states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. We prove a generalization of this conjecture to the case of connected manifolds of dimension at least 2 and give an explicit homological stability range.

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