Abstract. We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This has the following arithmetic consequence: let ℓ > 2 be prime and A a finite abelian ℓ-group. Then there exists Q = Q(A) such that, for q greater than Q, a positive fraction of quadratic extensions of Fq(t) have the ℓ-part of their class group isomorphic to A.

We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A∞-multiplication-we think of such algebras as A∞-algebras "with extra structure". As applications, we obtain an integral version of the Costello-Kontsevich-Soibelman moduli space action on the Hochschild complex of open TCFTs, the Tradler-Zeinalian action of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex.The Hochschild complex of an associative algebra A admits a degree 1 self-map, Connes-Rinehart's boundary operator B. If A is Frobenius, the (proven) cyclic Deligne conjecture says that B is the ∆-operator of a BV-structure on the Hochschild complex of A. In fact B is part of much richer structure, namely an action by the chain complex of Sullivan diagrams on the Hochschild complex [57,26,28,30]. A weaker version of Frobenius algebras, called here A ∞ -Frobenius algebras, yields instead an action by the chains on the moduli space of Riemann surfaces [11,37,26,28]. Most of these results use a very appealing recipe for constructing such operations introduced by Kontsevich in [38]. Starting from a model for the moduli of curves in terms of the combinatorial data of fatgraphs, the graphs can be used to guide the local-to-global construction of an operation on the Hochschild complex of an A ∞ -Frobenius algebra A -at every vertex of valence n, an n-ary trace is performed.In this paper we develop a general method for constructing explicit operations on the Hochschild complex of A ∞ -algebras "with extra structure", which contains these theorems as special cases. In constrast to the above, our method is global-to-local: we give conditions on a composable collection of operations that ensures that it acts on the Hochschild complex of algebras of a given type; by fiat these operations preserve composition, something that can be hard to verify in the setting of [38]. After constructing the operations globally, we then show how to read-off the action explicitly, so that formulas for individual operations can also be obtained. Doing this we recover the same formuli as in the local-to-global approach. Our construction can be seen as a formalization and extension of the method of [11] which considered the case of A ∞ -Frobenius algebras.Our main result, which we will explain now in more details, gave rise to new computations, including a complete description of the operations on the Hochschild complex of commutative algebras [35], a description of a large complex of operations on the Hochschild complex of commutative Frobenius algebras [34] and a description of the universal operations given any type of algebra [61].An A ∞ -algebra can be described as an enriched symmetric monoidal functor from a certain dg-category A ∞ to Ch, the dg-category of chain complexes over Z. The category A ∞ is what is called a dg-prop, a symmetric monoidal dg-category w...

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ABSTRACT. From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of C is trivial. Applying this theory to the little disk operad C 2 (which is an S 1 -operad) we obtain variations on Getzler's gravity operad, which we show governs the Chas-Sullivan string bracket. INTRODUCTIONIn their foundational paper [CS01], Chas and Sullivan constructed a variety of algebraic structures on the singular and S 1 -equivariant homology of the free loop spaceIn the beautiful [CJ02], Cohen and Jones showed that the Chas-Sullivan operations on H * (LM) are governed by an operad -the cactus operad, or equivalently, the two-dimensional framed little disk operad. One purpose of this paper is to do likewise for the Chas-Sullivan operations in the Borel equivariant homology, HRecall that Chas and Sullivan introduced the string bracket, a graded Lie bracket of dimension 2 − d:This was defined as follows: for classes a, b ∈ Hwhere · is the Chas-Sullivan loop product on H * (LM), τ is the S 1 -transferand p is the projection to the quotient, H * (LM) → H S 1 * (LM). It is remarkable that bracketing a ring multiplication with τ and p produces a Lie bracket. One is led to wonder whether this is an example of a construction of a more general nature. Furthermore, Chas and Sullivan define a family of k-ary operations m k :From an operadic point of view, this construction of operations of higher "arity" is very natural. The main application of this paper to string homology will be to give the action of an operad which governs these operations. In [Get94,Get95], Getzler defined the gravity operad Grav in the category of graded groups using the (open) moduli spaces of points in CP 1 :Grav(k) := ΣH * (M 0,k+1 ).