A symplectic groupoid G.: = (G1 ⇉ G0) determines a Poisson structure on G0. In this case, we call G. a symplectic groupoid of the Poisson manifold G0. However, not every Poisson manifold M has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing Morita equivalence in the category of all Poisson manifolds. In this paper, we construct symplectic Weinstein groupoids which provide a solution to the above problem (Theorem 1.1). More precisely, we show that a symplectic Weinstein groupoid induces a Poisson structure on its base manifold, and that to every Poisson manifold there is an associated symplectic Weinstein groupoid. 1

Orbispaces are spaces with extra structure. The main examples come from topological group actions X G and are denoted [X/G], their underlying space, or coarse moduli space being X/G. By definition, every orbispace is locally of the form [X/G], but the group G might vary. We shall work with orbispaces whose coarse moduli spaces are CW-complexes, and whose stabilizer groups are compact Lie groups. We also require the stratification of the coarse moduli space by the type of stabilizer group to be compactible with the CW structure. A convenient model for an orbispace is then given by a topological groupoid (see [2], [3]). An orbispace always comes with a map to its coarse moduli space. By a suborbispace X ′ ⊂ X, we shall mean an orbispace obtained by pulling back along a subspace of the coarse moduli space. If X is an orbispace modeled by a topological groupoid G, then a vector bundle over X is a vector bundle over the space of objects of G equipped with an action of the arrows of G. It is tempting to define K-theory as the Grothendieck

Abstract. This is the first in a series of papers devoted to foundations of topological stacks. We begin developing a homotopy theory for topological stacks along the lines of classical homotopy theory of topological spaces. In this paper we go as far as introducing the homotopy groups and establishing their basic properties. We also develop a Galois theory of covering spaces for a (locally connected semilocally 1-connected) topological stack. Built into the Galois theory is a method for determining the stacky structure (i.e., inertia groups) of covering stacks. As a consequence, we get for free a characterization of topological stacks that are quotients of topological spaces by discrete group actions. For example, this give a handy characterization of good orbifolds. Orbifolds, graphs of groups, and complexes of groups are examples of topological (Deligne-Mumford) stacks. We also show that any algebraic stack (of finite type over C) gives rise to a topological stack. We also prove a Riemann Existence Theorem for stacks. In particular, the algebraic fundamental group

Abstract. The identities for elliptic gamma functions discovered by A. Varchenko and one of us are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three (it is a stack). Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curve. Contents

We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S 1-bundles and S 1-gerbes over differentiable stacks. In particular, we establish the relationship between S 1-gerbes and groupoid S 1-central extensions. We define connections and curvings for groupoid S 1-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S 1-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S 1-bundles and S 1-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of S 1-central extensions with prescribed curvature-like data.