Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. Using the recognition principle of Fiedorowicz, we prove that spineless cacti are equivalent as operads to the little discs operad. It turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and Fiedorowicz' braided operad structure are given by the same explicit chains. We also prove that spineless cacti and cacti are homotopy equivalent to their normalized versions as quasi-operads by showing that both types of cacti are semi-direct products of the quasi-operad of their normalized versions with a re-scaling operad based on R >0 . Furthermore, we introduce the notion of bi-crossed products of quasi-operads and show that the cacti proper are a bi-crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S 1 . We also prove that this particular bi-crossed operad product is homotopy equivalent to the semi-direct product of the spineless cacti with the group S 1 . This implies that cacti are equivalent to the framed little discs operad. These results lead to new CW models for the little discs and the framed little discs operad.

For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, ``quantum,'' product, which respects the natural group grading. We prove there is a ring isomorphism, the stringy Chern character, from stringy K-theory to stringy cohomology, and a ring homomorphism from full orbifold K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy Grothendieck-Riemann-Roch for etale maps. We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's construction. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results simplify the definitions of Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of Abramovich-Graber-Vistoli's orbifold Chow. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler Resolution Conjecture holds for symmetric products. Our results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.Comment: Exposition improved and additional details provided. To appear in Inventiones Mathematica

This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co-chains of a Frobenius algebra. We also prove that a there is dg-PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co-cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply-connected manifold.In this second part, we discretize the operadic and PROPic structures of the first part. We also introduce the notion of operadic correlation functions and use them in conjunction with operadic maps from the cell level to the discretized objects to define the relevant actions.

Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne's conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology of Chas and Sullivan.From this description one obtains several useful corollaries [18]. The ones relevant to the present discussion are listed below. Corollary 2.16The quasi-operad of normalized spineless cacti is homotopy associative and thus its homology quasi-operad is an operad.Moreover as quasi-operads normalized spineless cacti and spineless cacti are homotopy equivalent via a homotopy of quasi-operads. And finally:Corollary 2.17 Normalized spineless cacti are operadically quasi-isomorphic spineless cacti. I.e. their homology operads are isomorphic.

Abstract. Using five basic principles we treat Gerstenhaber/Lie brackets, BV operators and Master equations appearing in mathematical and physical contexts in a unified way. The different contexts for this are given by the different types of (Feynman) graphs that underlie the particular situation.Two of the maxims we bring forth are (1) that extending to the non-connected graphs gives a commutative multiplication forming a part of the BV structure and (2) that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs, and is responsible for the BV operator being a differential.Our treatment results in uniform, general theorems. These allow us to prove new results and recover and connect many constructions that have appeared independently throughout the literature. The more general point of view also allows us to disentangle the necessary from the circumstantial.