1.1. Simplicial sets 5 1.2. Symmetric spectra 6

: We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K- and L-theory of integral group rings and to the Baum-Connes Conjecture on the topological K-theory of reduced group C -algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and the study of the associated generalized homology and cohomology theories and homotopy limits. Key words: Algebraic K and L-theory, Baum-Connes Conjecture, assembly maps, spaces and spectra over a category AMS-classification number: 57 Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G=H, considered as left G-sets, and morphisms are G-maps. This is a useful construct for o...

Étale groupoids arise naturally as models for leaf spaces of foliations, for orbifolds, and for orbit spaces of discrete group actions. In this paper we introduce a sheaf homology theory for etale groupoids. We prove its invariance under Morita equivalence, as well as Verdier duality between Haefliger cohomology and this homology. We also discuss the relation to the cyclic and Hochschild homologies of Connes' convolution algebra of the groupoid, and derive some spectral sequences which serve as a tool for the computation of these homologies.

We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the Feigin-Tsygan-Nistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the non-Hausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...

Abstract. A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson K-ring are proved and the Poisson K-rings are calculated for a number of examples. In particular, for the zero Poisson structure the K-ring is the ordinary K 0-ring of the manifold and for the dual space to a Lie algebra the K-ring is the ring of virtual representations of the Lie algebra. It is also shown that the K-ring is an invariant of Morita equivalence. Moreover, the K-ring is a functor on a category, the weak

This paper arose from an attempt to identify and understand the underlying algebraic topological aspects of Floer theory. In studying the homotopy theoretic aspects of this type of infinite dimensional Morse theory, one is naturally led to re-examine finite dimensional Morse theory. The purpose of this paper is to describe a method of processing the data provided by finite dimensional Morse theory in a way that generalises naturally to these infinite dimensional settings. In a sequel we will describe how this method allows us to associate to the data provided by Floer theory suitable spaces whose homotopy types yield invariants which include and generalise the various forms of Floer homology. The method also gives a particularly clean way of viewing finite dimensional Morse theory. The idea is to associate to a Morse function f : M ! R on a closed Riemannian manifold M a category C f whose objects are the critical points of f . The morphisms between two critical points a and b are, in a natural sense, "piecewise flow lines" of the gradient flow of f which connect a to b. Given a piecewise flow line connecting critical points a and b and one connecting critical points b and c there is an obvious way of joining them to get a piecewise flow line connecting a to c. This defines the composition law in the category C f . The goal of this paper is to show how to explicitly recover the topology of M from the category C f . More precisely, given a topological category C one can construct its classifying space BC, see [7]. We will describe the classifying space BC f in detail in x3 but for now recall that it is a simplicial space whose k-simplices are parameterized by the space of

Perhaps one of the most influential ideas in Smale’s seminal 1967 paper [20] is that a general dynamical system should have a structure something like that of a gradient dynamical system. In pursuit of this idea Smale defined Axiom A No-Cycle systems and showed that they had three gradient like properties:

Abstract. In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen’s Theorem A.

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