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: 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...

This paper is a progress report on our efforts to understand the homotopy theory underlying Floer homology; its objectives are as follows:

Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.

Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of G-valued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category

... categories. The associated cellular nerve of an o-category extends the well-known simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of A-algebras for each o-operad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of A-algebras (i.e. weak o-categories) is

this paper we develop combinatorial and representation-theoretic methods in the theory of arrangements of linear subspaces in R , which can be applied to the study of the cohomology of spaces of ordered and unordered point configurations in d-dimensional real space. In particular we provide and apply tools for determining representations of finite groups on the rational cohomology of the complement of an arrangement of linear subspaces. The connection between arrangements of linear subspaces and configuration spaces was first suggested by Bjorner [Bj4, 8.5], who was himself inspired by the work of Arnol'd [Ar1], [Ar2], [Ar3] and Vassiliev [Va]. Although the relation between hyperplane arrangements and configuration spaces had been established a long time ago by Fadell and Neuwirth [Fa-Ne], the idea of using subspace arrangements as a unifying approach in this context seems to be recent. In this paper we consider the following examples : (A) Let M be the set of all n = k + q tuples (x 1 ; : : : ; xn ) of points x i 2 R such that there is no subset E of f1; : : : ; ng of cardinality k satisfying x t = x s for all t; s 2 E. In particular M is the pure braid space (see [Fa-Ne], [Ar1]) for k = d = 2. (See [Co-La-Ma] for k = 2 and d 2: For general k the space M is first mentioned in [Co-Lu] in connection with a generalisation of the Borsuk-Ulam Theorem.) The cohomology of the spaces M was determined in [Bj-We] for d = 1; 2 (and implicitly for general d). We study the action of the symmetric group Sn on M by permuting the coordinates. For d = 2 the orbit space M ) =Sn is homeomorphic to the space of monic polynomials of degree n with no root of multiplicity k. For k = 2 this is the complement of the discriminant; it is thus the braid space...

We provide a "toolkit " of basic lemmas for the comparison of homotopy types of homotopy colimits of diagrams of spaces over small categories. We show how this toolkit can be used on quite different fields of applications. We demonstrate this with respect to 1. Bjorner's "Generalized Homotopy Complementation Formula" [4], 2. the topology of toric varieties, 3. the study of homotopy types of arrangements of subspaces, 4. the analysis of homotopy types of subgroup complexes.

In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids. Unlike previous work that focused on 2-manifolds with restricted geometry, our theory is general and localizes arbitrary-dimensional attributes in arbitrary spaces. We implement our algorithm to validate our approach in practice. 1