1.1. Simplicial sets 5 1.2. Symmetric spectra 6

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.

2. Determinants, differential cocycles and statement of results 5

: 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...

Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3-sphere are combinatorially given. 1.

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Abstract. Let M be a closed topological n–manifold, and let S(M) be the moduli space of closed topological manifolds equipped with a homotopy equivalence to M. We give an algebraic description of S(M) in the h-cobordism stable range, assuming n ≥ 5. (That is, we produce a highly connected map from S(M) to another space having an algebraic description.) The algebraic description is in terms of L–theory, Waldhausen’s algebraic K–theory of spaces, and a natural transformation Ξ (constructed in our paper [WW2]) from L–theory to the Tate cohomology of Z2 acting on K–theory. We develop a parallel theory for the moduli space S(τ) of Rn –bundles on M equipped with an ”equivalence ” to the tangent bundle τ of M. (The equivalence is a stable fiber homotopy equivalence of the corresponding spherical fibrations.) Results about moduli spaces of smooth manifolds can be obtained by combining the calculations of S(M) and S(τ). We have attempted to make this paper as self–contained as possible by summarizing results from the earlier papers in the series where necessary.

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

Abstract. We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen-Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.

The maps from loop suspensions to loop spaces are investigated using group representations in this article. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. Moreover the obstructions to the classical exponent problem in homotopy theory are displayed in the extension groups of the dual of the important symmetric group modules Lie(n), as well as in the top cohomology of the Artin braid groups with coefficients in the top homology of the Artin pure braid groups.