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54
Quadratic functions in geometry, topology,and mtheory
"... 2. Determinants, differential cocycles and statement of results 5 ..."
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Cited by 51 (5 self)
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2. Determinants, differential cocycles and statement of results 5
Combinatorial operad actions on cochains
, 2001
"... A classical Einfinity 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 FKconstruction for infinite loop spaces. The purpose of this article i ..."
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Cited by 51 (18 self)
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A classical Einfinity 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 FKconstruction 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 BarrattEccles operad. We also prove that differential graded algebras over the BarrattEccles 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 BarrattEccles operad which is equivalent to the classical little squares operad.
Spaces over a Category and Assembly Maps in Isomorphism Conjectures in Kand LTheory
"... : We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete ..."
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Cited by 49 (12 self)
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: We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory 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 Ltheory, BaumConnes Conjecture, assembly maps, spaces and spectra over a category AMSclassification number: 57 Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G=H, considered as left Gsets, and morphisms are Gmaps. This is a useful construct for o...
Topology and Data
, 2008
"... An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that ..."
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Cited by 30 (0 self)
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An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data
Combinatorial descriptions of the homotopy groups of certain spaces
 Math. Proc. Camb. Philos. Soc
"... Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3sphere are combinatorially given. 1. ..."
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Cited by 28 (21 self)
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Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3sphere are combinatorially given. 1.
Homology of generalized partition posets
 Journal of Pure and Applied Algebra, Volume 208, Issue
, 2007
"... 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 CohenMacaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the ..."
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Cited by 20 (4 self)
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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 CohenMacaulay. 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.
Automorphisms of manifolds and algebraic Ktheory: I, KTheory 1
 Stanford University, Stanford
, 1988
"... 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 hcobordism stable range, assuming n ≥ 5. (That is, we produce a highly connected map from S ..."
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Cited by 16 (2 self)
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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 hcobordism 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.
Cubical Sets And Their Site
 Theory Appl. Categ
, 2003
"... Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicia ..."
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Cited by 15 (3 self)
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Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid ; in fact, K is the classifying category of a monoidal algebraic theory of such monoids. Analogous results are given for the restricted cubical site I of ordinary cubical sets (just faces and degeneracies) and for the intermediate site J (including connections). We also consider briefly the reversible analogue, !K.