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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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: 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...
Homotopy Coherent Category Theory
, 1996
"... this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on: ..."
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Cited by 22 (6 self)
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this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:
Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using ..."
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Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.
Flexible sheaves
, 1996
"... This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector b ..."
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Cited by 3 (0 self)
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This is an unfinished explanation of the notion of “flexible sheaf”, that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See “Homotopy over the complex numbers and generalized de Rham cohomology ” (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to JardineIllusie’s theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in “Homotopy over the complex numbers and generalized de Rham cohomology”. 1 Added in August 1996: I am finally sending this to “Duke eprints ” because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopysheafification (by doing a certain operation n+2 times) seems to be useful, for example I need
Hecke algebras and cohomotopical Mackey functors
 Trans. Amer. Math. Soc
, 1999
"... Dedicated to Professor Hirosi Toda on his 70th birthday Abstract. In this paper, we define the concept of the cohomotopical Mackey functor, which is more general than the usual cohomological Mackey functor, and show that Hecke algebra techniques are applicable to cohomotopical Mackey functors. Our t ..."
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Dedicated to Professor Hirosi Toda on his 70th birthday Abstract. In this paper, we define the concept of the cohomotopical Mackey functor, which is more general than the usual cohomological Mackey functor, and show that Hecke algebra techniques are applicable to cohomotopical Mackey functors. Our theory is valid for any (possibly infinite) discrete group. Some applications to topology are also given. 1.
A DISCRETE MODEL OF S 1HOMOTOPY THEORY
, 2004
"... Abstract. We construct a discrete model of the homotopy theory of S 1spaces. We define a category P with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. P inherits a model structure from the model structures on the categories of simplicial sets and cycl ..."
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Abstract. We construct a discrete model of the homotopy theory of S 1spaces. We define a category P with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. P inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between P and the model category of S 1spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets. 1.
EQUIVARIANT ORIENTATION THEORY
"... Abstract. We give a long overdue theory of orientations of Gvector bundles, topological Gbundles, and spherical Gfibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equ ..."
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Abstract. We give a long overdue theory of orientations of Gvector bundles, topological Gbundles, and spherical Gfibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equivariant orientation such that every orientable Gvector bundle admits an orientation. Our focus here is on the geometric and homotopical aspects, rather than the cohomological aspects, of orientation theory. Orientations are described in terms of functors defined on equivariant fundamental groupoids of base Gspaces, and the essence of the theory is to construct an appropriate universal target category of Gvector bundles over orbit spaces G/H. The theory requires new categorical concepts and constructions that should be of interest in other subjects where analogous structures arise.
THE GROMOVLAWSONROSENBERG CONJECTURE FOR COCOMPACT FUCHSIAN GROUPS
, 2003
"... Abstract. We prove the GromovLawsonRosenberg conjecture for cocompact Fuchsian groups, thereby giving necessary and sufficient conditions for a closed spin manifold of dimension greater than four with fundamental group cocompact Fuchsian to admit a metric of positive scalar curvature. Given a smoo ..."
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Abstract. We prove the GromovLawsonRosenberg conjecture for cocompact Fuchsian groups, thereby giving necessary and sufficient conditions for a closed spin manifold of dimension greater than four with fundamental group cocompact Fuchsian to admit a metric of positive scalar curvature. Given a smooth closed manifold M n, it is a longstanding question to determine whether or not M admits a Riemannian metric of positive scalar curvature. Work of GromovLawson and SchoenYau shows that if N n admits positive scalar curvature and M is obtained from N by ksurgeries of codimension n −k ≥ 3, then M admits positive scalar curvature as well. In the case when M is spin, this surgery result implies the following. Bordism Theorem. ([10], [27]) Let Mn be a closed spin manifold, n ≥ 5, G = π1(M), and suppose u: M → BG induces the identity on the fundamental group. If there is a positively scalar curved spin manifold Nn and a map v: N → BG such that [M, u] = [N, v] ∈ ΩSpin n (BG), then M admits a metric of positive scalar curvature.
Equivariant Weak NEquivalences
"... first author acknowledges the Sao Paulo University hospitality and the FAPESPSao Paulo (Brasil) financial support during the time this work was completed. Received by the editors January 1996. Communicated by Y. Felix. 1991 Mathematics Subject Classification : Primary 55N91, 55P15, 55P91; Second ..."
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first author acknowledges the Sao Paulo University hospitality and the FAPESPSao Paulo (Brasil) financial support during the time this work was completed. Received by the editors January 1996. Communicated by Y. Felix. 1991 Mathematics Subject Classification : Primary 55N91, 55P15, 55P91; Secondary 55U35, 55Q91, 57S17. Key words and phrases : GCWspace, weak Gnequivalence, Gntype, isotropy ring, local coe#cient system, orbit category, Postnikov decomposition, universal covering. Bull. Belg. Math. Soc. 4 (1997), 265276 whilst an equivariant theory of ntypes from a Quillen model category theory viewpoint has recently been given by A.R. Garzon and J.G. Miranda ([8]). As yet however homological criteria for equivariant weak nequivalences have not been given and one of the aims of the present paper is to develop an equivariant version of a truncated Whitehead Theorem (cf. [16, 19] for its nontruncated form) with a list of some equivalent conditions for equivariant weak neq
Elmendorf’s Theorem for Cofibrantly Generated Model Categories
, 2010
"... Elmendorf’s Theorem in equivariant homotopy theory states that ..."