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Geometric Baum-Connes assembly map for twisted Differentiable Stacks 扭曲可微堆栈的几何Baum-Connes装配映射
arXiv: K-Theory and Homology Pub Date : 2014-02-14 DOI: 10.24033/ASENS.2283
P. C. Rouse, Bai-Ling Wang
{"title":"Geometric Baum-Connes assembly map for twisted Differentiable Stacks","authors":"P. C. Rouse, Bai-Ling Wang","doi":"10.24033/ASENS.2283","DOIUrl":"https://doi.org/10.24033/ASENS.2283","url":null,"abstract":"We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric twisted K-homology groups and the geometric assembly map are well defined for Lie groupoids is new, as it was only sketched by Connes in his book for general Lie groupoids without any restrictive hypothesis, in particular for non Hausdorff Lie groupoids. \u0000We also prove the Morita invariance of the assembly map, giving thus a precise meaning to the geometric assembly map for twisted differentiable stacks. We discuss the relation of the assembly map with the associated assembly map of the $S^1$-central extension. The relation with the analytic assembly map is treated, as well as some cases in which we have an isomorphism. One important tool is the twisted Thom isomorphism in the groupoid equivariant case which we establish in the appendix.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114277098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
K-theory of derivators revisited 衍生子的k理论
arXiv: K-Theory and Homology Pub Date : 2014-02-08 DOI: 10.2140/akt.2017.2.303
F. Muro, G. Raptis
{"title":"K-theory of derivators revisited","authors":"F. Muro, G. Raptis","doi":"10.2140/akt.2017.2.303","DOIUrl":"https://doi.org/10.2140/akt.2017.2.303","url":null,"abstract":"We define a $K$-theory for pointed right derivators and show that it agrees with Waldhausen $K$-theory in the case where the derivator arises from a good Waldhausen category. This $K$-theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator $K$-theory, as originally defined, is the best approximation to Waldhausen $K$-theory by a functor that is invariant under equivalences of derivators.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134086990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 14
Topological invariance of the homological index 同调指标的拓扑不变性
arXiv: K-Theory and Homology Pub Date : 2014-02-03 DOI: 10.1515/CRELLE-2014-0132
A. Carey, Jens Kaad
{"title":"Topological invariance of the homological index","authors":"A. Carey, Jens Kaad","doi":"10.1515/CRELLE-2014-0132","DOIUrl":"https://doi.org/10.1515/CRELLE-2014-0132","url":null,"abstract":"R. W. Carey and J. Pincus in [CaPi86] proposed and index theory for non-Fredholm bounded operators T on a separable Hilbert space H such that TT* - T*T is in the trace class. We showed in [CGK13] using Dirac-type operators acting on sections of bundles over R^{2n} that we could construct bounded operators T satisfying the more general condition that (1-TT*)^n - (1-T*T)^n is trace class. We proposed there a \"homological\" index for these Dirac-type operators given by Tr( (1-TT*)^n - (1-T*T)^n ). In this paper we show that the index introduced in [CGK13] represents the result of a pairing between a cyclic homology theory for the algebra generated by T and T* and its dual cohomology theory. This leads us to establish homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128526026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Controlled Algebra for Simplicial Rings and Algebraic K-theory 简单环的控制代数与代数k理论
arXiv: K-Theory and Homology Pub Date : 2014-01-30 DOI: 10.1017/9781316771327.011
Mark Ullmann
{"title":"Controlled Algebra for Simplicial Rings and Algebraic K-theory","authors":"Mark Ullmann","doi":"10.1017/9781316771327.011","DOIUrl":"https://doi.org/10.1017/9781316771327.011","url":null,"abstract":"We develop a version of controlled algebra for simplicial rings. This generalizes the methods which lead to successful proofs of the algebraic K- theory isomorphism conjecture (Farrell-Jones Conjecture) for a large class of groups. This is the first step to prove the algebraic K-theory isomorphism conjecture for simplicial rings. We construct a category of controlled simplicial modules, show that it has the structure of a Waldhausen category and discuss its algebraic K-theory. \u0000We lay emphasis on detailed proofs. Highlights include the discussion of a simplicial cylinder functor, the gluing lemma, a simplicial mapping telescope to split coherent homotopy idempotents, and a direct proof that a weak equivalence of simplicial rings induces an equivalence on their algebraic K-theory. Because we need a certain cofinality theorem for algebraic K-theory, we provide a proof and show that a certain assumption, sometimes omitted in the literature, is necessary. Last, we remark how our setup relates to ring spectra.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122321673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Equivariant K-theory of compact Lie groups with involution 具有对合的紧李群的等变k理论
arXiv: K-Theory and Homology Pub Date : 2014-01-30 DOI: 10.1017/IS014002004JKT254
P. Hu, I. Kríz, P. Somberg
{"title":"Equivariant K-theory of compact Lie groups with involution","authors":"P. Hu, I. Kríz, P. Somberg","doi":"10.1017/IS014002004JKT254","DOIUrl":"https://doi.org/10.1017/IS014002004JKT254","url":null,"abstract":"For a compact simply connected simple Lie group $G$ with an involution $alpha$, we compute the $Grtimes Z/2$-equivariant K-theory of $G$ where $G$ acts by conjugation and $Z/2$ acts either by $alpha$ or by $gmapsto alpha(g)^{-1}$. We also give a representation-theoretic interpretation of those groups, as well as of $K_G(G)$.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122326821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Homological Descent for Motivic Homology Theories 动机同源理论的同源下降
arXiv: K-Theory and Homology Pub Date : 2014-01-30 DOI: 10.4310/HHA.2014.V16.N2.A2
Thomas H. Geisser
{"title":"Homological Descent for Motivic Homology Theories","authors":"Thomas H. Geisser","doi":"10.4310/HHA.2014.V16.N2.A2","DOIUrl":"https://doi.org/10.4310/HHA.2014.V16.N2.A2","url":null,"abstract":"We show that motivic homology, motivic Borel-Moore homology and higher Chow groups satisfy homological descent for hyperenvelopes, and l-hyperenvelopes after inverting l.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130994336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
On the Baum-Connes conjecture for Gromov monster groups 关于Gromov怪物群的Baum-Connes猜想
arXiv: K-Theory and Homology Pub Date : 2014-01-27 DOI: 10.4171/JNCG/231
Martin Finn-Sell
{"title":"On the Baum-Connes conjecture for Gromov monster groups","authors":"Martin Finn-Sell","doi":"10.4171/JNCG/231","DOIUrl":"https://doi.org/10.4171/JNCG/231","url":null,"abstract":"We present a geometric approach to the Baum-Connes conjecture with coefficients for Gromov monster groups via a theorem of Khoskham and Skandalis. Secondly, we use recent results concerning the a-T-menability at infinity of large girth expanders to exhibit a family of coefficients for a Gromov monster group for which the Baum-Connes conjecture is an isomorphism.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124686940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Two- and three-cocycles for Laver tables 两圈和三圈的紫菜桌
arXiv: K-Theory and Homology Pub Date : 2014-01-10 DOI: 10.1142/S0218216514500175
Patrick Dehornoy, V. Lebed
{"title":"Two- and three-cocycles for Laver tables","authors":"Patrick Dehornoy, V. Lebed","doi":"10.1142/S0218216514500175","DOIUrl":"https://doi.org/10.1142/S0218216514500175","url":null,"abstract":"We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of finite structures obeying the left-selfdistributivity law; in particular, we describe simple explicit bases. This provides a number of new positive braid invariants and paves the way for further potential topological applications. Incidentally, we establish and study a partial ordering on Laver tables given by the right-divisibility relation.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125436378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
Compact operators and algebraic $K$-theory for groups which act properly and isometrically on Hilbert space Hilbert空间上适当等距作用群的紧算子和代数K理论
arXiv: K-Theory and Homology Pub Date : 2013-11-25 DOI: 10.1515/CRELLE-2014-0154
Guillermo Cortiñas, Gisela Tartaglia
{"title":"Compact operators and algebraic $K$-theory for groups which act properly and isometrically on Hilbert space","authors":"Guillermo Cortiñas, Gisela Tartaglia","doi":"10.1515/CRELLE-2014-0154","DOIUrl":"https://doi.org/10.1515/CRELLE-2014-0154","url":null,"abstract":"We prove the $K$-theoretic Farrell-Jones conjecture for groups as in the title with coefficient rings and $C^*$-algebras which are stable with respect to compact operators. We use this and Higson-Kasparov's result that the Baum-Connes conjecture with coefficients holds for such groups, to show that if $G$ is as in the title then the algebraic and the $C^*$-crossed products of $G$ with a stable $C^*$-algebra have the same $K$-theory.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114811491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Algebraic proofs of some fundamental theorems in algebraic $K$-theory 代数K理论中若干基本定理的代数证明
arXiv: K-Theory and Homology Pub Date : 2013-11-20 DOI: 10.4310/HHA.2015.V17.N1.A13
Tom Harris
{"title":"Algebraic proofs of some fundamental theorems in algebraic $K$-theory","authors":"Tom Harris","doi":"10.4310/HHA.2015.V17.N1.A13","DOIUrl":"https://doi.org/10.4310/HHA.2015.V17.N1.A13","url":null,"abstract":"We present news proofs of the additivity, resolution and cofinality theorems for the algebraic $K$-theory of exact categories. These proofs are entirely algebraic, based on Grayson's presentation of higher algebraic $K$-groups via binary complexes.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113966383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
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