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. We 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.
{"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}
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.
{"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}
Pub Date : 2014-02-03DOI: 10.1515/CRELLE-2014-0132
A. Carey, Jens Kaad
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.
R. W. Carey和J. Pincus在[CaPi86]中提出了可分离Hilbert空间H上的非fredholm有界算子T的和指标理论，使得TT* - T*T在迹类中。我们在[CGK13]中表明，使用作用于R^{2n}上束的部分上的狄拉克型算子，我们可以构造出满足(1-TT*)^n - (1-T*T)^n是迹类的更一般条件的有界算子T。我们提出了这些狄拉克型算子的“同调”指标，由Tr((1-TT*)^n - (1-T*T)^n给出。本文证明了[CGK13]中引入的指标表示由T和T*生成的代数的一个循环同调理论与其对偶上同调理论之间的配对结果。这使我们建立了同伦指标的同伦不变性(在循环理论的意义上)。然后，我们能够以非常一般的方式定义某些无界算子的同调指标，并证明该指标在一类无界扰动下的不变性。
{"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}
Pub Date : 2014-01-30DOI: 10.1017/9781316771327.011
Mark Ullmann
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. We 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.
{"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}
Pub Date : 2014-01-30DOI: 10.1017/IS014002004JKT254
P. Hu, I. Kríz, P. Somberg
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 $K_G(G)$.
{"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}
Pub Date : 2014-01-30DOI: 10.4310/HHA.2014.V16.N2.A2
Thomas H. Geisser
We show that motivic homology, motivic Borel-Moore homology and higher Chow groups satisfy homological descent for hyperenvelopes, and l-hyperenvelopes after inverting l.
{"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}
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.
{"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}
Pub Date : 2014-01-10DOI: 10.1142/S0218216514500175
Patrick Dehornoy, V. Lebed
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.
{"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}
Pub Date : 2013-11-25DOI: 10.1515/CRELLE-2014-0154
Guillermo Cortiñas, Gisela Tartaglia
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.
{"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}
Pub Date : 2013-11-20DOI: 10.4310/HHA.2015.V17.N1.A13
Tom Harris
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.
{"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}
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