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Transfers in coarse homology 粗同调中的转移
arXiv: K-Theory and Homology Pub Date : 2018-09-01 DOI: 10.17879/90169656968
U. Bunke, A. Engel, Daniel Kasprowski, Christoph Winges
{"title":"Transfers in coarse homology","authors":"U. Bunke, A. Engel, Daniel Kasprowski, Christoph Winges","doi":"10.17879/90169656968","DOIUrl":"https://doi.org/10.17879/90169656968","url":null,"abstract":"We enlarge the category of bornological coarse spaces by adding transfer morphisms and introduce the notion of an equivariant coarse homology theory with transfers. We then show that equivariant coarse algebraic $K$-homology and equivariant coarse ordinary homology can be extended to equivariant coarse homology theories with transfers. In the case of a finite group we observe that equivariant coarse homology theories with transfers provide Mackey functors. We express standard constructions with Mackey functors in terms of coarse geometry, and we demonstrate the usage of transfers in order to prove injectivity results about assembly maps.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114827789","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
Algebraic bivariant $K$-theory and Leavitt path algebras. 代数双变K理论与Leavitt路径代数。
arXiv: K-Theory and Homology Pub Date : 2018-06-24 DOI: 10.4171/JNCG/397
Guillermo Cortiñas, Diego Montero
{"title":"Algebraic bivariant $K$-theory and Leavitt path algebras.","authors":"Guillermo Cortiñas, Diego Montero","doi":"10.4171/JNCG/397","DOIUrl":"https://doi.org/10.4171/JNCG/397","url":null,"abstract":"This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a commutative ground ring $ell$. In this first article we consider Leavitt path algebras of general graphs over general ground rings; the second article will focus mostly on purely infinite simple unital Leavitt path algebras over a field. Bivariant algebraic $K$-theory $kk$ is the universal homology theory with the properties above; we prove a structure theorem for unital Leavitt path algebras in $kk$. We show that under very mild assumptions on $ell$, for a graph $E$ with finitely many vertices and reduced incidence matrix $A_E$, the structure of $L(E)$ depends only on the isomorphism classes of the cokernels of the matrix $I-A_E$ and of its transpose, which are respectively the $kk$ groups $KH^1(L(E))=kk_{-1}(L(E),ell)$ and $KH_0(L(E))=kk_0(ell,L(E))$. Hence if $L(E)$ and $L(F)$ are unital Leavitt path algebras such that $KH_0(L(E))cong KH_0(L(F))$ and $KH^1(L(E))cong KH^1(L(F))$ then no homology theory with the above properties can distinguish them. We also prove that for Leavitt path algebras, $kk$ has several properties similar to those that Kasparov's bivariant $K$-theory has for $C^*$-graph algebras, including analogues of the Universal coefficient and Kunneth theorems of Rosenberg and Schochet.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130740877","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: 11
First and second $K$-groups of an elliptic curve over a global field of positive characteristic 椭圆曲线在正特征整体域上的第一和第二K群
arXiv: K-Theory and Homology Pub Date : 2017-11-15 DOI: 10.5802/aif.3202
S. Kondo, S. Yasuda
{"title":"First and second $K$-groups of an elliptic curve over a global field of positive characteristic","authors":"S. Kondo, S. Yasuda","doi":"10.5802/aif.3202","DOIUrl":"https://doi.org/10.5802/aif.3202","url":null,"abstract":"In this paper, we show that the maximal divisible subgroup of groups $K_1$ and $K_2$ of an elliptic curve $E$ over a function field is uniquely divisible. Further those $K$-groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of $E$, which is an elliptic surface over a finite field.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117347636","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
A new approach to twisted K–theory of compact Lie groups 紧李群扭曲k理论的一种新方法
arXiv: K-Theory and Homology Pub Date : 2017-08-18 DOI: 10.2140/agt.2020.20.135
Jonathan Rosenberg
{"title":"A new approach to twisted K–theory of compact Lie groups","authors":"Jonathan Rosenberg","doi":"10.2140/agt.2020.20.135","DOIUrl":"https://doi.org/10.2140/agt.2020.20.135","url":null,"abstract":"This paper explores further the computation of the twisted K-theory and K-homology of compact simple Lie groups, previously studied by Hopkins, Moore, Maldacena-Moore-Seiberg, Braun, and Douglas, with a focus on groups of rank 2. We give a new method of computation based on the Segal spectral sequence which seems to us appreciably simpler than the methods used previously, at least in many key cases. The exposition has been clarified and one mistake in the previous version has been fixed. Also the references have been updated.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123393198","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: 2
The algebraic and topological K-theory of the Hilbert Modular Group Hilbert模群的代数和拓扑k理论
arXiv: K-Theory and Homology Pub Date : 2017-06-14 DOI: 10.4310/HHA.2018.V20.N2.A19
Luis Jorge S'anchez Saldana, Mario Vel'asquez
{"title":"The algebraic and topological K-theory of the Hilbert Modular Group","authors":"Luis Jorge S'anchez Saldana, Mario Vel'asquez","doi":"10.4310/HHA.2018.V20.N2.A19","DOIUrl":"https://doi.org/10.4310/HHA.2018.V20.N2.A19","url":null,"abstract":"In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Connes conjecture respectively. We also construct a model for the classifying space of the Hilbert modular group for the family of virtually cyclic subgroups.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128171741","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
Coarse assembly maps 粗装配图
arXiv: K-Theory and Homology Pub Date : 2017-06-07 DOI: 10.4171/jncg/410
U. Bunke, A. Engel
{"title":"Coarse assembly maps","authors":"U. Bunke, A. Engel","doi":"10.4171/jncg/410","DOIUrl":"https://doi.org/10.4171/jncg/410","url":null,"abstract":"A coarse assembly map relates the coarsification of a generalized homology theory with a coarse version of that homology theory. In the present paper we provide a motivic approach to coarse assembly maps. To every coarse homology theory $E$ we naturally associate a homology theory $Emathcal{O}^{infty}$ and construct an assembly map $$mu_{E} :mathrm{Coarsification}(Emathcal{O}^{infty})to E .$$ For sufficiently nice spaces $X$ we relate the value $Emathcal{O}^{infty}(X)$ with the locally finite homology of $X$ with coefficients in $E(*)$. In the example of coarse $K$-homology we discuss the relation of our motivic constructions with the classical constructions using $C^{*}$-algebra techniques.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121216388","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: 17
Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property 相对几何装配与映射锥，第二部分:Chern特征与Novikov性质
arXiv: K-Theory and Homology Pub Date : 2017-05-23 DOI: 10.17879/85169762441
R. Deeley, M. Goffeng
{"title":"Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property","authors":"R. Deeley, M. Goffeng","doi":"10.17879/85169762441","DOIUrl":"https://doi.org/10.17879/85169762441","url":null,"abstract":"We study Chern characters and the assembly mapping for free actions using the framework of geometric $K$-homology. The focus is on the relative groups associated with a group homomorphism $phi:Gamma_1to Gamma_2$ along with applications to Novikov type properties. In particular, we prove a relative strong Novikov property for homomorphisms of hyperbolic groups and a relative strong $ell^1$-Novikov property for polynomially bounded homomorphisms of groups with polynomially bounded cohomology in $C$. As a corollary, relative higher signatures on a manifold with boundary $W$, with $pi_1(partial W)to pi_1(W)$ belonging to the class above, are homotopy invariant.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129160568","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
Bounds for the rank of the finite part of operator $K$-theory 算子K有限部分的秩界
arXiv: K-Theory and Homology Pub Date : 2017-05-21 DOI: 10.4171/jncg/333
Süleyman Kağan Samurkaş
{"title":"Bounds for the rank of the finite part of operator $K$-theory","authors":"Süleyman Kağan Samurkaş","doi":"10.4171/jncg/333","DOIUrl":"https://doi.org/10.4171/jncg/333","url":null,"abstract":"We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds for the structure group $S(M)$ and the group of positive scalar curvature metrics $P(M)$ for an oriented manifold $M$. \u0000We define a class of groups called \"polynomially full groups\" for which the upper bound and the lower bound we derive are the same. We show that the class of polynomially full groups contains all virtually nilpotent groups. As example, we give explicit formulas for the ranks of the finite parts of operator $K$-theory groups for the finitely generated abelian groups, the symmetric groups and the dihedral groups.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123821756","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: 6
Bivariant KK -Theory and the Baum–Connes conjecure 双变KK理论与Baum-Connes猜想
arXiv: K-Theory and Homology Pub Date : 2017-03-31 DOI: 10.1007/978-3-319-59915-1_3
S. Echterhoff
{"title":"Bivariant KK -Theory and the Baum–Connes conjecure","authors":"S. Echterhoff","doi":"10.1007/978-3-319-59915-1_3","DOIUrl":"https://doi.org/10.1007/978-3-319-59915-1_3","url":null,"abstract":"","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114283571","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: 10
The K -Theory of Toric Schemes Over Regular Rings of Mixed Characteristic 混合特征正则环上环形格式的K -理论
arXiv: K-Theory and Homology Pub Date : 2017-03-22 DOI: 10.1007/978-3-319-96827-8_19
Guillermo Cortiñas, C. Haesemeyer, M. Walker, C. Weibel
{"title":"The K -Theory of Toric Schemes Over Regular Rings of Mixed Characteristic","authors":"Guillermo Cortiñas, C. Haesemeyer, M. Walker, C. Weibel","doi":"10.1007/978-3-319-96827-8_19","DOIUrl":"https://doi.org/10.1007/978-3-319-96827-8_19","url":null,"abstract":"","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126883352","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
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