{"title":"Geometric obstructions for Fredholm boundary conditions for manifolds with corners","authors":"P. C. Rouse, J. Lescure","doi":"10.2140/akt.2018.3.523","DOIUrl":"https://doi.org/10.2140/akt.2018.3.523","url":null,"abstract":"For every connected manifold with corners we use a homology theory called conormal homology, defined in terms of faces and incidences and whose cycles correspond geometrically to corner's cycles. Its Euler characteristic (over the rationals, dimension of the total even space minus the dimension of the total odd space), $chi_{cn}:=chi_0-chi_1$, is given by the alternated sum of the number of (open) faces of a given codimension. The main result of the present paper is that for a compact connected manifold with corners $X$ given as a finite product of manifolds with corners of codimension less or equal to three we have that \u00001) If $X$ satisfies the Fredholm Perturbation property (every elliptic pseudodifferential b-operator on $X$ can be perturbed by a b-regularizing operator so it becomes Fredholm) then the even Euler corner character of $X$ vanishes, i.e. $chi_0(X)=0$. \u00002) If the even Periodic conormal homology group vanishes, i.e. $H_0^{pcn}(X)=0$, then $X$ satisfies the stably homotopic Fredholm Perturbation property (i.e. every elliptic pseudodifferential b-operator on $X$ satisfies the same named property up to stable homotopy among elliptic operators). \u00003) If $H_0^{pcn}(X)$ is torsion free and if the even Euler corner character of $X$ vanishes, i.e. $chi_0(X)=0$ then $X$ satisfies the stably homotopic Fredholm Perturbation property. For example for every finite product of manifolds with corners of codimension at most two the conormal homology groups are torsion free. \u0000The main theorem behind the above result is the explicit computation in terms of conormal homology of the $K-$theory groups of the algebra $mathcal{K}_b(X)$ of $b$-compact operators for $X$ as above. Our computation unifies the only general cases covered before, for codimension zero (smooth manifolds) and for codimension 1 (smooth manifolds with boundary).","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2018.3.523","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41840965","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}
{"title":"Topological K-theory of affine Hecke algebras","authors":"M. Solleveld","doi":"10.2140/akt.2018.3.395","DOIUrl":"https://doi.org/10.2140/akt.2018.3.395","url":null,"abstract":"Let H(R,q) be an affine Hecke algebra with a positive parameter function q. We are interested in the topological K-theory of H(R,q), that is, the K-theory of its C*-completion C*_r (R,q). We will prove that $K_* (C*_r (R,q))$ does not depend on the parameter q. For this we use representation theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras. \u0000Thus, for the computation of these K-groups it suffices to work out the case q=1. These algebras are considerably simpler than for q not 1, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine $K_* (C*_r (R,q))$ for all classical root data R, and for some others as well. This will be useful to analyse the K-theory of the reduced C*-algebra of any classical p-adic group. \u0000For the computations in the case q=1 we study the more general situation of a finite group Gamma acting on a smooth manifold M. We develop a method to calculate the K-theory of the crossed product $C(M) rtimes Gamma$. In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2016-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2018.3.395","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67938811","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}
{"title":"Localization C∗-algebras and K-theoretic\u0000duality","authors":"M. Dadarlat, R. Willett, Jianchao Wu","doi":"10.2140/akt.2018.3.615","DOIUrl":"https://doi.org/10.2140/akt.2018.3.615","url":null,"abstract":"Based on the localization algebras of Yu, and their subsequent analysis by Qiao and Roe, we give a new picture of KK-theory in terms of time-parametrized families of (locally) compact operators that asymptotically commute with appropriate representations.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2016-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/akt.2018.3.615","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67938860","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}