{"title":"Framed motives of relative motivic spheres","authors":"G. Garkusha, A. Neshitov, I. Panin","doi":"10.1090/TRAN/8386","DOIUrl":"https://doi.org/10.1090/TRAN/8386","url":null,"abstract":"The category of framed correspondences $Fr_*(k)$, framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. The aim of this paper is to prove the following results stated in [GP1]: for any $k$-smooth scheme $X$ and any $ngeq 1$ the map of simplicial pointed sheaves $(-,mathbb A^1//mathbb G_m)^{wedge n}_+to T^n$ induces a Nisnevich local level weak equivalence of $S^1$-spectra \u0000$$M_{fr}(Xtimes (mathbb A^1//mathbb G_m)^{wedge n})to M_{fr}(Xtimes T^n)$$ and the sequence of $S^1$-spectra \u0000$$M_{fr}(X times T^n times mathbb G_m) to M_{fr}(X times T^n timesmathbb A^1) to M_{fr}(X times T^{n+1})$$ is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of [GP1]. This computation is crucial for the whole machinery of framed motives [GP1].","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129315330","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":"A decomposition of equivariant K-theory in twisted equivariant K-theories","authors":"J. M. G'omez, B. Uribe","doi":"10.1142/S0129167X17500161","DOIUrl":"https://doi.org/10.1142/S0129167X17500161","url":null,"abstract":"For G a finite group and X a G-space on which a normal subgroup A acts trivially, we show that the G-equivariant K-theory of X decomposes as a direct sum of twisted equivariant K-theories of X parametrized by the orbits of the conjugation action of G on the irreducible representations of A. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of A to the subgroup of G which fixes the isomorphism class of the irreducible representation.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134093639","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":"On the Strong Novikov Conjecture of Locally Compact Groups for Low Degree Cohomology Classes","authors":"Yoshiyasu Fukumoto","doi":"10.14989/DOCTOR.K20046","DOIUrl":"https://doi.org/10.14989/DOCTOR.K20046","url":null,"abstract":"The main result of this paper is non-vanishing of the image of the index map from the $G$-equivariant $K$-homology of a proper $G$-compact $G$-manifold $X$ to the $K$-theory of the $C^{*}$-algebra of the group $G$. Under the assumption that the Kronecker pairing of a $K$-homology class with a low-dimensional cohomology class is non-zero, we prove that the image of this class under the index map is non-zero. Neither discreteness of the locally compact group $G$ nor freeness of the action of $G$ on $X$ are required. The case of free actions of discrete groups was considered earlier by B. Hanke and T. Schick.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133884257","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":"A1-homotopy invariants of corner skew Laurent polynomial algebras","authors":"Gonçalo Tabuada","doi":"10.4171/jncg/11-4-12","DOIUrl":"https://doi.org/10.4171/jncg/11-4-12","url":null,"abstract":"In this note we prove some structural properties of all the A1-homotopy invariants of corner skew Laurent polynomial algebras. As an application, we compute de mod-l algebraic K-theory of Leavitt path algebras using solely the kernel/cokernel of the incidence matrix. This leads naturally to some vanishing and divisibility properties of the algebraic K-theory of these algebras.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121249073","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, Whitehead groups and the Atiyah conjecture","authors":"P. Linnell, W. Luck","doi":"10.2140/akt.2018.3.33","DOIUrl":"https://doi.org/10.2140/akt.2018.3.33","url":null,"abstract":"Let Wh^w(G) be the K_1-group of square matrices over the integral group ring ZG which are not necessarily invertible but induce weak isomorphisms after passing to Hilbert space completions. Let D(G) be the division closure of ZG in the algebra U(G) of operators affiliated to the group von Neumann algebra. Let C be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let G be a torsionfree group which belongs to C. Then we prove that Wh^w(G) is isomorphic to K_1(D(G)). Furthermore we show that D(G) is a skew field and henc K_1(D(G)) is the abelianization of the multiplicative group of units in D(G).","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121045380","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":"An equivariant index for proper actions II: properties and applications","authors":"P. Hochs, Yanli Song","doi":"10.4171/JNCG/273","DOIUrl":"https://doi.org/10.4171/JNCG/273","url":null,"abstract":"In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of this index. We prove that it has an induction property that can be used to deduce various other properties of the index. In the case of compact orbit spaces, we show how it is related to the analytic assembly map from the Baum-Connes conjecture, and an index used by Mathai and Zhang. We use the index to define a notion of K-homological Dirac induction, and show that, under conditions, it satisfies the quantisation commutes with reduction principle.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121709817","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":"A fixed point theorem on noncompact manifolds","authors":"P. Hochs, Han Wang","doi":"10.2140/akt.2018.3.235","DOIUrl":"https://doi.org/10.2140/akt.2018.3.235","url":null,"abstract":"We generalise the Atiyah-Segal-Singer fixed point theorem to noncompact manifolds. Using $KK$-theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah-Segal-Singer's result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant $K$-theory and $K$-homology, and a non-localised expression for the index we use, in terms of deformations of principal symbols. The latter result is one of several links we find to indices of deformed symbols and operators studied by various authors.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130529464","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":"Excellence of function fields of conics","authors":"A. Merkurjev, J. Tignol","doi":"10.4171/LEM/62-3/4-3","DOIUrl":"https://doi.org/10.4171/LEM/62-3/4-3","url":null,"abstract":"For every generalized quadratic form or hermitian form over a division algebra, the anisotropic kernel of the form obtained by scalar extension to the function field of a smooth projective conic is defined over the field of constants. The proof does not require any hypothesis on the characteristic.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132885371","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":"Hopf-dihedral (co)homology and $L$-theory","authors":"A. Kaygun, S. Sutlu","doi":"10.4171/JNCG/271","DOIUrl":"https://doi.org/10.4171/JNCG/271","url":null,"abstract":"We develop an appropriate dihedral extension of the Connes-Moscovici characteristic map for Hopf *-algebras. We then observe that one can use this extension together with the dihedral Chern character to detect non-trivial $L$-theory classes of a *-algebra that carry a Hopf symmetry over a Hopf *-algebra. Using our machinery we detect a previously unknown $L$-class of the standard Podle's sphere.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123300701","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":"The K-theory of endomorphisms of spaces","authors":"Filipp Levikov","doi":"10.4310/HHA.2016.V18.N1.A17","DOIUrl":"https://doi.org/10.4310/HHA.2016.V18.N1.A17","url":null,"abstract":"We prove a non-linear version of a theorem of Grayson which is an analogue of the Fundamental Theorem of Algebraic $K$-theory and identify the $K$-theory of the endomorphism category over a space $X$ in terms of reduced $K$-theory of a certain localisation of the category of $NN$-spaces over $X$. In particular we generalise the result of Klein and Williams describing the nil-terms of $A$-theory in terms of $K$-theory of nilpotent endomorphisms.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132540841","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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