{"title":"The universal six-functor formalism","authors":"B. Drew, Martin Gallauer","doi":"10.2140/akt.2022.7.599","DOIUrl":"https://doi.org/10.2140/akt.2022.7.599","url":null,"abstract":"We prove that Morel-Voevodsky's stable $mathbb{A}^1$-homotopy theory affords the universal six-functor formalism.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45449640","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}
Given a Galois covering of complete spin manifolds where the base metric has PSC near infinity, we prove that for small enough epsilon > 0, the epsilon spectral projection of the Dirac operator has finite trace in the Atiyah von Neumann algebra. This allows us to define the L2 index in the even case and we prove its compatibility with the Xie-Yu higher index. We also deduce L2 versions of the classical Gromov-Lawson relative index theorems. Finally, we briefly discuss some Gromov-Lawson L2 invariants.
给定一个完全自旋流形的伽罗瓦覆盖，其中基本度规的PSC接近无穷大，我们证明了对于足够小的epsilon > 0，狄拉克算子的epsilon谱投影在Atiyah von Neumann代数中具有有限迹。这允许我们在偶数情况下定义L2指数，并证明其与协余高指数的兼容性。我们还推导了经典Gromov-Lawson相对指数定理的L2版本。最后，我们简要讨论了一些Gromov-Lawson L2不变量。
{"title":"The relative L2 index theorem for Galois\u0000coverings","authors":"M. Benameur","doi":"10.2140/akt.2021.6.503","DOIUrl":"https://doi.org/10.2140/akt.2021.6.503","url":null,"abstract":"Given a Galois covering of complete spin manifolds where the base metric has PSC near infinity, we prove that for small enough epsilon > 0, the epsilon spectral projection of the Dirac operator has finite trace in the Atiyah von Neumann algebra. This allows us to define the L2 index in the even case and we prove its compatibility with the Xie-Yu higher index. We also deduce L2 versions of the classical Gromov-Lawson relative index theorems. Finally, we briefly discuss some Gromov-Lawson L2 invariants.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43490719","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 generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.
{"title":"K-theoretic torsion and the zeta\u0000function","authors":"John R. Klein, Cary Malkiewich","doi":"10.2140/akt.2022.7.77","DOIUrl":"https://doi.org/10.2140/akt.2022.7.77","url":null,"abstract":"We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49415401","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 spectrum of equivariant Kasparov theory for cyclic groups of prime order","authors":"Ivo Dell’Ambrogio, R. Meyer","doi":"10.2140/akt.2021.6.547","DOIUrl":"https://doi.org/10.2140/akt.2021.6.547","url":null,"abstract":"We compute the Balmer spectrum of the equivariant bootstrap category of separable $G$-C*-algebras when $G$ is a group of prime order.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44655916","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 prove an equivariant localized and norm-controlled version of the Pimsner-Popa-Voiculescu theorem. As an application, we deduce a proof of the Paschke-Higson duality for transformation groupoids.
{"title":"An equivariant PPV theorem and Paschke–Higson\u0000duality","authors":"M. Benameur, Indrava Roy","doi":"10.2140/akt.2022.7.237","DOIUrl":"https://doi.org/10.2140/akt.2022.7.237","url":null,"abstract":"We prove an equivariant localized and norm-controlled version of the Pimsner-Popa-Voiculescu theorem. As an application, we deduce a proof of the Paschke-Higson duality for transformation groupoids.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45472857","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}
For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.
{"title":"Positive scalar curvature and an equivariant Callias-type index theorem for proper actions","authors":"Haoyang Guo, P. Hochs, V. Mathai","doi":"10.2140/akt.2021.6.319","DOIUrl":"https://doi.org/10.2140/akt.2021.6.319","url":null,"abstract":"For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47236204","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 provide a concrete introduction to the topologised, graded analogue of an algebraic structure known as a plethory, originally due to Tall and Wraith. Stacey and Whitehouse showed this structure is present on the cohomology operations for a suitable generalised cohomology theory. We compute an explicit expression for the plethory of operations for complex topological K-theory. This is formulated in terms of a plethory enhanced with structure corresponding to the looping of operations. In this context we show that the familiar lambda operations generate all the operations.
{"title":"The plethory of operations in complex\u0000topological K-theory","authors":"William Mycroft, Sarah Whitehouse","doi":"10.2140/akt.2021.6.481","DOIUrl":"https://doi.org/10.2140/akt.2021.6.481","url":null,"abstract":"We provide a concrete introduction to the topologised, graded analogue of an algebraic structure known as a plethory, originally due to Tall and Wraith. Stacey and Whitehouse showed this structure is present on the cohomology operations for a suitable generalised cohomology theory. We compute an explicit expression for the plethory of operations for complex topological K-theory. This is formulated in terms of a plethory enhanced with structure corresponding to the looping of operations. In this context we show that the familiar lambda operations generate all the operations.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41726061","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}
This paper provides a generalization of excision theorems in controlled algebra in the context of equivariant G-theory with fibred control and families of bounded actions. It also states and proves several characteristic features of this theory such as existence of the fibred assembly and the fibrewise trivialization.
{"title":"Excision in equivariant fibred G-theory","authors":"G. Carlsson, B. Goldfarb","doi":"10.2140/akt.2020.5.721","DOIUrl":"https://doi.org/10.2140/akt.2020.5.721","url":null,"abstract":"This paper provides a generalization of excision theorems in controlled algebra in the context of equivariant G-theory with fibred control and families of bounded actions. It also states and proves several characteristic features of this theory such as existence of the fibred assembly and the fibrewise trivialization.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42399597","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}
Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners $X$ of any codimension, there is a natural and explicit morphism $$K_*(mathcal{K}_b(X)) stackrel{T}{longrightarrow} H^{pcn}_*(X,mathbb{Q})$$ between the $K-$theory group of the algebra $mathcal{K}_b(X)$ of $b$-compact operators for $X$ and the periodic conormal homology group with rational coeficients, and that $T$ is a rational isomorphism. As shown by the first two authors in a previous paper this computation implies that the rational groups $H^{pcn}_{ev}(X,mathbb{Q})$ provide an obstruction to the Fredholm perturbation property for compact connected manifold with corners. The difference with respect to the previous article of the first two authors in which they solve this problem for low codimensions is that we overcome in the present article the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the $K-$theory groups of the algebra $mathcal{K}_b(X)$.
{"title":"On Fredholm boundary conditions on manifolds with corners, I: Global corner cycles obstructions","authors":"P. C. Rouse, J. Lescure, Mario Velásquez","doi":"10.2140/akt.2021.6.607","DOIUrl":"https://doi.org/10.2140/akt.2021.6.607","url":null,"abstract":"Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. \u0000Our main theorem is that, for any manifold with corners $X$ of any codimension, there is a natural and explicit morphism $$K_*(mathcal{K}_b(X)) stackrel{T}{longrightarrow} H^{pcn}_*(X,mathbb{Q})$$ between the $K-$theory group of the algebra $mathcal{K}_b(X)$ of $b$-compact operators for $X$ and the periodic conormal homology group with rational coeficients, and that $T$ is a rational isomorphism. \u0000As shown by the first two authors in a previous paper this computation implies that the rational groups $H^{pcn}_{ev}(X,mathbb{Q})$ provide an obstruction to the Fredholm perturbation property for compact connected manifold with corners. \u0000The difference with respect to the previous article of the first two authors in which they solve this problem for low codimensions is that we overcome in the present article the problem of computing the higher spectral sequence K-theory differentials associated to the canonical filtration by codimension by introducing an explicit topological space whose singular cohomology is canonically isomorphic to the conormal homology and whose K-theory is naturally isomorphic to the $K-$theory groups of the algebra $mathcal{K}_b(X)$.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43306336","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}
Let $R$ be a regular semi-local ring, essentially of finite type over a perfect field of characteristic $p ge 3$. We show that the cycle class map with modulus from an earlier work of the authors induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and the relative $K$-theory of truncated polynomial rings over $R$. This settles the problem of equating 0-cycles with modulus and relative $K$-theory of such rings in all characteristics $neq 2$.
{"title":"Relative K-theory via 0-cycles in finite\u0000characteristic","authors":"Rahul Gupta, A. Krishna","doi":"10.2140/akt.2021.6.673","DOIUrl":"https://doi.org/10.2140/akt.2021.6.673","url":null,"abstract":"Let $R$ be a regular semi-local ring, essentially of finite type over a perfect field of characteristic $p ge 3$. We show that the cycle class map with modulus from an earlier work of the authors induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and the relative $K$-theory of truncated polynomial rings over $R$. This settles the problem of equating 0-cycles with modulus and relative $K$-theory of such rings in all characteristics $neq 2$.","PeriodicalId":42182,"journal":{"name":"Annals of K-Theory","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45650515","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：