We prove a Galois correspondence for $n$-stacks. It gives a correspondence between the $infty$-category of Deligne-Mumford $n$-stacks finite 'etale over a connected scheme $X$ and the $infty$-category of $n$-stacks of finite sets with an action of the fundamental group of $X$.
{"title":"The Galois Correspondence for n-Stacks","authors":"Yuxiang Yao","doi":"arxiv-2408.00281","DOIUrl":"https://doi.org/arxiv-2408.00281","url":null,"abstract":"We prove a Galois correspondence for $n$-stacks. It gives a correspondence\u0000between the $infty$-category of Deligne-Mumford $n$-stacks finite 'etale over\u0000a connected scheme $X$ and the $infty$-category of $n$-stacks of finite sets\u0000with an action of the fundamental group of $X$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886793","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 $G$ be a compact simple Lie group equipped with the left invariant framing $L$. It is known that there are several groups $G$ such that $(G, L)$ is non-null framed bordant. Previously we gave an alternative proof of these results using the decomposition formula of its bordism class into a Kronecker product by E. Ossa. In this note we propose a verification formula by reconsidering it, through a little more ingenious in the use of this product formula, and try to apply it to the non-null bordantness results above.
让 $G$ 是一个紧凑的简单李群，具有左不变构型 $L$。众所周知，有几个组$G$使得$(G, L)$是非空有边框的。在此之前，我们曾利用 E. Ossa 将其边际类分解为 Kroneckerproduct 的分解公式，给出了上述结果的另一种证明。在本注释中，我们通过重新考虑它，提出了一个验证公式，通过更巧妙地使用这个乘积公式，并尝试将它应用于上述非空边界性结果。
{"title":"Non-null framed bordant simple Lie groups","authors":"Haruo Minami","doi":"arxiv-2408.02682","DOIUrl":"https://doi.org/arxiv-2408.02682","url":null,"abstract":"Let $G$ be a compact simple Lie group equipped with the left invariant\u0000framing $L$. It is known that there are several groups $G$ such that $(G, L)$\u0000is non-null framed bordant. Previously we gave an alternative proof of these\u0000results using the decomposition formula of its bordism class into a Kronecker\u0000product by E. Ossa. In this note we propose a verification formula by\u0000reconsidering it, through a little more ingenious in the use of this product\u0000formula, and try to apply it to the non-null bordantness results above.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944859","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}
Recall that topological complex $K$-theory associates to an isomorphism class of a complex vector bundle $E$ over a space $X$ an element of the complex $K$-theory group of $X$. Or from algebraic $K$-theory perspective, one assigns a homotopy class $[X to K (mathcal{K})]$, where $mathcal{K}$ is the ring of compact operators on the Hilbert space. We show that there is an analogous story for algebraic $K$-theory of a general commutative ring $k$, replacing complex vector bundles by certain Hamiltonian fiber bundles. The construction actually first assigns elements in a certain categorified algebraic $K$-theory, analogous to To"en's secondary $K$-theory of $k$. And there is a natural map from this categorified algebraic $K$-theory to the classical variant.
{"title":"Hamiltonian elements in algebraic K-theory","authors":"Yasha Savelyev","doi":"arxiv-2407.21003","DOIUrl":"https://doi.org/arxiv-2407.21003","url":null,"abstract":"Recall that topological complex $K$-theory associates to an isomorphism class\u0000of a complex vector bundle $E$ over a space $X$ an element of the complex\u0000$K$-theory group of $X$. Or from algebraic $K$-theory perspective, one assigns\u0000a homotopy class $[X to K (mathcal{K})]$, where $mathcal{K}$ is the ring of\u0000compact operators on the Hilbert space. We show that there is an analogous\u0000story for algebraic $K$-theory of a general commutative ring $k$, replacing\u0000complex vector bundles by certain Hamiltonian fiber bundles. The construction\u0000actually first assigns elements in a certain categorified algebraic $K$-theory,\u0000analogous to To\"en's secondary $K$-theory of $k$. And there is a natural map\u0000from this categorified algebraic $K$-theory to the classical variant.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862862","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 that the morphisms from a minimal Sullivan algebra of non-finite type to the algebra of polynomial differential forms on its realization cannot be quasi-isomorphic. This provides a positive answer to a question posed by F'elix, Halperin and Thomas. Furthermore, we give some discussion about the relationship between the homotopy groups of a topological space and its minimal Sullivan model.
{"title":"No quasi-isomorphism between a minimal Sullivan algebra of non-finite type and its realization","authors":"Jiawei Zhou","doi":"arxiv-2407.20881","DOIUrl":"https://doi.org/arxiv-2407.20881","url":null,"abstract":"We prove that the morphisms from a minimal Sullivan algebra of non-finite\u0000type to the algebra of polynomial differential forms on its realization cannot\u0000be quasi-isomorphic. This provides a positive answer to a question posed by\u0000F'elix, Halperin and Thomas. Furthermore, we give some discussion about the\u0000relationship between the homotopy groups of a topological space and its minimal\u0000Sullivan model.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862863","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}
Cobordism offers an unique perspective into the non-perturbative sector of string theory by demanding the absence of higher form global symmetries for quantum gravitational consistency. In this work we compute the spin cobordism groups of the classifying space of $Spin(32)/mathbb{Z}_2$ relevant to describing type I/heterotic string theory and explore their (shared) non-perturbative sector. To facilitate this we leverage our knowledge of type I D-brane physics behind the related ko-homology. The computation utilizes several established tools from algebraic topology, the focus here is on two spectral sequences. First, the Eilenberg-Moore spectral sequence is used to obtain the cohomology of the classifying space of the $Spin(32)/mathbb{Z}_2$ with $mathbb{Z}_2$ coefficients. This will enable us to start the Adams spectral sequence for finally obtaining our result, the spin cobordism groups. We conclude by providing a string theoretic interpretation to the cobordism groups.
{"title":"Spin cobordism and the gauge group of type I/heterotic string theory","authors":"Christian Kneissl","doi":"arxiv-2407.20333","DOIUrl":"https://doi.org/arxiv-2407.20333","url":null,"abstract":"Cobordism offers an unique perspective into the non-perturbative sector of\u0000string theory by demanding the absence of higher form global symmetries for\u0000quantum gravitational consistency. In this work we compute the spin cobordism\u0000groups of the classifying space of $Spin(32)/mathbb{Z}_2$ relevant to\u0000describing type I/heterotic string theory and explore their (shared)\u0000non-perturbative sector. To facilitate this we leverage our knowledge of type I\u0000D-brane physics behind the related ko-homology. The computation utilizes\u0000several established tools from algebraic topology, the focus here is on two\u0000spectral sequences. First, the Eilenberg-Moore spectral sequence is used to\u0000obtain the cohomology of the classifying space of the $Spin(32)/mathbb{Z}_2$\u0000with $mathbb{Z}_2$ coefficients. This will enable us to start the Adams\u0000spectral sequence for finally obtaining our result, the spin cobordism groups.\u0000We conclude by providing a string theoretic interpretation to the cobordism\u0000groups.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862864","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 that for commutative rings whose underlying abelian group is flat and in which $2$ is invertible, the homotopy groups at the trivial orbit of the equivariant Loday construction of the one-point compactification of the sign-representation are isomorphic to reflexive homology as studied by Graves and to involutive Hochschild homology defined by Fern`andez-al`encia and Giansiracusa. We also show a relative version of these results for commutative $k$-algebras $R$ with involution, whenever $2$ is invertible in $R$ and $R$ is flat as a $k$-module.
{"title":"Reflexive homology and involutive Hochschild homology as equivariant Loday constructions","authors":"Ayelet Lindenstrauss, Birgit Richter","doi":"arxiv-2407.20082","DOIUrl":"https://doi.org/arxiv-2407.20082","url":null,"abstract":"We prove that for commutative rings whose underlying abelian group is flat\u0000and in which $2$ is invertible, the homotopy groups at the trivial orbit of the\u0000equivariant Loday construction of the one-point compactification of the\u0000sign-representation are isomorphic to reflexive homology as studied by Graves\u0000and to involutive Hochschild homology defined by Fern`andez-al`encia and\u0000Giansiracusa. We also show a relative version of these results for commutative\u0000$k$-algebras $R$ with involution, whenever $2$ is invertible in $R$ and $R$ is\u0000flat as a $k$-module.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872805","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 aims to determine the ring structure of the torus equivariant cohomology of odd-dimensional complex quadrics by computing the graph equivariant cohomology of their corresponding GKM graphs. We show that its graph equivariant cohomology is generated by three types of subgraphs in the GKM graph, which are subject to four different types of relations. Furthermore, we consider the relationship between the two graph equivariant cohomology rings induced by odd- and even-dimensional complex quadrics.
{"title":"Equivariant cohomology of odd-dimensional complex quadrics from a combinatorial point of view","authors":"Shintaro Kuroki, Bidhan Paul","doi":"arxiv-2407.17921","DOIUrl":"https://doi.org/arxiv-2407.17921","url":null,"abstract":"This paper aims to determine the ring structure of the torus equivariant\u0000cohomology of odd-dimensional complex quadrics by computing the graph\u0000equivariant cohomology of their corresponding GKM graphs. We show that its\u0000graph equivariant cohomology is generated by three types of subgraphs in the\u0000GKM graph, which are subject to four different types of relations. Furthermore,\u0000we consider the relationship between the two graph equivariant cohomology rings\u0000induced by odd- and even-dimensional complex quadrics.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775500","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}
Chuan-Shen Hu, Rishikanta Mayengbam, Kelin Xia, Tze Chien Sum
With remarkable stability and exceptional optoelectronic properties, two-dimensional (2D) halide layered perovskites hold immense promise for revolutionizing photovoltaic technology. Presently, inadequate representations have substantially impeded the design and discovery of 2D perovskites. In this context, we introduce a novel computational topology framework termed the quotient complex (QC), which serves as the foundation for the material representation. Our QC-based features are seamlessly integrated with learning models for the advancement of 2D perovskite design. At the heart of this framework lies the quotient complex descriptors (QCDs), representing a quotient variation of simplicial complexes derived from materials unit cell and periodic boundary conditions. Differing from prior material representations, this approach encodes higher-order interactions and periodicity information simultaneously. Based on the well-established New Materials for Solar Energetics (NMSE) databank, our QC-based machine learning models exhibit superior performance against all existing counterparts. This underscores the paramount role of periodicity information in predicting material functionality, while also showcasing the remarkable efficiency of the QC-based model in characterizing materials structural attributes.
{"title":"Quotient complex (QC)-based machine learning for 2D perovskite design","authors":"Chuan-Shen Hu, Rishikanta Mayengbam, Kelin Xia, Tze Chien Sum","doi":"arxiv-2407.16996","DOIUrl":"https://doi.org/arxiv-2407.16996","url":null,"abstract":"With remarkable stability and exceptional optoelectronic properties,\u0000two-dimensional (2D) halide layered perovskites hold immense promise for\u0000revolutionizing photovoltaic technology. Presently, inadequate representations\u0000have substantially impeded the design and discovery of 2D perovskites. In this\u0000context, we introduce a novel computational topology framework termed the\u0000quotient complex (QC), which serves as the foundation for the material\u0000representation. Our QC-based features are seamlessly integrated with learning\u0000models for the advancement of 2D perovskite design. At the heart of this\u0000framework lies the quotient complex descriptors (QCDs), representing a quotient\u0000variation of simplicial complexes derived from materials unit cell and periodic\u0000boundary conditions. Differing from prior material representations, this\u0000approach encodes higher-order interactions and periodicity information\u0000simultaneously. Based on the well-established New Materials for Solar\u0000Energetics (NMSE) databank, our QC-based machine learning models exhibit\u0000superior performance against all existing counterparts. This underscores the\u0000paramount role of periodicity information in predicting material functionality,\u0000while also showcasing the remarkable efficiency of the QC-based model in\u0000characterizing materials structural attributes.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"262 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775510","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 introduce and study the proper topological complexity of a given configuration space, a version of the classical invariant for which we require that the algorithm controlling the motion is able to avoid any possible choice of ``unsafe'' area. To make it a homotopy functorial invariant we characterize it as a particular instance of the exterior sectional category of an exterior map, an invariant of the exterior homotopy category which is also deeply analyzed.
{"title":"Proper topological complexity","authors":"Jose M. Garcia-Calcines, Aniceto Murillo","doi":"arxiv-2407.16679","DOIUrl":"https://doi.org/arxiv-2407.16679","url":null,"abstract":"We introduce and study the proper topological complexity of a given\u0000configuration space, a version of the classical invariant for which we require\u0000that the algorithm controlling the motion is able to avoid any possible choice\u0000of ``unsafe'' area. To make it a homotopy functorial invariant we characterize\u0000it as a particular instance of the exterior sectional category of an exterior\u0000map, an invariant of the exterior homotopy category which is also deeply\u0000analyzed.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775503","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}
In the present paper, we construct a $mathbb{Z}/p$-equivariant analog of the $mathbb{Z}/2$-equivariant spectrum $BPmathbb{R}$ previously constructed by Hu and Kriz. We prove that this spectrum has some of the properties conjectured by Hill, Hopkins, and Ravenel. Our main construction method is an $mathbb{Z}/p$-equivariant analog of the Brown-Peterson tower of $BP$, based on a previous description of the $mathbb{Z}/p$-equivariant Steenrod algebra with constant coefficients by the authors. We also describe several variants of our construction and comparisons with other known equivariant spectra.
{"title":"The $mathbb{Z}/p$-equivariant spectrum $BPmathbb{R}$ for an odd prime $p$","authors":"Po Hu, Igor Kriz, Petr Somberg, Foling Zou","doi":"arxiv-2407.16599","DOIUrl":"https://doi.org/arxiv-2407.16599","url":null,"abstract":"In the present paper, we construct a $mathbb{Z}/p$-equivariant analog of the\u0000$mathbb{Z}/2$-equivariant spectrum $BPmathbb{R}$ previously constructed by Hu\u0000and Kriz. We prove that this spectrum has some of the properties conjectured by\u0000Hill, Hopkins, and Ravenel. Our main construction method is an\u0000$mathbb{Z}/p$-equivariant analog of the Brown-Peterson tower of $BP$, based on\u0000a previous description of the $mathbb{Z}/p$-equivariant Steenrod algebra with\u0000constant coefficients by the authors. We also describe several variants of our\u0000construction and comparisons with other known equivariant spectra.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775501","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：