Let $PU_n$ denote the projective unitary group of rank $n$ and $BPU_n$ be its classifying space, for $n>1$. By using the Serre spectral sequence induced by the fibration $BU_nto BPU_nto K(mathbb{Z},3)$, we compute the integral cohomology of $BPU_n$ in dimensions less than $15$ except for $4mid n$ in dimension $14$.
{"title":"The cohomology of $BPU_n$ in dimensions less than $15$","authors":"Jiaxi Zha, Zhilei Zhang","doi":"arxiv-2407.16297","DOIUrl":"https://doi.org/arxiv-2407.16297","url":null,"abstract":"Let $PU_n$ denote the projective unitary group of rank $n$ and $BPU_n$ be its\u0000classifying space, for $n>1$. By using the Serre spectral sequence induced by\u0000the fibration $BU_nto BPU_nto K(mathbb{Z},3)$, we compute the integral\u0000cohomology of $BPU_n$ in dimensions less than $15$ except for $4mid n$ in\u0000dimension $14$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775507","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 attack the question of E_2-formality of differential graded algebras over prime fields via obstruction theory. We are able to prove that E_2-algebras whose cohomology ring is a polynomial algebra on even degree classes are intrinsically formal. As a consequence we prove E_2-formality of the classifying space of some compact Lie group or of Davis-Januszkiewicz spaces.
{"title":"E2 formality via obstruction theory","authors":"Geoffroy Horel","doi":"arxiv-2407.16236","DOIUrl":"https://doi.org/arxiv-2407.16236","url":null,"abstract":"We attack the question of E_2-formality of differential graded algebras over\u0000prime fields via obstruction theory. We are able to prove that E_2-algebras\u0000whose cohomology ring is a polynomial algebra on even degree classes are\u0000intrinsically formal. As a consequence we prove E_2-formality of the\u0000classifying space of some compact Lie group or of Davis-Januszkiewicz spaces.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775502","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 reduced crystallographic root system with a fixed simple system, it is associated to a Weyl group $W$, parabolic subgroups $W_K$'s and a polytope $P$ which is the convex hull of a dominant weight. The quotient $P/W_K$ can be identified with a polytope. Polytopes $P$ and $P/W_K$ are associated to toric varieties $X_P$ and $X_{P/W_K}$ respectively. It turns out the underlying topological spaces $X_P/W_K$ and $X_{P/W_K}$ are homotopy equivalent, when considering the polytopes in the real span of the root lattice or of the weight lattice.
{"title":"Homotopy Types Of Toric Orbifolds From Weyl Polytopes","authors":"Tao Gong","doi":"arxiv-2407.16070","DOIUrl":"https://doi.org/arxiv-2407.16070","url":null,"abstract":"Given a reduced crystallographic root system with a fixed simple system, it\u0000is associated to a Weyl group $W$, parabolic subgroups $W_K$'s and a polytope\u0000$P$ which is the convex hull of a dominant weight. The quotient $P/W_K$ can be\u0000identified with a polytope. Polytopes $P$ and $P/W_K$ are associated to toric\u0000varieties $X_P$ and $X_{P/W_K}$ respectively. It turns out the underlying\u0000topological spaces $X_P/W_K$ and $X_{P/W_K}$ are homotopy equivalent, when\u0000considering the polytopes in the real span of the root lattice or of the weight\u0000lattice.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775504","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 submission is a PhD dissertation. It constitutes the summary of the author's work concerning the relations between cohomology rings of algebraic varieties and rings of functions on zero schemes and fixed point schemes. It includes the results from the co-authored article arXiv:2212.11836. They are complemented by: an introduction to the theory of group actions on algebraic varieties, with particular focus on vector fields; a historical overview of the field; a few newer results of the author. The fundamental theorem from arXiv:2212.11836 says that if the principal nilpotent has a unique zero, then the zero scheme over the Kostant section is isomorphic to the spectrum of the equivariant cohomology ring, remembering the grading in terms of a $mathbb{C}$ action. In this thesis, we also tackle the case of a singular variety. As long as it is embedded in a smooth variety with regular action, we are able to study its cohomology as well by means of the zero scheme. In largest generality, this allows us to see geometrically a subring of the cohomology ring. We also show that the cohomology ring of spherical varieties appears as the ring of functions on the zero scheme. Lastly, the K-theory conjecture is studied, with some results attained for GKM spaces.
本论文为博士论文。它是作者关于代数变量的同调环与零方案和定点方案上的函数环之间关系的工作总结。它包括合著文章 arXiv:2212.11836 中的结果。此外还有：代数变量上的群作用理论简介，尤其侧重于向量场；该领域的历史概述；作者的一些新成果。arXiv:2212.11836的基本定理指出，如果主无势有一个唯一的零，那么在Kostant部分上的零方案与等变同调环的谱同构，记得用$mathbb{C}$作用来表示等级。在本论文中，我们还处理了奇异品种的情况。只要它嵌入到具有规则作用的光滑综中，我们就能通过零方案来研究它的同调。在最大广义上，这使我们可以几何地看到同调环的下环。最后，我们研究了 K 理论猜想，并取得了 GKM 空间的一些结果。
{"title":"Equivariant cohomology and rings of functions","authors":"Kamil Rychlewicz","doi":"arxiv-2407.14659","DOIUrl":"https://doi.org/arxiv-2407.14659","url":null,"abstract":"This submission is a PhD dissertation. It constitutes the summary of the\u0000author's work concerning the relations between cohomology rings of algebraic\u0000varieties and rings of functions on zero schemes and fixed point schemes. It\u0000includes the results from the co-authored article arXiv:2212.11836. They are\u0000complemented by: an introduction to the theory of group actions on algebraic\u0000varieties, with particular focus on vector fields; a historical overview of the\u0000field; a few newer results of the author. The fundamental theorem from arXiv:2212.11836 says that if the principal\u0000nilpotent has a unique zero, then the zero scheme over the Kostant section is\u0000isomorphic to the spectrum of the equivariant cohomology ring, remembering the\u0000grading in terms of a $mathbb{C}$ action. In this thesis, we also tackle the\u0000case of a singular variety. As long as it is embedded in a smooth variety with\u0000regular action, we are able to study its cohomology as well by means of the\u0000zero scheme. In largest generality, this allows us to see geometrically a\u0000subring of the cohomology ring. We also show that the cohomology ring of\u0000spherical varieties appears as the ring of functions on the zero scheme.\u0000Lastly, the K-theory conjecture is studied, with some results attained for GKM\u0000spaces.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141775505","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 $mathrm{R}$ be a real closed field, $S subset mathrm{R}^n$ a closed and bounded semi-algebraic set and $mathbf{f} = (f_1,ldots,f_p):S rightarrow mathrm{R}^p$ a continuous semi-algebraic map. We study the poset module structure in homology induced by the simultaneous filtrations of $S$ by the sub-level sets of the functions $f_i$ from an algorithmic and quantitative point of view. For fixed dimensional homology we prove a singly exponential upper bound on the complexity of these modules which are encoded as certain semi-algebraically constructible functions on $mathrm{R}^p times mathrm{R}^p$. We also deduce for semi-algebraic filtrations of bounded complexity, upper bounds on the number of equivalence classes of finite poset modules that such a filtration induces -- establishing a tight analogy with a well-known graph theoretical result on the "speed'' of algebraically defined graphs.
{"title":"Complexity and speed of semi-algebraic multi-persistence","authors":"Arindam Banerjee, Saugata Basu","doi":"arxiv-2407.13586","DOIUrl":"https://doi.org/arxiv-2407.13586","url":null,"abstract":"Let $mathrm{R}$ be a real closed field, $S subset mathrm{R}^n$ a closed\u0000and bounded semi-algebraic set and $mathbf{f} = (f_1,ldots,f_p):S rightarrow\u0000mathrm{R}^p$ a continuous semi-algebraic map. We study the poset module\u0000structure in homology induced by the simultaneous filtrations of $S$ by the\u0000sub-level sets of the functions $f_i$ from an algorithmic and quantitative\u0000point of view. For fixed dimensional homology we prove a singly exponential\u0000upper bound on the complexity of these modules which are encoded as certain\u0000semi-algebraically constructible functions on $mathrm{R}^p times\u0000mathrm{R}^p$. We also deduce for semi-algebraic filtrations of bounded\u0000complexity, upper bounds on the number of equivalence classes of finite poset\u0000modules that such a filtration induces -- establishing a tight analogy with a\u0000well-known graph theoretical result on the \"speed'' of algebraically defined\u0000graphs.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740213","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}
The Thom polynomial of a singularity $eta$ expresses the cohomology class of the $eta$-singularity locus of a map in terms of the map's simple invariants. In this informal survey -- based on two lectures given at the Isaac Newton Institute in 2024 -- we explore various Thom polynomial concepts with examples.
奇点$eta$的托姆多项式（Thom polynomial of a singularity $eta$）用映射的简单不变式表达了映射的奇点位置的同调类。在这个非正式的调查中--基于2024年在艾萨克-牛顿研究所（Isaac NewtonInstitute）的两次讲座--我们用实例探讨了各种托姆多项式的概念。
{"title":"Thom polynomials. A primer","authors":"Richard Rimanyi","doi":"arxiv-2407.13883","DOIUrl":"https://doi.org/arxiv-2407.13883","url":null,"abstract":"The Thom polynomial of a singularity $eta$ expresses the cohomology class of\u0000the $eta$-singularity locus of a map in terms of the map's simple invariants.\u0000In this informal survey -- based on two lectures given at the Isaac Newton\u0000Institute in 2024 -- we explore various Thom polynomial concepts with examples.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740212","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}
A classical result in Morse theory is the determination of the homotopy type of the loop space of a manifold. In this paper, we study this result through the lens of discrete Morse theory. This requires a suitable simplicial model for the loop space. Here, we use Milnor's $textrm{F}^+textrm{K}$ construction to model the loop space of the sphere $S^2$, describe a discrete gradient on it, and identify a collection of critical cells. We also compute the action of the boundary operator in the Morse complex on these critical cells, showing that they are potential homology generators. A careful analysis allows us to recover the calculation of the first homology of $Omega S^2$.
{"title":"Discrete Morse theory on $ΩS^2$","authors":"Lacey Johnson, Kevin Knudson","doi":"arxiv-2407.12156","DOIUrl":"https://doi.org/arxiv-2407.12156","url":null,"abstract":"A classical result in Morse theory is the determination of the homotopy type\u0000of the loop space of a manifold. In this paper, we study this result through\u0000the lens of discrete Morse theory. This requires a suitable simplicial model\u0000for the loop space. Here, we use Milnor's $textrm{F}^+textrm{K}$ construction\u0000to model the loop space of the sphere $S^2$, describe a discrete gradient on\u0000it, and identify a collection of critical cells. We also compute the action of\u0000the boundary operator in the Morse complex on these critical cells, showing\u0000that they are potential homology generators. A careful analysis allows us to\u0000recover the calculation of the first homology of $Omega S^2$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740218","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 show that the loop space of a moment-angle complex associated to a $2$-dimensional simplicial complex decomposes as a finite type product of spheres, loops on spheres, and certain indecomposable spaces which appear in the loop space decomposition of Moore spaces. We also give conditions on certain subcomplexes under which, localised away from sufficiently many primes, the loop space of a moment-angle complex decomposes as a finite type product of spheres and loops on spheres.
{"title":"Loop space decompositions of moment-angle complexes associated to two dimensional simplicial complexes","authors":"Lewis Stanton","doi":"arxiv-2407.10781","DOIUrl":"https://doi.org/arxiv-2407.10781","url":null,"abstract":"We show that the loop space of a moment-angle complex associated to a\u0000$2$-dimensional simplicial complex decomposes as a finite type product of\u0000spheres, loops on spheres, and certain indecomposable spaces which appear in\u0000the loop space decomposition of Moore spaces. We also give conditions on\u0000certain subcomplexes under which, localised away from sufficiently many primes,\u0000the loop space of a moment-angle complex decomposes as a finite type product of\u0000spheres and loops on spheres.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717507","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 calculate the mod $2$ spin$^c$-cobordism ring up to uniform $F$-isomorphism (i.e., inseparable isogeny). As a consequence we get the prime ideal spectrum of the mod $2$ spin$^c$-cobordism ring. We also calculate the mod $2$ spin$^c$-cobordism ring ``on the nose'' in degrees $leq 33$. We construct an infinitely generated nonunital subring of the $2$-torsion in the spin$^c$-cobordism ring. We use our calculations of product structure in the spin and spin$^c$ cobordism rings to give an explicit example, up to cobordism, of a compact $24$-dimensional spin manifold which is not cobordant to a sum of squares, which was asked about in a 1965 question of Milnor.
我们计算了模 2$ 自旋$^c$-同调环的均匀$F$-同构（即不可分割的同源性）。因此，我们得到了 mod $2$ 自旋$^c$-同调环的质谱。我们还计算了度数为 $leq 33$ 的 mod $2$ 自旋$^c$-共轭环的 "鼻子上"。我们在自旋$^c$-共轭环中构建了一个无限生成的 2$-扭转的非空心子环。我们利用对自旋和自旋^c$共弦环中乘积结构的计算，给出了一个紧凑的$24$维自旋流形不与平方和共弦的明确例子。
{"title":"Products in spin$^c$-cobordism","authors":"Hassan Abdallah, Andrew Salch","doi":"arxiv-2407.10045","DOIUrl":"https://doi.org/arxiv-2407.10045","url":null,"abstract":"We calculate the mod $2$ spin$^c$-cobordism ring up to uniform\u0000$F$-isomorphism (i.e., inseparable isogeny). As a consequence we get the prime\u0000ideal spectrum of the mod $2$ spin$^c$-cobordism ring. We also calculate the\u0000mod $2$ spin$^c$-cobordism ring ``on the nose'' in degrees $leq 33$. We\u0000construct an infinitely generated nonunital subring of the $2$-torsion in the\u0000spin$^c$-cobordism ring. We use our calculations of product structure in the\u0000spin and spin$^c$ cobordism rings to give an explicit example, up to cobordism,\u0000of a compact $24$-dimensional spin manifold which is not cobordant to a sum of\u0000squares, which was asked about in a 1965 question of Milnor.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717508","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}
Rayna Andreeva, Benjamin Dupuis, Rik Sarkar, Tolga Birdal, Umut Şimşekli
We present a novel set of rigorous and computationally efficient topology-based complexity notions that exhibit a strong correlation with the generalization gap in modern deep neural networks (DNNs). DNNs show remarkable generalization properties, yet the source of these capabilities remains elusive, defying the established statistical learning theory. Recent studies have revealed that properties of training trajectories can be indicative of generalization. Building on this insight, state-of-the-art methods have leveraged the topology of these trajectories, particularly their fractal dimension, to quantify generalization. Most existing works compute this quantity by assuming continuous- or infinite-time training dynamics, complicating the development of practical estimators capable of accurately predicting generalization without access to test data. In this paper, we respect the discrete-time nature of training trajectories and investigate the underlying topological quantities that can be amenable to topological data analysis tools. This leads to a new family of reliable topological complexity measures that provably bound the generalization error, eliminating the need for restrictive geometric assumptions. These measures are computationally friendly, enabling us to propose simple yet effective algorithms for computing generalization indices. Moreover, our flexible framework can be extended to different domains, tasks, and architectures. Our experimental results demonstrate that our new complexity measures correlate highly with generalization error in industry-standards architectures such as transformers and deep graph networks. Our approach consistently outperforms existing topological bounds across a wide range of datasets, models, and optimizers, highlighting the practical relevance and effectiveness of our complexity measures.
{"title":"Topological Generalization Bounds for Discrete-Time Stochastic Optimization Algorithms","authors":"Rayna Andreeva, Benjamin Dupuis, Rik Sarkar, Tolga Birdal, Umut Şimşekli","doi":"arxiv-2407.08723","DOIUrl":"https://doi.org/arxiv-2407.08723","url":null,"abstract":"We present a novel set of rigorous and computationally efficient\u0000topology-based complexity notions that exhibit a strong correlation with the\u0000generalization gap in modern deep neural networks (DNNs). DNNs show remarkable\u0000generalization properties, yet the source of these capabilities remains\u0000elusive, defying the established statistical learning theory. Recent studies\u0000have revealed that properties of training trajectories can be indicative of\u0000generalization. Building on this insight, state-of-the-art methods have\u0000leveraged the topology of these trajectories, particularly their fractal\u0000dimension, to quantify generalization. Most existing works compute this\u0000quantity by assuming continuous- or infinite-time training dynamics,\u0000complicating the development of practical estimators capable of accurately\u0000predicting generalization without access to test data. In this paper, we\u0000respect the discrete-time nature of training trajectories and investigate the\u0000underlying topological quantities that can be amenable to topological data\u0000analysis tools. This leads to a new family of reliable topological complexity\u0000measures that provably bound the generalization error, eliminating the need for\u0000restrictive geometric assumptions. These measures are computationally friendly,\u0000enabling us to propose simple yet effective algorithms for computing\u0000generalization indices. Moreover, our flexible framework can be extended to\u0000different domains, tasks, and architectures. Our experimental results\u0000demonstrate that our new complexity measures correlate highly with\u0000generalization error in industry-standards architectures such as transformers\u0000and deep graph networks. Our approach consistently outperforms existing\u0000topological bounds across a wide range of datasets, models, and optimizers,\u0000highlighting the practical relevance and effectiveness of our complexity\u0000measures.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611636","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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