arXiv 1808.073422018-08-22math.AT

Borel cohomology and the relative Gorenstein condition for classifying spaces of compact Lie groups

J. P. C. Greenlees

英文摘要：

For a compact Lie group G we show that if the representing spectrum for Borel cohomology generates its category of modules if G is connected. For a closed subgroup H of G we consider the map C^*(BG)--->C^*(BH) and establish the sense in which it is relatively Gorenstein. Throughout, we pay careful attention to the importance of connectedness of the groups.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

String topology on Gorenstein spaces

The purpose of this paper is to describe a general and simple setting for defining $(g,p+q)$-string operations on a Poincaré duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincaré duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each $(g,p+q)$-string operation as well as it leads to explicit computations.

2008-06-17math.ATmath.QA

色群上同调中的局部 Gorenstein 对偶性

Local Gorenstein duality in chromatic group cohomology

本文研究了紧 Lie 群分类空间上链谱的局部 Gorenstein 对偶性，证明其在多种环谱（如 Lubin-Tate 理论、拓扑 K-理论和拓扑模形式）中系统成立，并通过下降结果扩展了适用范围。

2023-05-15math.ACmath.AT

Vector bundles over classifying spaces of p-local finite groups and Benson-Carlson duality

In this paper we obtain a description of the Grothendieck group of complex vector bundles over the classifying space of a p-local finite group in terms of representation rings of subgroups of its Sylow. We also prove a stable elements formula for generalized cohomological invariants of p-local finite groups, which is used to show the existence of unitary embeddings of p-local finite groups. Finally, we show that the augmentation map for the cochains of the classifying space of a p-local finite group is Gorenstein in the sense of Dwyer-Greenlees-Iyengar and obtain some consequences about the co…

2019-09-27math.ATmath.GR

Orbispaces and Orbifolds from the Point of View of the Borel Construction, a new Definition

``An orbifold is a space which is locally modeled on the quotient of a vector space by a finite group.'' This sentence is so easily said or written that more than one person has missed some of the subtleties hidden by orbifolds. Orbifolds were first introduced by Satake under the name ``V-manifold'' and rediscovered by Thurston who called them ``orbifolds''. Both of them used only faithful actions to define their orbifolds (these are the so called reduced orbifolds). This intuitive restriction is very unnatural mathematically if we want for example to study suborbifolds. Orbispaces are to topo…

2001-12-01math.ATmath.GT

Topological K-(co-)homology of classifying spaces of discrete groups

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology of the classifying space BG up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculat…

2012-01-23math.ATmath.KT

Dimension functions for spherical fibrations

Given a spherical fibration $ξ$ over the classifying space $BG$ of a finite group we define a dimension function for the $m-$fold fiber join of $ξ$ where $m$ is some large positive integer. We show that the dimension functions satisfy the Borel-Smith conditions when $m$ is large enough. As an application we prove that there exists no spherical fibration over the classifying space of $\text{Qd}(p)= (\mathbb{Z}/p)^2\rtimes\text{SL}_2(\mathbb{Z}/p)$ with $p-$effective Euler class, generalizing the result of Özgün Ünlü about group actions on finite complexes homotopy equivalent to a sphere. We hav…

2018-06-29math.AT

更多相关论文收起更多论文

Torelli spaces of high-dimensional manifolds

The Torelli group of a manifold is the group of all diffeomorphisms which act as the identity on the homology of the manifold. In this paper, we calculate the invariant part (invariant under the action of the automorphisms of the homology) of the cohomology of the classifying space of the Torelli group of certain high-dimensional, highly connected manifolds, with rational coefficients and in a certain range of degrees. This is based on Galatius--Randal-Williams' work on the diffeomorphism groups of these manifolds, Borel's classical results on arithmetic groups, and methods from surgery theory…

2014-06-02math.AT

Formality in an Equivariant Setting

We define and discuss G-formality for certain spaces endowed with an action by a compact Lie group. This concept is essentially formality of the Borel construction of the space in a category of commutative differential graded algebras over R, where R is the cohomology of the classifying space BG. These results may be applied in computing the equivariant cohomology of their loop spaces.

2003-04-29math.AT

Equivariant cohomology for differentiable stacks

We construct and analyse models of equivariant cohomology for differentiable stacks with Lie group actions extending classical results for smooth manifolds due to Borel, Cartan and Getzler. We also derive various spectral sequences for the equivariant cohomology of a differentiable stack generalising among others Bott's spectral sequence which converges to the cohomology of the classifying space of a Lie group.

2020-11-01math.ATmath.DG

On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces

We describe various equivalent ways of associating to an orbifold, or more generally a higher étale differentiable stack, a weak homotopy type. Some of these ways extend to arbitrary higher stacks on the site of smooth manifolds, and we show that for a differentiable stack X arising from a Lie groupoid G, the weak homotopy type of X agrees with that of BG. Using this machinery, we are able to find new presentations for the weak homotopy type of certain classifying spaces. In particular, we give a new presentation for the Borel construction of an almost free action of a Lie group G on a smooth …

2015-04-09math.ATmath.CTmath.DG

不变性形式与右角Coxeter群子群的上同调

Equivariant formality and the cohomology of subgroups of right-angled Coxeter groups

Carleton University（卡尔顿大学）；University of Western Ontario（韦仕敦大学）

本文研究了右角Coxeter群的coabelian子群的分类空间模型，通过实moment-angle复形的同调轨道空间构造，并利用Borel等效上同调理论分析其上同调结构，提出图论判据判断coabelian子群的上同调自由性。

2026-04-23math.ATmath.GR

半直积作用的等变上同调

The equivariant cohomology for semidirect product actions

本文证明任意紧致李群在特定闭子群条件下等变上同调可约化，并研究半直积情形，为计算环面作用及对合空间的合系阶提供新方法。

2024-02-13math.AT

Support varieties for transporter category algebras

Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G-space BP, while the k-category algebra of the transporter category is the (Gorenstein) skew group algebra on the G-incidence algebra kP. We introduce a support variety theory for the category algebras of transporter categories. It extends Carlson's support variety theory on group cohomology rings to equivariant cohomology rings. In the mean time it provides a class of (usually…

2011-08-26math.ATmath.GRmath.RA

On the Duflot filtration for equivariant cohomology rings and applications to group cohomology

We study the Duflot filtration on the Borel equivariant cohomology of smooth manifolds with a smooth $p$-torus action. We axiomatize the filtration and prove analog of several structural results about equivariant cohomology rings in this setting. We apply this abstract theory to study the $\mathbb{F}_p$ cohomology rings of classifying spaces of compact Lie groups, and show how to recover geometric results about the cohomology of $BG$ using equivariant cohomology. This includes some results about detection on subgroups and restrictions on associated primes that were previously only known for fi…

2018-10-16math.AT

Ganea分解的分类空间

Ganea decompositions of classifying spaces

Cornell University（康奈尔大学）；Indiana University（印第安纳大学）

本文研究了通过纤维-纤维束构造获得的紧连通李群分类空间BG的Ganea分解，确定了纤维束的上同调条件，证明了分解的sharp性，并构造了多个例子。

2026-03-08math.ATmath.CTmath.KT

有限反射群的准不变量代数的拓扑实现，I（附录由M.V. Feigin和K.E. Feldman撰写)

Topological realization of algebras of quasi-invariants, I (with an Appendix by M. V. Feigin and K. E. Feldman)

Cornell University（康奈尔大学）；Indiana University（印第安纳大学）

本文通过构造m-准旗 manifold及其商空间，实现了有限反射群的准不变量代数的拓扑表示，并计算了其K-理论和椭圆上同调。

2026-02-14math.ATmath.RT

关于神经与自由环路空间的乘性谱序列

On multiplicative spectral sequences for nerves and the free loop spaces

本研究构造了收敛于双单纯集对角复形上同调代数的乘性谱序列，应用于Borel构造得到实射影空间自由环路空间的模$p$上同调代数，并尝试用迭代积分表示非单连通流形自由环路空间的上同调生成元。

2024-05-17math.ATmath.KT

反同调

Reflexive homology

本文研究反同调与自由环路空间 $C_2$-等变同调的关系，通过函子同调定义反同调并给出计算双复形，证明其满足 Morita 不变性且与对合 Hochschild 同调在特定条件下一致。

2023-07-17math.AT

同调不变量的可定义内容 II：Čech 上同调与同伦分类

The definable content of homological invariants II: Čech cohomology and homotopy classification

本文将描述集合论应用于同调代数与代数拓扑，提出可定义的Čech上同调函子并证明其能精化经典函子，解决了映射望远镜同伦分类与Borsuk-Eilenberg问题的非光滑性。

2024-04-25math.ATmath.GTmath.KT