Oihana Garaialde Ocaña, Jon González-Sánchez
Let p be a prime number and let c, d be natural numbers. Then, the number of possible isomorphism types for the mod p cohomology algebra of a d-generated finite p-group of nilpotency class c is bounded by a function depending only on p, c and d.
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We provide a nilpotency criterion for fusion systems in terms of the vanishing of its cohomology with twisted coefficients.
Homotopical Nilpotency and Homology Nilpotency: A Dimension--Connectivity Bound
本文证明有理空间同伦幂零性等于同调幂零性的维数-连通性充分界，并给出精细界及最优常数。研究问题为两类幂零性的等价条件，核心方法是极小模型与本原元附加，主要贡献是界的最优性证明。
On the topological complexity of non-simply connected spaces
本文研究了非单连通空间的拓扑复杂性，通过引入上同调类和谱序列，扩展了现有结果并确定了某些3流形的拓扑复杂性。
In this paper, we establish the rationality conjecture raised in \cite{FKS} for any $(r-1)$-connected ($r\geq 2$) $kr$-dimensional CW-complex $X$ ($k\geq 2$) having a unique spherical cohomology class $u\in \tilde{H}^r(X, \mathbb{Z})$ such that $u^k\not =0$. %which is nilpotent with order of nilpotency equal to $k+1$. Next, we illustrate (topologically) our result by giving the minimal cell structure of such a CW-complex whose cohomology is a truncated polynomial algebra.
Let $M$ be a nilmanifold with a fundamental group which is free $2$-step nilpotent on at least 4 generators. We will show that for any nonnegative integer $n$ there exists a self-diffeomorphism $h_n$ of $M$ such that $h_n$ has exactly $n$ fixed points and any self-map $f$ of $M$ which is homotopic to $h_n$ has at least $n$ fixed points. We will also shed some light on the situation for less generators and also for higher nilpotency classes.
Fixed point properties and cohomology of Banach representations of arithmetic groups
本文研究半单群格的固定点定理，通过证明高秩单群格在无不变向量时秩以下$L^p$上同调消失，推出低维无环复形作用有有限轨道，并证明Farb与Gromov的猜想。
Let $X$ be a finitistic space with its rational cohomology isomorphic to that of the wedge sum $P^2(n)\vee S^{3n} $ or $S^{n} \vee S^{2n}\vee S^{3n}$. We study continuous $\mathbb{S}^1$ actions on $X$ and determine the possible fixed point sets up to rational cohomology depending on whether or not $X$ is totally non-homologous to zero in $X_{\mathbb{S}^1}$ in the Borel fibration $X\hookrightarrow X_{\mathbb{S}^1} \longrightarrow B_{\mathbb{S}^1}$. We also give examples realizing the possible cases.
Geometric cohomology frames on Hausmann-Holm-Puppe conjugation spaces
对于某些具有对合的流形，不动点集的模2上同调环同构于流形的上同调环，直到其度数除以2。例子包括复射影空间和具有标准反全纯对合的格拉斯曼空间（以实射影空间和格拉斯曼空间作为不动点集）。Hausmann，Holm和Puppe把这个观察放在等变上同调的框架中，并提出了“共轭空间”的概念，其中环同态自然地产生于他们所谓的“上同调框架”的存在。更早的时候，Borel和Haefliger研究了度减半同构之间的上同调环的复杂和实射影空间和格拉斯曼使用理论的复杂和真正的分析循环和循环映射到上同调。在本说明中的主要结果给出了这两个结果之间的（纯拓扑）连接，并提供了一个几何直观的概念之间的上同调框架。特别地，我们看到，如果具有对合的流形X上的每一个上同调类是余维数为其不动点余维数的两倍（在X的不动点集内）的等变拓扑环的Thom类，则这些拓扑环将产生上同调框架。
Let $p$ be a fixed prime number and let $R$ denote a uniserial $p$-adic space group of dimension $d_x=(p-1)p^{x-1}$ and with cyclic point group of order $p^x$. In this short note we prove that all the quotients of $R$ of size bigger than or equal to $p^{d_x+x}$ have isomorphic mod $p$ cohomology groups. In particular, we show that the cohomology groups of sufficiently large quotients of the unique maximal class pro-$p$ group are isomorphic as $\F_p$-modules.
Let $C_2$ denote the cyclic group of order 2. We compute the $RO(C_2)$-graded cohomology of all $C_2$-surfaces with constant integral coefficients. We show when the action is nonfree, the answer depends only on the genus, the orientability of the underlying surface, the number of isolated fixed points, the number of fixed circles with trivial normal bundles, and the number of fixed circles with nontrivial normal bundles. When the action on the surface is free, we show the answer depends only on the genus, the orientability of the underlying surface, whether the action is orientation preserving…
Spectrum of equivariant cohomology as a fixed point scheme
本文证明了复约化群G在光滑射影簇X上的正则作用使得X的G-等变上同调环同构于某个正则不动点概形的坐标环，并推广到GKM空间如环面簇。
The mod 2 cohomology of fixed point sets of anti-symplectic involutions
设$M$是一个紧致的连通辛流形，其Hamilton作用在一个紧致$n$维环面$G=T^n$上.设$σ$是与$G$ -作用相容的反辛对合. $M$的实轨迹是$X$，即$σ$的不动点集。Duistermaat使用莫尔斯理论，根据$M$的上同调，给出了$X$的普通上同调的描述。在$X$上有一个剩余的$\G=(\Zt)^n$作用，我们可以利用Duistermaat的结果，以及关于等变上同调的一些一般事实，证明一个类似于Duistermaat定理的等变类似。在某些情况下，我们也可以将Goresky-Kottwitz-MacPherson和Goldin-Holm的定理推广到实轨迹。
When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah--Bott--Berline--Vergne converts the integral of an equivariantly closed form to a finite sum over the fixed points, providing a powerful tool for computing integrals on a manifold. This article seeks to give an accessible exposition of the equivariant localization formula.
The Gysin map of a map between compact oriented manifolds is the map in cohomology induced by the push-forward map in homology. In enumerative algebraic geometry, formulas for the Gysin map of a flag bundle play a vital role. These formulas are usually proven by algebraic or combinatorial means. This article shows how the localization formula in equivariant cohomology provides a systematic method for calculating the Gysin homomorphism in the ordinary cohomology of a fiber bundle. As examples, we recover classical pushforward formulas for generalized flag bundles. Our method extends the classic…
A polyhedral product is a natural subspace of a Cartesian product, which is specified by a simplicial complex K. The automorphism group Aut(K) of K induces a group action on the polyhedral product. In this paper we study this group action and give a formula for the fixed point set of the polyhedral product for any subgroup H of Aut(K). We use the fixed point data to compute examples of Bredon cohomolohgy for small non-Abelian groups such as $D_8$ and $Σ_4$.
This paper generalizes the concept of index and co-index and some related results for free actions of G = S0 on a paracompact Hausdorff space which were introduced by Conner and Floyd. We define the index and co-index of a finitistic free G-space X, where G = Sd , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)-sphere (resp. (4n+3)-sphere) for the action of componentwise multiplication of G = S1 (resp. S3) is n. We also determine the orbit spaces of free actions of G = S3 on a finitistic space X wit…
This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.
We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava $E$-theory, $H^*(\mathbb{G}_2, E_t)$, at $p=2$, for $0\leq t < 12$, using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the $d_3$-differentials in the homotopy fixed point spectral sequence for the $K(2)$-local sphere spectrum. These cohomology groups and differentials play a central role in $K(2)$-local stable homotopy theory.
Let $G$ be a Lie group and $H$ be a subgroup of it. We can construct a bisimplicial manifold $NG(*) \rtimes NH(*)$ and the de Rham complex $Ω^*(NG(*) \rtimes NH(*))$ on it. This complex is a triple complex and the cohomology of its total complex is isomorphic to $H^*(B(G \rtimes H))$. In this paper, we show that the total complex of the double complex $Ω^q(NG(*) \rtimes NH(*))$ is isomorphic to the continuous cohomology $H_c^*(G \rtimes H;S^q{\mathcal G}^* \otimes S^q{\mathcal H}^*)$ for any fixed $q$.
In this paper, we compute the $RO(C_{pq})$-graded cohomology of $C_{pq}$-orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong and Lewis for the group $C_p$. The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group $C_p$ was proved by Lewis. We demonstrate that certain complex projective sp…
We show that the well-known fact that the equivariant cohomology of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with maximal isotropy rank. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivarian…
It is well known that cohomology with compact supports is not a homotopy invariant but only a proper homotopy one. However, as the proper category lacks of general categorical properties, a Brown representability theorem type does not seem reachable. However, by proving such a theorem for the so called exterior cohomology in the complete and cocomplete exterior category, we show that the $n$-th cohomology with compact supports of a given countable, locally finite, finite dimensional relative CW-complex $(X,\mathbb{R}_+)$ is naturally identified with the set $[X,K_n]^{\mathbb{R}_+}$ of exterior…
Let F be an arbitrary family of subgroups of a group G and let Orb be the associated orbit category. We investigate interpretations of low dimensional F-Bredon cohomology of G in terms of abelian extensions of Orb. Specializing to fixed point functors as coefficients, we derive several group theoretic applications and introduce Bredon-Galois cohomology. We prove an analog of Hilbert's Theorem 90 and show that the second Bredon-Galois cohomology is a certain intersection of relative Brauer groups. As applications, we realize the relative Brauer group Br(L/K) of a finite separable non-normal ext…
Equivariant cohomology for cyclic groups of square-free order
本文计算了无平方因子阶循环群 $G=C_n$ 的 $G$-轨道的 $RO(G)$-分次上同调，针对常数和 Burnside 环 Mackey 函子揭示了群结构与环结构，并据此证明了某些 $G$-复形的自由性结果。
Equivariant cohomology and analytic descriptions of ring isomorphisms
本文考虑一类连通闭$G$ -流形，其中每个$M$在$M_G$（或$G$ -等变形式）中与零完全非同调，且具有非空的有限不动点集，其中$G={\Bbb Z}_2$ .借助于等变指标，我们给出了这类$G$ -流形的等变上同调的代数描述，从而得到了这类$G$ -流形的等变上同调环之间的环同构的解析描述，以及这两个$G$ -流形的等变上同调环同构的充要条件.这也导致我们分析有多少等变上同调环同构等$G$ -流形在2-和3-维的情况下。
Reconstructing the orbit type stratification of a torus action from its equivariant cohomology
本文研究了如何从紧空间上环面作用的有理等变上同调代数中提取轨道类型分层信息，证明了其编码了分支元素子偏序集，并在等变形式及光滑条件下揭示了流形等变上同调对各分层信息的编码机制。
Cohomology of the Moduli Stacks of Real Vector Bundles on Type I Real Algebraic Curves
研究第一类实代数曲线上固定秩和度的实向量丛模空间，并用示性类确定其模2上同调代数。
The mod p cohomology of the Morava stabilizer group at large primes
该研究针对大素数下高度为n的扩展Morava稳定子群模p上同调问题，通过形变Ravenel李代数模型、结合导出不变循环定理与约化李代数上同调新方法，证明其结果为n个生成元的外代数。
O-minimal cohomology and definably compact definable groups
设N是一个实封闭的域的O-极小展开。我们发展了N-可定义流形和N-可定义映射范畴的上同调理论，并利用它解决了N-可定义紧N-可定义阿贝尔群上扭点存在性的Peterzil-Steinhorn问题.我们计算了N-可定义紧N-可定义群的上同调环，证明了Poincare对偶定理、Alexander对偶定理、Lefschetz对偶定理和Lefschetz不动点定理的一个o-极小模拟.
Let $C_2$ be the cyclic group of order two. We show that the $RO(C_2)$-graded Bredon cohomology of a finite Rep($C_2$)-complex is free as a module over the cohomology of a point when using coefficients in the constant Mackey functor $\underline{\mathbb{F}_2}$. This paper corrects some errors in Kronholm's proof of this freeness theorem. It also extends the freeness result to finite type complexes, those with finitely many cells of each fixed-set dimension. We give a counterexample showing the theorem does not hold for locally finite complexes.
We prove that the cohomology of the moduli space of morphisms of a fixed finite degree from a smooth projective curve $C$ of genus $g$ to a complete simplicial toric variety $\mathbb{P}(Σ)$, denoted by the rational polyhedral fan $Σ$, stabilizes. As an arithmetic consequence we obtain a resolution of the Batyrev-Manin conjecture for toric varieties over global function fields in all but finitely many characteristics.
Equivariant operads, string topology, and Tate cohomology
从一个具有群G作用的操作数C出发，利用同伦不动点和轨道谱构造了新的操作数。当C的广义G-Tate上同调是平凡的时，这些新的运算是等价的。将这一理论应用于小圆盘操作数C_2（这是一个S^1操作数），我们得到了格兹勒引力操作数的变化，我们表明它支配着查斯-沙利文弦括号。
We study Cohen-Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang-Skjelbred Lemma, and its stronger version, the exactness of an Atiyah-Bredon sequence, hold. The main difference is that the fixed point set is replaced by the union of lowest dimensional orbits. We find sufficient conditions for the Cohen-Macaulay property such as the existence of an invariant Morse-Bott function whose critical set is …
Group cohomology of universal ordinary distribution
对任意无平方因子的奇整数r，本文给出了泛常分布U_r的群上同调G_r=Gal（Q（mu_r）/Q）的完全刻画.此外，如果M是一个固定整数，对r的所有素因子l，它能整除l-1，我们还研究了上同调群H^{*}（G_r，U_r/MU_r）.特别地，我们解释了元素kappa_{r'}对于r'的神秘构造|r是Rubin构造Kolyvagin元中的一个元素，它是由上同调群H^{0}（G_r，U_r/MU_r）的某个Z/M Z-基通过一个赋值映射构造的.
String topology prospectra and Hochschild cohomology
我们研究弦拓扑的分类空间的连通紧李群，绘图连接Hochschild上同调和等变同伦理论。首先，对于紧李群$G$，我们证明了弦拓扑原谱$LBG^{-TBG}$等价于$G$对自身的共轭作用$G^{hG}$的同伦不动点原谱。同时，我们用同伦轨道谱$(DG)_{hG}$来标识$LBG^{-ad}$，并研究了这些同伦轨道谱上的环和余环结构。最后，我们表明，在同源性，这些产品可以确定与Gerstenhaber上积的Hochschild上同调的$C^*(BG)$和$C_*(G)$，分别。这些，反过来，同构通过Koszul对偶。
On the cohomology rings of Hamiltonian T-spaces
设$M$是一个具有环面$T$的Hamilton作用的辛流形.令$F$表示$T$ -作用的不动点集，令$i:F\hookrightarrow M$表示包含。利用F. Kirwan \cite{K}在等变上同调中的诱导映射$i^*:H_T^*(M) \to H_T^*(F)$是一个内射。给出了映射$i^*$的象的Goresky-Kottwitz-MacPherson \cite{GKM}公式的一个简单证明.
Equivariant real cycle class map and Witt-sheaf cohomology of classifying spaces
本文研究等变实循环类映射与Witt-sheaf上同调的关系，通过分类空间的上同调分析，探讨了实代数群的几何分类空间与强实形式之间的联系，并计算了旋群的上同调不变量数量。
Mod 2 magnitude cohomology ring of real hyperplane arrangements
本文针对有限中心实超平面排列的顶图，改进Koizumi的模长同调结果，利用典范上同调基确定其交叉分次模长上同调环，明确了直和项的秩性质。
Let PConf^n M be the configuration space of ordered n-tuples of distinct points on a smooth manifold M admitting a nowhere-vanishing vector field. We show that the ith cohomology group with coefficients in a field H^i(PConf^n M, k) is an N-module, where N is the category of noncommutative finite sets introduced by Pirashvili and Richter. Studying the representation theory of N, we obtain new polynomiality results for the cohomology groups H^i(PConf^n M, k). In the case of unordered configuration space Conf^n M = (PConf^n M)/S_n and rational coefficients, we show that cohomology dimension in fi…
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