Igor Kriz, Yunze Lu
We completely calculate the $RO(G)$-graded coefficients of ordinary equivariant cohomology where $G$ is the dihedral group of order $2p$ for a prime $p>2$ both with constant and Burnside ring coefficients. The authors first proved it for $p=3$ and then the second author generalized it to arbitrary $p$. These are the first such calculations for a non-abelian group.
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The category of rational O(2)-equivariant spectra splits as a product of cyclic and dihedral parts. Using the classification of rational G-equivariant spectra for finite groups G, we classify the dihedral part of rational O(2)-equivariant spectra in terms of an algebraic model.
In this paper, we calculate $RO(C_2)$-graded cohomology of $C_2$-equivariant Eilenberg-Mac Lane spaces $K(\underline{Z/2}, n+σ)$ for $n\geq 0$. These can be used to give the relation between equivariant lambda algebra and equivariant Adams resolution and equivariant unstable Adams spectral sequence, which are defined in author`s dissertation.
We compute the $RO(C_2)$-graded Bredon cohomology of certain families of real and complex $C_2$-equivariant Grassmannians.
In this paper, we calculate the $RO(G)$-graded coefficients of $H\underline{\mathbb{Z}}$, the Eilenberg-MacLane spectrum of constant Mackey functor for quaternion group $Q_8$.
Pirashvili--Richter-type theorems for the reflexive and dihedral homology theories
本文证明自反同调与二面体同调可视为非交换集合范畴上的函子同调，从而建立了对应理论的 Pirashvili–Richter 型定理。
We compute the complete RO(G)-graded coefficients of "ordinary" cohomology with coefficients in Z/2 for G=(Z/2)^n.
We prove an $O(2)$-equivariant version of the Jones isomorphism relating the Borel $O(2)$-equivariant cohomology of the free loop space to the dihedral homology of the cochain algebra. We discuss polynomial forms and a variation of the de Rham isomorphism and use these to do a computation for the 2-sphere.
In $G$-equivariant stable homotopy theory, it is known that the equivariant Eilenberg-Mac Lane spectra representing ordinary equivariant cohomology have nontrivial $RO(G)$-graded homotopy corresponding to the equivariant (co)homology of representation spheres. We will compute the universal case of this ordinary $RO(G)$-graded homotopy in the case of $G=\mathcal{K}$, where $\mathcal{K}$ is the Klein-four group. In particular, we will compute a subring of the $RO(\mathcal{K})$-graded homotopy of $H\underline{A}$ for $\underline{A}$ the Burnside Mackey functor.
Let $C_2$ denote the cyclic group of order two. Given a manifold with a $C_2$-action, we can consider its equivariant Bredon $RO(C_2)$-graded cohomology. In this paper, we develop a theory of fundamental classes for equivariant submanifolds in $RO(C_2)$-graded cohomology in constant $\mathbb{Z}/2$ coefficients. We show the cohomology of any $C_2$-surface is generated by fundamental classes, and these classes can be used to easily compute the ring structure. To define fundamental classes we are led to study the cohomology of Thom spaces of equivariant vector bundles. In general the cohomology o…
The dihedral homology functor $HD:A_\infty^{\rm inv}(K)\to GrM(K)$ from the category $A_\infty^{\rm inv}(K)$ of involutive $A_\infty$-algebras over any commutative unital ring $K$ to the category $GrM(K)$ of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of involutive $A_\infty$-algebras into isomorphisms of graded modules.
Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups. Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of Özaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups.
The reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems
研究$RO(C_2)$-graded $C_2$-equivariant 稳定茎的环结构，通过生成元和关系描述，确定其在去除所有 nilpotent 元素后的结构，并记录有理 $C_2$-equivariant 球面的同调群结构。
This thesis consists of two main parts. In the second part, we recall how a description of local coefficients that Eilenberg introduced in the 1940s leads to spectral sequences for the computation of homology and cohomology with local coefficients. We then show how to construct new equivariant analogues of these spectral sequences for RO(G)-graded Bredon homology and cohomology. Finally, we use these spectral sequences to complete a sample calculation, in which we use the equivariant Serre spectral sequence and the equivariant cohomology of complex projective spaces to compute the cohomology o…
We construct hyper-homology spectral sequences of Z-graded and ROG-graded Mackey functors for Ext and Tor over G-equivariant S-algebras (A-infty ring spectra) for finite groups G. These specialize to universal coefficient and Kunneth spectral sequences.
In this paper we establish a formula for computing $d_2(sq^i(x))$ where $x$ is a permanent cycle in the $C_2$-equivariant Adams spectral sequence or the motivic Adams spectral sequence over $Spec(\mathbb{R})$. This requires establishing that the Adams towers have an $H_{\infty}$-structure as well as determining the attaching maps for $C_2$-equivariant projective spaces. The attaching maps of $C_2$-equivariant projective spaces can then be used to determine the coefficients of differentials in both the equivariant and motivic case. At the end some sample computations are given.
We describe a general method for algorithmic construction of G-equivariant chain homotopy equivalences from non-equivariant ones. As a consequence, we obtain an algorithm for computing equivariant (co)homology of Eilenberg-MacLane spaces K(pi,n), where pi is a finitely generated ZG-module. This result will be used in a forthcoming paper to construct equivariant Postnikov towers of simply connected spaces with free actions of a finite group G and further to compute stable equivariant homotopy classes of maps between such spaces. The methods of this paper work for modules over any non-negatively…
The $RO(C_2)$-graded Hurewicz image of Atiyah's Real K-theory
本研究确定了$RO(C_2)$分次的阿蒂亚实K-理论的胡雷维茨像，得到四类等变稳定茎中的无限族，还构造了捕获相关周期现象的等变亚当斯周期算子。
The $RO(ΠB)$-graded $C_2$-equivariant ordinary cohomology of $B_{C_2} U(1)$
本文将 $B_{C_2}U(1)$ 的 $C_2$-等变 Bredon 上同调从 $RO(C_2)$ 分次推广到更大的 $RO(\Pi B)$ 分次，从而纳入万有丛 Euler 类等更自然的生成元。
The Green functor structure of $RO(D_{2p})$-graded cohomology of a point
本文计算了常系数下一点的 $RO(D_{2p})$-分次上同调及其 Green 函子结构，推广了 Kriz-Lu 的加性计算结果。
In this paper, we compute the $RO(\mathbb{Z}/2)$-graded equivariant cohomology of rotation groups and Stiefel manifolds with particular involutions.
We study the action of the dihedral group on the (equivariant) cohomology of the toric manifolds associated with cycle graphs.
We compute the $RO(C_2)$-graded Green functor $\underlineπ_\star L_{KU_{C_2}/(2)}S_{C_2}$.
We compute the additive structure of the $\mathrm{RO}(C_n)$-graded Bredon equivariant cohomology of the equivariant classifying space $B_{C_n}\mathrm{SU}(2)$, for any $n$ that is either prime or a product of distinct odd primes, and we also compute its multiplicative structure for $n=2$. In particular, as an algebra over the cohomology of a point, we show that the cohomology of $B_{C_2}\mathrm{SU}(2)$ is generated by two elements subject to a single relation: writing $σ$ for the sign representation of $C_2$ in $\mathrm{RO}(C_2)$, the generators are an element $c$ in dimension $4σ$ and an eleme…
We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of $C_2$, using ordinary cohomology graded on representations of the fundamental groupoid, with coefficients in the Burnside ring Mackey functor. This extension of the $RO(C_2)$-graded theory allows for the definition of Euler classes, which are used as generators of the cohomology of the projective spaces. As an application, we give an equivariant version of Bezout's theorem.
The main result of this paper is the computation of TR^n_α(F_p;p) for αin R(S^1). These R(S^1)-graded TR-groups are the equivariant homotopy groups naturally associated to the S^1-spectrum THH(F_p), the topological Hochschild S^1-spectrum. This computation, which extends a partial result of Hesselholt and Madsen, provides the first example of the R(S^1)-graded TR-groups of a ring. These groups arise in algebraic K-theory computations, and are particularly important to the understanding of the algebraic K-theory of non-regular schemes.
This paper determines the RO(G)-graded Eilenberg-MacLane cohomology of the real, infinite, equivariant Grassmannians in the case G=Z/2. Possible connections with motivic characteristic classes for quadratic bundles are briefly discussed.
Let $G$ be a finite group. We re-analyze the splitting of rational $G$-spectra, which is discussed in Barne's thesis and traces back to Greenlees and May. We will study how the splitting behaves under a localization which is weaker than rationalization and discuss its application in computing equivariant cohomology of $G$-spectra. In particular, we will explicitly compute $π_\bigstar(H\underline{\mathbb{Z}})$ and $π_\bigstar(HA_G)$ for the dihedral group $G=D_{2p}$. The additive structures and partial multiplicative structures are already computed by Kriz, Lu, and Zou. Our new method will prov…
Poincare duality lies at the heart of the homological theory of manifolds. In the presence of the action of a group it is well-known that Poincare duality fails in Bredon's ordinary, integer-graded equivariant homology. We give here a detailed account of one way around this problem, which is to extend equivariant ordinary homology to a theory graded on representations of fundamental groupoids. Versions of this theory have appeared previously for actions of finite groups, but this is the first account that works for all compact Lie groups. The first part of this work is a detailed discussion of…
In this paper we compute $RO(G)$-graded homotopy Mackey functors of $H\underline{\mathbb{Z}}$, the Eilenberg-Mac Lane spectrum of the constant Mackey functor of integers for cyclic p-groups and give a complete computation for $G = C_{p^2}$ . We also discuss homological algebra of $\underline{\mathbb{Z}}$-modules for cyclic $p$-groups, and interactions between these two. The goal of this paper is to understand various slice spectral sequences as $RO(G)$-graded spectral sequences of Mackey functors.
$RO(C_p \times C_p)$-graded cohomology of universal spaces and the coefficient ring
本研究计算了子群族相关等变通用与分类空间的$RO(C_p \times C_p)$分次Bredon上同调，得到含乘法结构的系数环显式描述，并将其应用于等变复射影空间Bredon上同调的上同调运算提升研究。
A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres
本文确定了 $C_2$-等变Eilenberg-MacLane谱的 $RO(C_2)$-分次Hurewicz像，并发现其与球面上向量场问题紧密相关，进而利用广义Leibniz规则和广义Mahowald技巧证明了真 $C_2$-等变Adams谱序列中存在任意长度的非零Adams微分。
We compute the $RO(A)$-graded coefficients of $A$-equivariant complex and real topological $K$-theory for $A$ a finite elementary abelian $2$-group, together with all products, transfers, restrictions, power operations, and Adams operations.
For a compact Lie group G, we use G-equivariant Poincaré duality for ordinary RO(G)-graded homology to define an equivariant intersection product, the dual of the equivariant cup product. Using this, we give a homological construction of the equivariant Lefschetz number and a simple proof of the equivariant Lefschetz fixed point theorem. With similar techniques, an equivariant Fuller index with values in the rationalized Burnside ring is constructed.
The $RO(C_3)$-graded Bredon cohomology of $C_3$-surfaces in $\underline{\mathbb{Z}/3}$-coefficients
本文借助等变手术分类，计算常值 $\mathbb{Z}/3$ 系数下所有 $C_3$-曲面的 $RO(C_3)$-分次 Bredon 上同调，并证明其可由少数拓扑不变量及手术构造确定。
Real Hochschild homology as an equivariant Loday construction
本文研究了通过等变Loday构造解释真实Hochschild同调的方法，利用D_{2m}群作用于2m边形及对称群作用于排列多面体1骨架，探讨了适合子群H的Tambara函子应用。
This note contains a generalization to $p>2$ of the authors' previous calculations of the coefficients of $(\mathbb{Z}/2)^n$-equivariant ordinary cohomology with coefficients in the constant $\mathbb{Z}/2$-Mackey functor. The algberaic results by S.Kriz allow us to calculate the coefficients of the geometric fixed point spectrum $Φ^{(\mathbb{Z}/p)^n}H\mathbb{Z}/p$, and more generally, the $\mathbb{Z}$-graded coefficients of the localization of $H\mathbb{Z}/p_{(\mathbb{Z}/p)^n}$ by inverting any chosen set of embeddings $S^0\rightarrow S^{α_i}$ where $α_i$ are non-trivial irreducible representa…
We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit small and practical algebraic model, thereby providing a universal de Rham model for rational G-equivariant cohomology theories. The result builds on the first author's Adams spectral sequence, the second author's functors making rational spectra algebraic. There are several steps, some perhaps of wider interest (1) isotropy separation (replacing the category of G-spectra by modules over a diagram of isotropically simple ring G-spectra) (2) passage to fixed points on ring and module categories (r…
We give minimal presentations for the $RO(C_2)$-graded Bredon cohomology of the equivariant classifying spaces $B_{C_2}U(n), B_{C_2}SO(n)$ and $B_{C_2}Sp(n)$ with coefficients in the rational Burnside Green functor $A_{\mathbf Q}$. This results in an efficient description of rational $C_2$ equivariant Chern, Pontryagin and symplectic characteristic classes. These classes are then related to each other using the inclusions of maximal tori.
We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. This serves as the input to the $C_2$-equivariant Adams spectral sequence converging to the $RO(C_2)$-graded homotopy groups of an equivariant spectrum $ko_{C_2}$. Our approach is to use simpler $\mathbb{C}$-motivic and $\mathbb{R}$-motivic calculations as stepping stones.
We propose a new method to compute the $C_{2^n}$-equivariant homotopy groups of the Eilenberg-Mac Lane spectrum $H\underline{\mathbb{Z}}$ as a $RO(C_{2^n})$-graded Green functor using the generalized Tate squares. As an example, we completely compute the $C_4$ case and investigate two $\mathscr{P}$-homotopy limit spectral sequences for the family $\mathscr{P}=\{e,C_2\}$.
The $\mathbb{Z}/p$-equivariant dual Steenrod algebra for an odd prime $p$
该研究完全计算了常Mackey函子ℤ/p的RO(ℤ/p)-分次系数Hℤ/p_⋆Hℤ/p，属于代数拓扑领域的相关计算。
We prove that if G is the circle group or a profinite group, then the all of the homotopical information of the category of rational G-spectra is captured by triangulated structure of the rational G-equivariant stable homotopy category. That is, for G profinite or S1, the rational G-equivariant stable homotopy category is rigid. For the case of profinite groups this rigidity comes from an intrinsic formality statement, so we carefully relate the notion of intrinsic formality of a differential graded algebra to rigidity.
The Bredon equivariant cohomology of a point for cyclic groups
本文研究了奇数阶循环群下点的Bredon等变上同调，通过代数方法将其转化为稳定球面同调群问题。
A Guide to Equivariant Parametrized Cohomology
本文系统介绍Costenoble-Waner等变参数化胞腔上同调理论，重点讨论有限群（尤其$C_2$）情形，给出新计算并阐明其与局部系数上同调的关系。
In this paper, we investigate certain graded-commutative rings which are related to the reciprocal plane compactification of the coordinate ring of a complement of a hyperplane arrangement. We give a presentation of these rings by generators and defining relations. This presentation was used by Holler and I. Kriz to calculate the $\mathbb{Z}$-graded coefficients of localizations of ordinary $RO((\mathbb{Z}/p)^n)$-graded equivariant cohomology at a given set of representation spheres, and also more recently by the author in a generalization to the case of an arbitrary finite group. We also give…
We introduce a notion of freeness for $RO$-graded equivariant generalized homology theories, considering spaces or spectra $E$ such that the $R$-homology of $E$ splits as a wedge of the $R$-homology of induced virtual representation spheres. The full subcategory of these spectra is closed under all of the basic equivariant operations, and this greatly simplifies computation. Many examples of spectra and homology theories are included along the way. We refine this to a collection of spectra analogous to the pure and isotropic spectra considered by Hill--Hopkins--Ravenel. For these spectra, the …
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.
Let M be a simply-connected closed manifold and consider the (ordered) configuration space of $k$ points in M, F(M,k). In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F(M,k). We prove that our model it is at least a Sigma_k-equivariant differential graded model. We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold.
We compute the $G$-equivariant homotopy type of the real topological cyclic homology of spherical group rings with anti-involution induced by taking inverses in the group, where $G$ denotes the group $Gal(\mathbb{C}/\mathbb{R})$. The real topological Hochschild homology of a spherical group ring $\mathbb{S}[Γ]$, with anti-involution as described, is an $O(2)$-cyclotomic spectrum and we construct a map commuting with the cyclotomic structures from the $O(2)$-equivariant suspension spectrum of the dihedral bar construction on $Γ$ to the real topological Hochschild homology of $\mathbb{S}[Γ]$, wh…
$C_3$-equivariant stable stems
论文计算了$C_3$-equivariant稳定同调群$π_{i,j}^{C_3}$，针对茎小于25和权重范围-16到16的情况，并描述了几何固定点映射和底层映射。
Equivalences of the form $Σ^V X \simeq Σ^W X$ in equivariant stable homotopy theory
本文研究紧李群等变稳定同伦论中 $Σ^{V}X\simeq Σ^{W}X$ 型等价所编码的周期性现象，通过构造 $RO(G)$-分次 $J$-同态控制此类周期性并产生无限周期族，进而应用于等变谱序列中普遍周期性与微分的构造。
For $G$ a finite group, a normalized 2-cocycle $α\in Z^{2}\big(G,{\mathbb S}^{1}\big)$ and $X$ a $G$-space on which a normal subgroup $A$ acts trivially, we show that the $α$-twisted $G$-equivariant $K$-theory of $X$ decomposes as a direct sum of twisted equivariant $K$-theories of $X$ parametrized by the orbits of an action of $G$ on the set of irreducible $α$-projective representations of $A$. This generalizes the decomposition obtained in [Gómez J.M., Uribe B., Internat. J. Math. 28 (2017), 1750016, 23 pages, arXiv:1604.01656] for equivariant $K$-theory. We also explore some examples of thi…
The $RO(\mathcal{K})$-graded homotopy of Klein-four normed Mackey functors
计算Klein四元群$\mathcal{K}$的常值$\mathbb{F}_2$ Mackey函子的Hill-Hopkins-Ravenel范数的等变Eilenberg-Mac Lane谱的$RO(\mathcal{K})$-分次系数，并分析这些$RO(\mathcal{K})$-分次Tambara函子的乘法结构。
The $C_2$-effective spectral sequence for $C_2$-equivariant connective real $K$-theory
本文利用动机有效切片滤化和 $C_2$-等变 Betti 实现构造了用于 RO$(C_2)$-分次同伦群的 $C_2$-等变谱序列，并以此计算完备化 $C_2$-等变连通实 $K$-理论谱的同伦群，重新证明了已有的 Adams 谱序列结果。
Let $T$ be a compact torus. We prove that, up to equivariant rational equivalence, the category of $T$-simply connected, $T$-finite type $T$-spaces with finitely many isotropy types is completely described by certain finite systems of commutative differential graded algebras with consistent choices of degree $2$ cohomology classes. We show that the algebraic systems corresponding to finite $T$-CW-complexes are exactly those which satisfy the necessary condition imposed by the Borel localization theorem along with certain finiteness conditions. We derive an algebraic characterization of when an…
Let G be a general (not necessarily finite dimensional compact) Lie group, let g be its Lie algebra, let Cg be the cone on g in the category of differential graded Lie algebras, and consider the functor which assigns to a chain complex V the V-valued total de Rham complex of G. We describe the G-equivariant de Rham cohomology in terms of a suitable relative differential graded Ext, defined on the appropriate category of (G,Cg)-modules. The meaning of "relative" is made precise via the dual standard construction associated with the monad involving the aforementioned functor and the associated f…
An old result of the first author and David Lieberman says that if a compact Kaehler manifold X admits a holomorphic vector field V having at least one zero, then the Dolbeault cohomology algebra H^*(X, Ω^*) of X is isomorphic with the bigraded algebra associated to a certain filtered graded ring. If V has isolated zeros this is the graded algebra associated to a certain filtration of the coordinate ring of the scheme Z defined by zero(V). The purpose of this note is to show that if V is generated by a torus action then equivariant Dolbeault cohomology can be used to give direct geometric proo…
Algebraic models for equivariant rational homotopy theory for discrete groups
本文提出一种框架，将空间同调理论的代数模型推广到真正的equivariant情况。讨论了交换微分格雷德代数模型和完备微分格雷德李代数模型在有理空间中的应用，并与其他模型类别进行比较。
In this short note, we compute the rational $C_{2^n}$-equivariant stable stems and give minimal presentations for the $RO(C_{2^n})$-graded Bredon cohomology of the equivariant classifying spaces $B_{C_{2^n}}S^1$ and $B_{C_{2^n}}Σ_2$ over the rational Burnside functor $A_{\mathbf Q}$. We also examine for which compact Lie groups $L$ the maximal torus inclusion $T\to L$ induces an isomorphism from $H^*_{C_{2^n}}(B_{C_{2^n}}L;A_{\mathbf Q})$ onto the fixed points of $H^*_{C_{2^n}}(B_{C_{2^n}}T;A_{\mathbf Q})$ under the Weyl group action. We prove that this holds for $L=U(m)$ and any $n,m\ge 1$ bu…
A localization theorem for equivariant connective K theory
本文证明了循环群情形下Greenlees等变连通K理论谱可表示为等变拓扑K理论谱连通覆盖的RO(C_n)-分次局部化。
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