Steven Lillywhite
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.
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Equivariant formality for rational $\C_{p^k}$-spaces
本文研究带有素数幂阶循环群作用的空间的等变形式性，分析其拓扑构造下的性质，处理内射性相关代数复杂性，最终探讨等变内蕴形式性。
In this note we present an analogue of equivariant formality in $K$-theory and show that it is equivalent to equivariant formality \emph{à la} Goresky-Kottwitz-MacPherson. We also apply this analogue to give alternative proofs of equivariant formality of conjugation action on compact Lie groups, left translation action on generalized flag manifolds, and compact Lie group actions with maximal rank isotropy subgroups.
Equivariant formality of corank-one isotropy actions and products of rational spheres
本文完全刻画了满足秩差为1且左作用等变形式化的连通李群对，并修正与扩展了具有奇偶维球面乘积有理同伦型的齐性商分类。
Equivariant formality of istropy actions
本文研究紧致连通李群子群在齐性空间上的迷向作用的等变形式性，通过K-理论类比给出表示论刻画，并统一证明若干齐性空间的等变形式性。
Equivariant formality of isotropic torus actions
本文研究连通 Lie 群在齐性空间上左作用的等变形式性，通过约化将问题归结为 $G$ 紧单连通且 $K$ 为环面的情形，分类了紧连通 Lie 群与圆子群的所有合格对，并证明了相关的上同调环结构。
Equivariant formality and representation theory
该论文利用$K$-理论版本的等变形式性，得到迷向作用等变形式性的两种刻画，并据此对单李群相关情形给出等变形式性的完整分类。
Equivariant Intrinsic Formality
本文研究等变有理同伦型按上同调分类的问题，证明在适当条件下 $\mathbb{Z}_p$ 作用的等变情形可归结为非等变情形。
We show that the isotropy action of a homogeneous space $G/K$, where $G$ and $K$ are compact, connected Lie groups and $K$ is defined by an automorphism on $G$, is equivariantly formal and that $(G, K)$ is a Cartan pair.
Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalisation, it is even known that their isotropy action is equivariantly formal. In this article we show that $(\mathbb{Z}_2\oplus \mathbb{Z}_2)$-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.
Equivariant formality of the little disks operad
本文证明了小n-圆盘操作元在有理数上关于SO(n)和O(n)是等变形式的，等价于有向和无向框架小圆盘操作元作为∞-操作元是有理形式的。
An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive. In this article, also motivated by the lack of junction between the notion of equivariant formality and the concept of formality of spaces (surging from rational homotopy theory) we suggest two new variations of equivariant formality: "MOD-formal actions" and "actions of formal core". We investigate and characterize these new terms in many different ways involving various tools from rational ho…
K-theory and formality
本文计算了等变形式等迷作用的整系数等变K-理论，通过构造谱序列映射给出证明，并由此得出等变形式性的新刻画及弱等变形式性的等价条件。
Equivariant formality and the cohomology of subgroups of right-angled Coxeter groups
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。
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…
Equivariant formality in complex-oriented theories
该研究将有理上同调的等变形式性结果推广至任意复定向上同调理论，借助新的上同调分裂结果完成证明，还建立了相关定理的类似结论并给出其在Morava K-理论中的应用。
The classification of Seifert manifolds was given in terms of numeric data by Seifert in 1933, and then generalized by Orlik and Raymond in 1968 to circle actions on closed 3d manifolds. In this paper, we further generalize the classification to circle actions on 3d manifolds with boundaries by adding a numeric parameter and a union of cycle graphs. Then we describe the equivariant cohomology of 3d manifolds with circle actions in terms of ring, module and vector-space structures. We also compute equivariant Betti numbers and Poincaré series for these manifolds and discuss the equivariant form…
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 equivariant cohomology $H^*_{T_I}(\mathcal Z_K)$ of moment-angle complexes $\mathcal Z_K$ with respect to the action of coordinate subtori $T_I \subset T^m$. We give a criterion for the equivariant formality of $\mathcal Z_K$ and obtain specifications for the cases of flag complexes and graphs.
Spaces of homomorphisms, formality and Hochschild homology
本文研究酉表示与同态簇的 $\mathbb{E}_\infty$ 结构，证明带阿贝尔因子的分解公式、自由群情形的形式性，并以高阶 Hochschild 同调刻画等变同调。
A stroll in equivariant $K$-theory
本文综述等变 $K$-理论的基础，重点介绍 Atiyah–Segal 局部化定理与公式，并讨论其在等变形式性中的应用及近期进展。
Formality is a topological property, defined in terms of Sullivan's model for a space. In the simply-connected setting, a space is formal if its rational homotopy type is determined by the rational cohomology ring. In the general setting, the weaker 1-formality property allows one to reconstruct the rational pro-unipotent completion of the fundamental group, solely from the cup products of degree 1 cohomology classes. In this note, we survey various facets of formality, with emphasis on the geometric and algebraic implications of 1-formality, and its relations to the cohomology jump loci and t…
We compute the equivariant complex K-theory ring of a cohomogeneity-one action of a compact Lie group at the level of generators and relations and derive a characterization of K-theoretic equivariant formality for these actions. Less explicit expressions survive for a range of equivariant cohomology theories including Bredon cohomology and Borel complex cobordism. The proof accordingly involves elements of equivariant homotopy theory, representation theory, and Lie theory. Aside from analysis of maps of representation rings and heavy use of the structure theory of compact Lie groups, a more cu…
Strong formality below middle degree implies strong formality
本文证明强形式性 manifold 的超平面截面继承强形式性，尤其针对由正线包定义的广义完全交集，以及具有中心上同调宽度 n/2-1 的紧致 Kahler 纳米，以及无非平凡乘法关系的紧致 ∂∂̄-manifolds 的强形式性。
In this paper we compute homotopical equivariant bordism for the group ${\bf Z/2}$, namely $MO^{\bf Z/2}$, geometric equivariant bordism $Ω^{\bf Z/2}_*$, and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. In doing these computations, we use the techniques the author developed in the complex setting. Because we are working in the real setting only with Z/2, these techniques simplify greatly.
We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck's splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed point functors and the motivic Adams isomorphism.
We generalize the Mahowald invariant to the $\mathbb{R}$-motivic and $C_2$-equivariant settings. For all $i>0$ with $i \equiv 2,3 \mod 4$, we show that the $\mathbb{R}$-motivic Mahowald invariant of $(2+ρη)^i \in π_{0,0}^{\mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Adams' classical $v_1$-periodic families, and for all $i > 0$, we show that the $\mathbb{R}$-motivic Mahowald invariant of $η^i \in π_{i,i}^{\mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Andrews' $\mathbb{C}$-motivic $w_1$-periodic families. We prove analogous results about the $C_2$-equivariant Mah…
In this paper we prove that the quotient of any real or complex moment-angle complex by any closed subgroup in the naturally acting compact torus on it is equivariantly homotopy equivalent to the homotopy colimit of a certain toric diagram. For any quotient we prove an equivariant homeomorphism generalizing the well-known Davis-Januszkiewicz construction for quasitoric manifolds and small covers. We deduce formality of the corresponding Borel construction space under the natural assumption on the group action in the complex case leading to the new description of the equivariant cohomology for …
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。
In this paper we give a model for equivariant $(\infty, 1)$-categories. We modify an approach of Shimakawa for equivariant $Γ$-spaces to the setting of simplicial spaces. We then adapt Rezk's Segal and completeness conditions to fit with this setting.
Equivariant cohomology epimorphisms and face ring quotients for Hamiltonian and complexity one GKM$_4$ manifolds
本文研究了GKM$_4$流形的等变上同调环结构，通过图论方法证明了扩展GKM图的限制映射是满射，并探讨了其在哈密顿和复杂度一作用中的应用。
The aim of this note is to provide a comprehensive treatment of the homotopy theory of $Γ$-$G$-spaces for $G$ a finite group. We introduce two level and stable model structures on $Γ$-$G$-spaces and exhibit Quillen adjunctions to $G$-symmetric spectra with respect to a flat level and a stable flat model structure respectively. Then we give a proof that $Γ$-$G$-spaces model connective equivariant stable homotopy theory along the lines of the proof in the non-equivariant setting given by Bousfield and Friedlander. Furthermore, we study the smash product of $Γ$-$G$-spaces and show that the functo…
We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N-infinity operads, equivariant generalizations of E-infinity operads. Algebras in equivariant spectra over an N-infinity operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N-infinity operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant in…
Orbit spaces of equivariantly formal torus actions of complexity one
本文研究了一类复杂度为一的 torus 作用的轨道空间，证明了其同调球面性质，并给出了等变形式的判据及 betti 数的表达式。
Cohomology operations for moment-angle complexes and resolutions of Stanley-Reisner rings
本文通过研究矩角复形上标准环面作用诱导的上同调运算，将Stanley-Reisner环的极小自由消解赋予拓扑解释，并利用所得工具完全刻画了矩角复形上环面作用的等变形式性。
Cyclotomic Structures in Symplectic Topology
本文推广Cohen-Jones-Segal构造至等变Morse函数，发展等变Floer理论，并在辛流形上建立辛上同调谱提升的分圆结构，连接Floer同调与p-adic Hodge理论。
We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of equivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting and prove a Cancellation Theorem.
Equivariant simplicial distributions and quantum contextuality
本文提出等变语境性框架，构造可检测语境性的上同调类，并建立其与基于测量的量子计算中计算原语的联系。
Local cohomology techniques in equivariant homotopy theory, introduced by John Greenlees, may be applied to understand homology of classifying spaces through other equivariant data. In this paper we relate the local cohomology filtration to the families filtration. By doing so, we may identify geometry codified by the local cohomology filtration in the setting of equivariant bordism. The constructions which arise are naturally analyzed by localized K-theory machinery due to Atiyah and Segal, which we review.
This PhD thesis deals with some new models of intensional type theory and the Univalence Axiom introduced by Vladimir Voevodsky. Our work takes place in the framework of the definitions of type-theoretic fibration categories (the notion of model under consideration in this thesis) and universe in a type-theoretic fibration category, definitions due to Michael Shulman. The goal of this thesis consists mainly in the exploration of the stability of the univalence axiom, in particular in the following sense: being given a type-theoretic fibration category C equipped with a univalent universe U, we…
In this paper, we develop the theory of equivariant motivic homotopy theory, both unstable and stable. While our original interest was in the case of profinite group actions on smooth schemes, we discuss our results in as broad a setting as possible so as to be applicable in a variety of contexts, for example to the case of smooth group scheme actions on schemes that are not necessarily smooth. We also discuss how ${\mathbb A}^1$-localization behaves with respect to mod-$\ell$-completion, where $\ell$ is a fixed prime.
In this article we extend the classical definitions of equivariant cohomotopy theory to the setting of proper actions of Lie groups. We combine methods originally developed in the analysis of nonlinear differential equations, mainly in connection with Leray-Schauder theory, and on the other hand from developments of equivariant $K$-Theory by N.C. Phillips. We prove the correspondence with a previous construction of W. Lück by constructing an index. As an illustration of these methods, we introduce a Burnside ring defined in analytical terms. With this definition, we extend a weak version of th…
We define Pin(2)-equivariant Seiberg-Witten Floer homology for rational homology 3-spheres equipped with a spin structure. The analogue of Froyshov's correction term in this setting is an integer-valued invariant of homology cobordism whose mod 2 reduction is the Rokhlin invariant. As an application, we show that there are no homology 3-spheres Y of Rokhlin invariant one such that Y # Y bounds an acyclic smooth 4-manifold. By previous work of Galewski-Stern and Matumoto, this implies the existence of non-triangulable high-dimensional manifolds.
We give a characterization of those Alexandrov spaces admitting a cohomogeneity one action of a compact connected Lie group $G$ for which the action is Cohen--Macaulay. This generalizes a similar result for manifolds to the singular setting of Alexandrov spaces where, in contrast to the manifold case, we find several actions which are not Cohen--Macaulay. In fact, we present results in a slightly more general context. We extend the methods in this field by a conceptual approach on equivariant cohomology via rational homotopy theory using an explicit rational model for a double mapping cylinder…
In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces X equipped with a torus action, the T-equivariant cohomology ring of X can be described by combinatorial data obtained from its orbit decomposition. Thus, their theory transforms calculations of the equivariant topology of X to those of the combinatorics of the orbit decomposition. Since then, many authors have studied this interplay between topology and combinatorics. In this paper, we generalize the theorem of Goresky, Kottwitz, and MacPherson to the (possibly infinite-dimensional) setting where X is any equivariant ce…
We prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups $X/G$ in the setting of derived loop spaces as well as Hochschild homology and its cyclic variants. We show that the derived loop spaces of the stack $X/G$ and its classical $z$-fixed point stack $π_0(X^z)/G^z$ become equivalent after completion along a semisimple parameter $[z] \in G//G$, implying the analogous statement for Hochschild and cyclic homology of the dg category of perfect complexes $\text{Perf}(X/G)$. We then prove an analogue of th…
We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial G-sets to symmetric G-spectra, where G is a finite group. We extend a notion of G-linearity suggested by Blumberg to define stably excisive and rho-analytic homotopy functors, as well as a G-differential, in this equivariant context. A main result of the paper is that analytic functors with trivial derivatives send highly connected G-maps to G-equivalences. It is analogous to the classical result of Goodwillie that "functors with zero derivative are locally constant". As main example we show that Hes…
In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S of T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen-Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen-Ruan cohomology in terms of orbifold fixed point data.
An Arithmetic Topology viewpoint on Descent theory and Equivariant Categories
本文通过算术拓扑视角，统一了降阶理论与等变范畴，建立了算术与拓扑不变量的类比，并展示了其在范畴层面的应用。
A twist is a datum playing a role of a local system for topological $K$-theory. In equivariant setting, twists are classified into four types according to how they are realized geometrically. This paper lists the possible types of twists for the torus with the actions of the point groups of all the 2-dimensional space groups (crystallographic groups), or equivalently, the torus with the actions of all the possible finite subgroups in its mapping class group. This is carried out by computing Borel's equivariant cohomology and the Leray-Serre spectral sequence. As a byproduct, the equivariant co…
Equivariant Floer homotopy via Morse-Bott theory
本研究将Cohen-Jones-Segal构造推广至Morse-Bott情形，定义流范畴Morse-Bott类似物的标架并关联稳定同伦型，进而在Floer理论中构造球谱系数辛上同调的圆周等变模型，证明余切丛零截面情形下的等变Viterbo同构定理。
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