Julia Weber
We define equivariant homology theories using bordism of stratifolds with a G-action, where G is a discrete group. Stratifolds are a generalization of smooth manifolds which were introduced by Kreck. He defines homology theories using bordism of suitable stratifolds. We develop the equivariant generalization of these ideas.
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Generalizing the ideas of $\mathbb{Z}_k$-manifolds from Sullivan and stratifolds from Kreck, we define $\mathbb{Z}_k$-stratifolds. We show that the bordism theory of $\mathbb{Z}_k$-stratifolds is sufficient to represent all homology classes of a $CW$-complex with coefficients in $\mathbb{Z}_k$. We present a geometric interpretation of the Bockstein long exact sequences and the Atiyah-Hirzebruch spectral sequence for $\mathbb{Z}_k$-bordism ($k$ an odd number). Finally, for $p$ an odd prime, we give geometric representatives of all classes in $H_*(B\mathbb{Z}_p;\mathbb{Z}_p)$ using $\mathbb{Z}_p…
In this paper we give a geometric construction of the Borel equivariant (co)homology for spaces with a $G$-action, where $G$ is a compact Lie group with the property that the adjoint representation is orientable. A nice feature of these constructions is that there are corresponding Poincaré dual (co)homology theories called backwards (co)homology. This gives rise to a third relative (co)homology theory which we call stratifold Tate (co)homology. These Tate groups agree with the original definition of Tate cohomology for finite groups given by Swan. All constructions in this paper are geometric…
We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring N_*(X) of a topological space X. This homology theory Eh_* has coefficients Z/2 in every nonnegative dimension. There exists a natural transformation N_*(X)->Eh_*(X) that for X=pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds. For discrete groups G, we also define an equivariant version of the homology theory Eh_*, generalizing the equivariant Euler characteristic.
A careful account is given of generalized equivariant homology theories on the category of topological pairs acted on by a group. In particular, upon restriction to the category of equivariant simplicial complexes, the equivalence of equivariant simplicial homology (also known as Bredon homology), the second derived term of the Atiyah--Hirzebruch spectral sequence, and equivariant singular homology is demonstrated.
In this paper, for a finite group, we discuss a method for calculating equivariant homology with constant coefficients. We apply it to completely calculate the geometric fixed points of the equivariant spectrum representing equivariant (co)homology with constant coefficients. We also treat a more complicated example of inverting the standard representation in the equivariant homology of split extraspecial groups at the prime 2.
We construct a genuine $G$-equivariant extension of factorization homology for $G$ a finite group, assigning a genuine $G$-spectrum to a manifold with $G$-action. We show that $G$-factorization homology is compatible with Hill-Hopkins-Ravenel norms and satisfies equivariant $\otimes$-excision. Following Ayala-Francis we prove an axiomatic characterization of genuine $G$-factorization homology. Applications include a description of real topological Hochschild homology and relative topological Hochschild homology of $C_n$-rings using genuine $G$-factorization homology.
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb R}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of ${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As a technical tool, we calculate the homology groups of configurat…
Applications of equivariant factorization homology
本文利用参数化高范畴论构造的等变因子化同调，证明其可描述系列论文中的相关结果，为相关研究提供了理论工具。
Equivariant nonabelian Poincaré duality and equivariant factorization homology of Thom spectra
该研究利用参数化高范畴论构造等变Thom谱，证明等变非阿贝尔庞加莱对偶定理，刻画Thom谱的等变因子同调并完成多类谱与球面的相关同调计算。
In this note we state corrected and expanded versions of our previous results on power operations for $C_2$-equivariant Bredon homology with coefficients in the constant Mackey functor on $\mathbb{F}_2$. In particular, we give a version of the Adem relations. The proofs rely on certain results in equivariant higher algebra which we will supply in a longer version of this paper.
A cocyclic construction of $S^1$-equivariant homology and application to string topology
本文通过上循环构造建立循环同调与$S^1$-等变同调的联系，并应用于弦拓扑，给出自由环空间上重力代数结构的链级细化。
Reflexive homology and involutive Hochschild homology as equivariant Loday constructions
该研究证明了在特定条件下，反演同调与反演Hochschild同调可通过等变Loday构造来表示，并探讨了其相对版本。
We introduce the notion of the space of parallel strings with partially summable labels, which can be viewed as a geometrically constructed group completion of the space of particles with labels. We utilize this to construct a machinery which produces equivariant generalized homology theories from such simple and abundant data as partial monoids.
In this paper, we study the equivariant homotopy type of a connected sum of linear actions on complex projective planes defined by Hambleton and Tanase. These actions are constructed for cyclic groups of odd order. We construct cellular filtrations on the connected sum using spheres inside unitary representations. A judicious choice of filtration implies a splitting on equivariant homology for general cyclic groups under a divisibility hypothesis, and in all cases for those of prime power order.
In this paper, we prove an equivariant version of the classical Dold-Thom theorem. Associated to a finite group, a CW-complex on which this group acts and a covariant coefficient system in the sense of Bredon, we functorially construct a topological abelian group by the coend construction. Then we prove that the homotopy groups of this topological abelian group are naturally isomorphic to the Bredon equivariant homology of the CW-complex. At the end we present several examples of this result.
The Homology of Complex Equivariant Bordism
研究阿贝尔紧致李群\(A\)及定向\(A -\)谱\(E\)，通过两种方式计算tom Dieck的同伦\(A -\)等变复配边谱\(MU_A\)的\(E -\)同调并修正错误，还计算了几何\(A -\)等变复配边谱\(mU_A\)的\(E -\)同调。
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。
Fix a finite group G and an n-dimensional orthogonal G-representation V. We define the equivariant factorization homology of a V-framed smooth G-manifold with coefficients in an $E_V$-algebra using a two-sided bar construction, generalizing [And10, KM18]. This construction uses minimal categorical background and aims for maximal concreteness, allowing convenient proofs of key properties, including invariance of equivariant factorization homology under change of tangential structures. Using a geometrically-seen scanning map, we prove an equivariant version (eNPD) of the nonabelian Poincare dual…
Let $M$ be a covariant coefficient system for a finite group $G$. In this paper we analyze several topological abelian groups, some of them new, whose homotopy groups are isomorphic to the Bredon-Illman $G$-equivariant homology theory with coefficients in $M$. We call these groups equivariant Dold-Thom topological groups and we show that they are unique up to homotopy. We use one of the new groups to prove that the Bredon-Illman homology satisfies the infinite-wedge axiom and to make some calculations of the 0th equivariant homology.
The theories in our zoo are all bordism groups, which generalize the case of smooth manifolds by allowing singularities. There are many concepts of manifolds with singularities one could use here. For our pupose the objects the author introduced some years ago, which are called stratifolds, work particularly well. The zoo comes from forcing certain strata indexed by the subset $A$ to be empty. Special cases are ordinary singular homology and singular bordism. Despite their simple construction computations of these groups seem to be very complicated. We give a few simple examples. Thus there ar…
Szczarba's twisted shuffle and equivariant path homology of directed graphs
本文研究了标记设定下路径同调的等变性质，通过扭曲洗牌定义了链同构，并应用于有向图的Borel构造。
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 introduce equivariant factorization homology, extending the axiomatic framework of Ayala-Francis to encompass multiplicative invariants of manifolds equipped with finite group actions. Examples of such equivariant factorization homology theories include Bredon equivariant homology and (twisted versions of) Hochschild homology. Our main result is that equivariant factorization homology satisfies an equivariant version of tensor excision, and is uniquely characterised by this property. We also discuss applications to representation theory, such as constructions of categorical braid group acti…
In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincare formula and an equivariant Lefschetz theorem.
We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) ==> [X,Y]^G_* for rational G-spectra X and Y.
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…
The $H \underline{\mathbb{F}}_2$-homology of $C_2$-equivariant Eilenberg-MacLane spaces
本文将 Ravenel-Wilson Hopf 环技术扩展至 $C_2$-等变同伦理论，计算了与常值 Mackey 函子关联的 $C_2$-等变 Eilenberg-MacLane 空间的 $RO(C_2)$-分次同调，并探讨了相关的扭曲 bar 谱序列。
Andre-Quillen (co)homology and Equivariant Stable Homotopy Theory
针对等变背景下交换化函子类似物的问题，本文基于Hill的真导子等概念扩展到不完全Tambara函子，引入余切复形并构造基本谱序列类似物，同时修正了生成传递长正合序列的假设。
We generalize the results of a previous paper of ours to compact Lie groups. Using a recently developed ordinary equivariant homology and cohomology, we define equivariant Poincare complexes with the properties that (1) every compact G-manifold is an equivariant Poincare complex, (2) every finite equivariant Poincare complex (with some mild additional hypotheses) has an equivariant spherical Spivak normal fibration, and (3) the Pi-Pi Theorem holds for equivariant Poincare pairs under suitable gap hypotheses. The nice behavior of the ordinary equivariant homology and cohomology theories allows …
Following ideas of Graeme Segal, we construct an equivariant con- figuration space that is a model of equivariant connective K-homology spec- trum for finite groups, as a consequence we obtain an induction structure for equivariant connective K-homology. We describe explicitly the homology with complex coefficients for the fixed points of this configuration space as a Hopf algebra.
Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the …
We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a gamma-equivariant G-module A, when a separate group "gamma" acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.
Equivariant homology of stacks
本文利用层论与六种运算定义并研究栈的等变同调，并证明该构造适用于代数、复解析和拓扑范畴。
We define the orbit category for transitive topological groupoids and their equivariant CW-complexes. By using these constructions we define equivariant Bredon homology and cohomology for actions of transitive topological groupoids. We show how these theories can be obtained by looking at the action of a single isotropy group on a fiber of the anchor map, extending equivariant results for compact group actions. We also show how this extension from a single isotropy group to the entire groupoid action can be applied to the structure of principal bundles and classifying spaces.
We obtain the equivariant K-homology of the classifying space \underline{E}SL(3,Z) from the computation of its Bredon homology with respect to finite subgroups and coefficients in the representation ring. We also obtain the corresponding results for GL(3,Z). Our calculations give therefore the topological side of the Baum-Connes conjecture for these groups.
Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer-Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also …
We calculate the equivariant instanton Floer homology, in the sense of Miller Eismeier, for the trivial $SO(3)$-bundle over the binary polyhedral spaces with coefficients in a PID $R$ for which $2\in R$ is invertible. Along the way we modify a part of the algebraic construction needed to define the equivariant instanton Floer groups.
Equivariant operations in topological Hochschild homology
本文发现拓扑 Hochschild 同调与上同调之间的新等变关系，并通过计算有限素域的拓扑 Hochschild 上同调的拓扑 Hochschild 同调，将其解释为结构化运算环。
We obtain the equivariant K-homology of the classifying space \underline{E}W for W a right-angled or, more generally, an even Coxeter group. The key result is a formula for the relative Bredon homology of \underline{E}W in terms of Coxeter cells. Our calculations amount to the K-theory of the reduced C^*-algebra of W, via the Baum-Connes assembly map.
Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum $MU_{\mathbb{R}}$. In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green f…
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 refine the intersection product in homology to an equivariant setting, which unifies several known constructions. As an application, we give a common generalisation of the Chas-Sullivan string product on a manifold and the Chataur-Menichi string product on the classifying space by defining a string product on the Borel construction of a manifold. We prove a vanishing result which enables us to define a secondary product. The secondary product is then used to construct secondary versions of the Chataur-Menichi string product, and the equivariant intersection product in the Borel equivariant …
This paper provides an introduction to equivariant cohomology and homology using the approach of Goresky, Kottwitz, and MacPherson. When a group G acts suitably on a variety X, the equivariant cohomology of X can be computed using the combinatorial data of a skeleton of G-orbits on X. We give both a geometric definition and the traditional definition of equivariant cohomology. We include a discussion of the moment map and an algorithm for finding a set of generators for the equivariant cohomology of X. Many examples and explicit calculations are provided.
Let $G/H$ be a homogeneous variety, and let $X$ be a $G$-equivariant embedding of $G/H$such that the number of $G$-orbits in $X$ is finite. We show that the equivariant Borel-Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel-Moore homologies of the $G$-orbits. If $T$ is a maximal torus of $G$ such that each $G$-orbit has a $T$-fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel-Moore homology of $X$. We apply our findings to certain wonderful compactifications as well as to double flag varie…
We prove a Poincare-Alexander-Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and non-orientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen-Macaulayness of relative equivariant cohomology modules arising from the orbit filtration.
We study the C_p-equivariant Tate construction on the topological Hochschild homology THH(B) of a symmetric ring spectrum B by relating it to a topological version R_+(B) of the Singer construction, extended by a natural circle action. This enables us to prove that the fixed and homotopy fixed point spectra of THH(B) are p-adically equivalent for B = MU and BP. This generalizes the classical C_p-equivariant Segal conjecture, which corresponds to the case B = S.
Computations in Equivariant Topological Hochschild Homology
本文对等变拓扑霍奇schild同调（ETHH）开展基础计算，包括奇素数下ETHH(H𝔽̲ₚ)及等变复配边谱MU_G、MU_ℝ的ETHH计算，为后续相关研究提供输入。
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。
For G a Lie group acting on a symplectic manifold $(M,ω)$ preserving a pair of Lagrangians $L_0$, $L_1$, under certain hypotheses not including equivariant transversality we construct a G-equivariant Floer cohomology of $L_0$ and $L_1$.
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 define $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.
Breaking symmetries for equivariant coarse homology theories
本文提出粗几何中的对称性破缺构造，通过限制到更小群和空间获取等变粗同调类信息，并在等变粗K-同调中给出解析解释，应用于不变微分算子谱理论。
Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant K-homology of M. The universal equivariant Euler characteristic of M, which lives in a group U^G(M), counts the equivariant cells of M, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from U^G(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Eul…
We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,\mathbb{Z})$-module.
Equivariant instanton homology
本文构造四类等变瞬子Floer同调，证明其配边函子性、与既有理论的一致性并计算I^∞，附录给出相应dg-模代数框架。
We enlarge the category of bornological coarse spaces by adding transfer morphisms and introduce the notion of an equivariant coarse homology theory with transfers. We then show that equivariant coarse algebraic $K$-homology and equivariant coarse ordinary homology can be extended to equivariant coarse homology theories with transfers. In the case of a finite group we observe that equivariant coarse homology theories with transfers provide Mackey functors. We express standard constructions with Mackey functors in terms of coarse geometry, and we demonstrate the usage of transfers in order to p…
We identify the equivariant structure of the filtered pieces of the motivic filtration defined by Bhatt, Morrow and Scholze on the topological Hochschild cohomology spectrum of polynomial algebras over $\mathbb{F}_p$.
We show that the equivariant chain complex associated to a minimal CW-structure X on the complement M(A) of a hyperplane arrangement A, is independent of X. When A is a sufficiently general linear section of an aspheric arrangement, we explain a new way for computing the twisted homology of M(A).
Equivariant Witt Complexes and Twisted Topological Hochschild Homology
本文研究扭曲拓扑霍奇同调（THH）的等变同伦群代数结构，定义等变Witt复形并证明其捕捉扭曲THH等变同伦的结构，为等变代数研究提供新贡献。
Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the Čech cohomology groups of a tiling space in a highly geometric way. In this paper we consider homology groups of PE infinite chains. We establish Poincaré duality between the PE cohomology and PE homology. The Penrose kite and dart tilings are taken as our central running example, we show how through this formalism one may give highly approachable geometric descriptions of the generators of the Čech cohomology of their tiling space. These invariants are also considered in the context of rotational symmet…
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