Ruben Sanchez-Garcia
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.
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In this paper we compute the Bredon homology of wallpaper groups with respect to the family of finite groups and with coefficients in the complex representation ring. We provide explicit bases of the homology groups in terms of irreducible characters of the representation rings of the stabilizers.
For Artin groups of dihedral type, we compute the Bredon homology groups of the classifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.
In this article we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum-Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the $K$-theory of the reduced $C^{*}$-algebra of the group. We show the power of this method giving explicit computations for the group $SL_2(\mathbb{Z}[i])$. In order to carry out these computations we use an Atiyah-Segal type spectral sequence together with the Bredon homol…
We provide some new computations of Farrell--Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell--Tate or Bredon homology, one needs cell complexes wherecell stabilizers fix their cells pointwise. We provide two algorithms computing an efficient subdivision of a complex to achieve this rigidity property. Applying these algorithms to available cell complexes for PSL4(Z) provides computations of Farrell--Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring.
Bredon sheaf cohomology
本文提出Bredon层上同调这一新的等变上同调理论，用以描述有限群作用下局部紧Hausdorff空间等变层的代数K-理论与连续函数C*-代数的等变E-理论，建立其性质并证明强唯一性定理，推广了Efimov与Clausen的相关结果。
Localization for coarse homology theories
本文引入Bredon风格等变粗同调理论，证明其满足局部化定理，并指出一般等变粗同调理论可被其逼近，进而得到Segal局部化定理的粗类比。
We develop an Eilenberg-Moore spectral sequence to compute Bredon cohomology of spaces with an action of a group given as a pullback. Using several other spectral sequences, and positive results on the Baum-Connes Conjecture, we are able to compute Equivariant K-theory and K-Homology of the reduced group C*-algebra of a 6-dimensional crystallographic group $Γ$ introduced by Vafa and Witten. We also use positive results on the Farrell-Jones Conjecture to give a vanishing result for the negative algebraic K-theory of the integral group ring of $Γ$.
Bredon motivic cohomology of the complex numbers
本文在复数域上计算了$C_2$-等变Bredon母题上同调环，并利用刚性将Suslin的经典结果推广到等变母题框架。
On homological Real trace methods
该研究开发了同调实迹方法以理解实拓扑周期同调与实拓扑负循环同调的连续模2 Bredon同调，推广了Bruner和Rognes的工作，并利用该方法完成了实配边相关的同调计算。
Bredon motivic cohomology of the real numbers
本文在实数域上以 $\Z/2$-系数计算 $C_2$-等变 Borel 动机上同调环与 Bredon 动机上同调群，证明其为点空间 $RO(C_2\times C_2)$-分次 Bredon 上同调环的真子环，并推广至任意实闭域。
We establish formulae for the part due to torsion of the equivariant K-homology of all the Bianchi groups (PSL\_2 of the imaginary quadratic integers), in terms of elementary number-theoretic quantities. To achieve this, we introduce a novel technique in the computation of Bredon homology: representation ring splitting, which allows us to adapt the recent technique of torsion subcomplex reduction from group homology to Bredon homology.
In this paper we define twisted equivariant K-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite CW-complexes with equivariant stable projective bundles. A classification of these bundles is shown. We also obtain a completion theorem and apply these results to proper actions of groups.
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.
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.
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.
We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramo…
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。
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…
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。
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 compute the equivariant $KO$-homology of the classifying space for proper actions of $\textrm{SL}_3(\mathbb{Z})$ and $\textrm{GL}_3(\mathbb{Z})$. We also compute the Bredon homology and equivariant $K$-homology of the classifying spaces for proper actions of $\textrm{PSL}_2(\mathbb{Z}[\frac{1}{p}])$ and $\textrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$ for each prime $p$. Finally, we prove the Unstable Gromov-Lawson-Rosenberg Conjecture for a large class of groups whose maximal finite subgroups are odd order and have periodic cohomology.
We consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the the…
Replaces Previous version. Includes comments on poincare duality for twisted equivariant in the context of proper and discrete actions and the Baum-Connes Conjecture. We use a spectral sequence proposed by C. Dwyer and previous work by Sanchez-Garcia and Soule to compute Twisted Equivariant K-theory groups of the classifying space for proper actions of Sl3(Z). After proving a Universal coefficient theorem in Bredon Cohomology with specific coefficients, we compute the twisted equivariant K-homology and state a relation to the Baum-Connes Conjecture with coefficients.
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。
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。
We define and study equivariant analytic and local cyclic homology for smooth actions of totally disconnected groups on bornological algebras. Our approach contains equivariant entire cyclic cohomology in the sense of Klimek, Kondracki and Lesniewski as a special case and provides an equivariant extension of the local cyclic theory developped by Puschnigg. As a main result we construct a multiplicative Chern-Connes character for equivariant KK-theory with values in equivariant local cyclic homology.
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…
Topological equivariant coarse K-homology
本文构造了具有严格群作用的$C^{*}$-范畴上的等变粗同调理论，通过引入受控于有界粗空间的Roe范畴并应用同调函子，验证了轨道范畴上的CP-函子性质，进而影响装配映射的单射性。
We consider groups G which have a cocompact, 3-manifold model for the classifying space \underline{E}G. We provide an algorithm for computing the rationalized equivariant K-homology of \underline{E}G. Under the additional hypothesis that the quotient 3-orbifold \underline{E}G/G is geometrizable, the rationalized K-homology groups coincide with the rationalized K-theory of the reduced C*-algebra of G. We illustrate our algorithm on some concrete examples.
We define and study equivariant periodic cyclic homology for locally compact groups. This can be viewed as a noncommutative generalization of equivariant de Rham cohomology. Although the construction resembles the Cuntz-Quillen approach to ordinary cyclic homology, a completely new feature in the equivariant setting is the fact that the basic ingredient in the theory is not a complex in the usual sense. As a consequence, in the equivariant context only the periodic cyclic theory can be defined in complete generality. Our definition recovers particular cases studied previously by various author…
We generalize several comparison results between algebraic, semi-topological and topological K-theories to the equivariant case with respect to a finite group.
This paper is our first step in establishing a de Rham model for equivariant twisted $K$-theory using machinery from noncommutative geometry. Let $G$ be a compact Lie group, $M$ a compact manifold on which $G$ acts smoothly. For any $α\in H^3_G (M, {\mathbb Z})$ we introduce a notion of localized equivariant twisted cohomology $H^\bullet ({\barΩ}^\bullet (M, G, L)_g, d^α_{G^g})$, indexed by $g\in G$. We prove that there exists a natural family of chain maps, indexed by $g\in G$, inducing a family of morphisms from the equivariant periodic cyclic homology $HP^G_\bullet ( C^\infty (M, α) )$, whe…
We present an Eilenberg-Steenrod-like axiomatic framework for equivariant coarse homology and cohomology theories. We also discuss a general construction of such coarse theories from topological ones and the associated transgression maps. A large part of this paper is devoted to showing how some well-established coarse (co-)homology theories, whose equivariant versions are either already known or will be introduced in this paper, fit into this setup. Furthermore, a new and more flexible notion of coarse homotopy is given which is more in the spirit of topological homotopies. Some, but not all,…
Equivariant coarse homotopy theory and coarse algebraic $\boldsymbol{K}$-homology
本文提出等变粗同调理论的公理框架，并引入等变Bornological粗空间和universal理论，探讨了等变粗代数K-上同调及其与锥 functor、组装映射的关系，为后续研究split-injectivity结果奠定基础。
We compute the equivariant $K$-homology of the classifying space for proper actions, for compact 3-dimensional hyperbolic reflection groups. This coincides with the topological $K$-theory of the reduced $C^\ast$-algebra associated to the group, via the Baum-Connes conjecture. We show that, for any such reflection group, the associated $K$-theory groups are torsion-free. As a result we can promote previous rational computations to integral compu- tations. Our proof relies on a new efficient algebraic criterion for checking torsion-freeness of K-theory groups, which could be applied to many othe…
We consider a invariant Dirac operator D on a manifold with a proper and cocompact action of a discrete group G. It gives rise to an equivariant K-homology class [D]. We show how the index of the induced orbifold Dirac operator can be calculated from [D] via the assembly map. We further derive a formula for this index in terms of the contributions of finite cyclic subgroups of G. According to results of W. Lueck, the equivariant K-homology can rationally be decomposed as a direct sum of contributions of finite cyclic subgroups of G. Our index formula thus leads to an explicit decomposition of …
Let G be a discrete group and let X be a G-finite, proper G-CW-complex. We prove that Kasparov's equivariant K-homology groups KK^G(C_0(X),\C) are isomorphic to the geometric equivariant K-homology groups of X that are obtained by making the geometric K-homology theory of Baum and Douglas equivariant in the natural way. This reconciles the original and current formulations of the Baum-Connes conjecture for discrete groups.
This is a further investigation of our approach to group actions in homological algebra in the settings of homology of Γ-simplicial groups, particularly of Γ-equivariant homology and cohomology of Γ-groups. This approach could be called Γ-homological algebra. The abstract kernel of non-abelian extensions of groups, its relation with the obstruction to the existence of non-abelian extensions and with the second group cohomology are extended to the case of non-abelian Γ-extensions of Γ-groups. We compute the rational Γ-equivariant (co)homology groups of finite cyclic Γ-groups. The isomorphism of…
Using a smooth version of the Connes--Thom isomorphism in Grensing's bivariant K-theory for locally convex algebras, we prove an equivariant version of the Connes--Thom isomorphism in periodic cyclic homology. As an application, we prove that periodic cyclic homology is invariant with respect to equivariant strict deformation quantization.
Equivariant periodic cyclic homology for ample groupoids
本文研究了ample群oid作用的双变等变周期性循环同调，证明了其在双变量中的同调不变性、稳定性与消去性，并证明了proper群oid作用的Green-Julg定理类比。
We compute the cyclic homology for the cross-product al- gebra $A(M)\rtimesΓ$ of the algebra of complete symbols on a compact man- ifold $M$ with action of a finite group $Γ$. A spectral sequence argument shows that these groups can be identified using deRham cohomology of the fixed point manifolds $S ^*M ^g$ . In the process we obtain new re- sults about the homologies of general cross-product algebras and provide explicit identification of the homologies for $C^{\infty}(M)\rtimes Γ$.
Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K^G_*(X), using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphism.
We define equivariant periodic cyclic homology for bornological quantum groups. Generalizing corresponding results from the group case, we show that the theory is homotopy invariant, stable and satisfies excision in both variables. Along the way we prove Radfords formula for the antipode of a bornological quantum group. Moreover we discuss anti-Yetter-Drinfeld modules and establish an analogue of the Takesaki-Takai duality theorem in the setting of bornological quantum groups.
We establish restrictions on Lagrangian embeddings of rational homology spheres into certain open symplectic manifolds, namely the (A_m) Milnor fibres of odd complex dimension. This relies on general considerations about equivariant objects in module categories (which may be applicable in other situations as well), as well as results of Ishii-Uehara and Ishii-Ueda-Uehara concerning the derived categories of coherent sheaves on the resolutions of (A_m) surface singularities.
Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{…
We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of the motivic cohomology of a real variety as homotopy groups of the complex of averaged equidimensional cycles. We also establish an equivariant Poincare dua…
Toeplitz Operators on Contact Manifolds and Equivariant K-homology
本文通过等变K-同调理论推广了Boutet de Monvel的Toeplitz算子索引定理，证明Dirac算子与Szego投影在等变K-同调中确定相同类。
Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each n > 0, a natural map from cyclic homology of an algebra to the GL_n-equivariant Deligne cohomology of the variety of n-dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by…
Equivariant $K$-homology of affine Grassmannian and $K$-theoretic double $k$-Schur functions
本文研究仿射格拉斯曼流形的环面等变 $K$-同调环，在 $A$ 型中引入 $K$-理论双重 $k$-Schur 函数作为 Schubert 基，并给出 $\mathrm{Gr}_{{SL}_n}$ 的等变 $K$-同调环的 Ginzburg-Peterson 型实现。
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