Po Hu, Igor Kriz
In this note, we calculate all untwisted and twisted (Z/2)^n-equivariant K-groups with compact supports of real finite-dimensional linear representations of (Z/2)^n. The question was motivated by the question of D-brane charges for orbifold type II string vacua.
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Magnetic Equivariant Graded Brauer Group
本文研究磁性有限群下的磁性等变中心简单分层代数的相似类，定义了磁性等变分层 Brauer 群，并确定其作为交换群的结构。通过 Karoubi 的方法，证明该群元素参数化了点的磁性等变 K-理论的扭结。
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.
We introduce an equivariant algebraic kk-theory for G-algebras and G-graded algebras. We study some adjointness theorems related with crossed product, trivial action, induction and restriction. In particular we obtain an algebraic version of the Green-Julg Theorem which gives us a computational tool.
In this paper, we study a generalization of twisted (groupoid) equivariant $\mathrm{K}$-theory in the sense of Freed-Moore for $\mathbb{Z}_2$-graded $\mathrm{C}^*$-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele's $\mathrm{K}$-theory for $\mathbb{Z}_2$-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant $\mathrm{K}$-group when the $\mathrm{C}^*$-algebra is trivially graded. It is applied for the bu…
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。
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。
The Galois-equivariant $K$-theory of finite fields
本文计算了有限域在 Galois 群作用下的 $RO(G)$-分次等变代数 $K$-群，将其分解为可计算项与等变 Eilenberg--MacLane 谱的系数群之和，并证明两者具有相同的 Tate 谱和几何不动点谱，且在素数阶情形给出显式表示。
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.
Let $G$ be a finite group, $X$ be a compact $G$-space. In this note we study the $(\mathbb{Z}_ + \times\mathbb{Z}/2\mathbb{Z})$-graded algebra $$\mathcal{F}^q_G(X) = \bigoplus_{n\geq0} q^n \cdot K_{G\wr\mathfrak{S}_n}(X^n)\otimes\mathbb{C},$$ defined in terms of equivariant K-theory with respect to wreath products as a symmetric algebra. More specifically, let $H$ be another finite group and $Y$ be a compact $H$-space, we give a decomposition of $\mathcal{F}^q_{G\times H}(X\times Y)$ in terms of $\mathcal{F}^q_G(X)$ and $\mathcal{F}^q_H(Y)$. For this, we need to study the representation theory…
The Künneth Theorem for equivariant (complex) K-theory K^*_G, in the form developed by Hodgkin and others, fails dramatically when G is a finite group, and even when G is cyclic of order 2. We remedy this situation in this very simplest case G = Z/2 by using the power of RO(G)-graded equivariant K-theory.
We develop equivariant KK-theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K-theory.
Given a finite group G and a G-space X, we show that a direct sum $F_G (X) = \bigoplus_{n \geq 0}K_{G_n} (X^n) \bigotimes \C$ admits a natural graded Hopf algebra and $λ$-ring structure, where $G_n$ denotes the wreath product $G \sim S_n$. $F_G (X)$ is shown to be isomorphic to a certain supersymmetric product in terms of $K_G(X)\bigotimes \C$ as a graded algebra. We further prove that $F_G (X)$ is isomorphic to the Fock space of an infinite dimensional Heisenberg (super)algebra. As one of several applications, we compute the orbifold Euler characteristic $e(X^n, G_n)$.
Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact Lie group acting on itself by conjugation, and relate the result to the Verlinde algebra and to the Kac numerator at q=1. Verlinde's formula is also discussed in this context.
A higher categorical approach to graded bivariant $K$-theories for Real $C^*$-algebras
该研究针对局部紧第二可数群G作用下的分次实C*代数，给出经典KK理论与E理论的∞范畴细化，得到连接K理论谱的纤维序列，分析K0群自同态并解答民间问题，证明两类范畴非分次C*代数的Dwyer-Kan局部化。
We relate Kellendonk and Putnam's pattern-equivariant (PE) cohomology to the inverse-limit structure of a tiling space. This gives a version of PE cohomology with integer coefficients, or with coefficients in any Abelian group. It also provides an easy proof of Kellendonk and Putnam's original theorem relating PE cohomology to the Cech cohomology of the tiling space. The inverse-limit structure also allows for the construction of a new non-Abelian invariant, the PE representation variety.
B-fields over a groupoid with involution are defined as Real graded Dixmier-Douady bundles. We use these to introduce the Real graded Brauer group which constitutes the set of twistings for Atiyah's KR-functor in the category of locally compact groupoids with involutions. We interpret this group in terms of groupoid extensions and elements of some equivariant Cech cohomology theory. The construction of the twisted KR-functor is outlined.
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