L. Gaunce Lewis, Michael A. Mandell
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.
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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。
We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K_0 of the category of "phi-endomorphisms of finitely generated free RPi(G,X)-modules". We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.
We study the fixed points of the universal G-equivariant n-dimensional complex vector bundle and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller dimension. We use this decomposition to describe the fixed points of the complex equivariant K-theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.
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…
We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classe…
Partial parametrized presentability and the universal property of equivariant spectra
本文引入参数化高阶范畴理论中的部分可呈现性概念，探讨其与参数化半可加性和稳定性之间的相互作用，并在不稳定、半可加和稳定背景下构造自由部分可呈现的T-范畴。在全局同调理论中，得出全局范畴的等价可呈现性概念，并证明真正的同调谱的全局范畴是自由的全局范畴，既可呈现又可稳定。
The theory of indices of Morse--Bott vector fields on a manifold is constructed and the famous localization problem for the transfer map is solved on its base in the present paper. As a consequence, we obtained addition theorems for the universal Pontrjagin characteristic classes in cobordisms. These results gave us a possibility to complete the construction, which was begun more than twenty years ago, of the universal characteristic classes' theory.
We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.
We develop a new method in the computation of equivariant homotopy, which is based on the splitting of cofiber sequences associated to universal spaces in the category of equivariant spectra. In particular, we use this method to compute the homotopy of $H\underline{\mathbb{Z}}$ for $G=D_{2p}$ and $A_5$.
The universal property of bordism of commuting involutions
本文提出oriented el$_2^{RO}$-代数这一形式体系以刻画交换对合流形的等变配边环结构，并证明初等阿贝尔2-群的等变配边是该代数的初始对象，同时推广至其他等变同调理论的泛性质刻画。
We show that the $\mathbb{Z}/2$-equivariant Morava K-theories with reality (as defined by Hu) are self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in Morava K-theory with reality. As a particular example, we recover the self-duality of the spectrum $KO$. The study of $\mathbb{Z}/2$-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of $RO(\mathbb{Z}/2)$-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.
$RO(C_p \times C_p)$-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。
We study the left adjoint $\mathbb{D}$ to the forgetful functor from the $\infty$-category of symmetric monoidal $\infty$-categories with duals and finite colimits to the $\infty$-category of symmetric monoidal $\infty$-categories with finite colimits, and related free constructions. The main result is that $\mathbb{D} \mathcal C$ always splits as the product of 3 factors, each characterized by a certain universal property. As an application, we show that, for any compact Lie group $G$, the $\infty$-category of genuine $G$-spectra is obtained from the $\infty$-category of Bredon (\emph{a.k.a} …
Let T be a torus. We show that Koszul duality can be used to compute the equivariant cohomology of topological T-spaces as well as the cohomology of pull backs of the universal T-bundle. The new features are that no further assumptions about the spaces are made and that the coefficient ring may be arbitrary. This gives in particular a Cartan-type model for the equivariant cohomology of a T-space with arbitrary coefficients. Our method works for intersection homology as well.
For every abelian compact Lie group A, we prove that the homotopical A-equivariant complex bordism ring, introduced by tom Dieck (1970), is isomorphic to the A-equivariant Lazard ring, introduced by Cole-Greenlees-Kriz (2000). This settles a conjecture of Greenlees. We also show an analog for homotopical real bordism rings over elementary abelian 2-groups. Our results generalize classical theorems of Quillen (1969) on the connection between non-equivariant bordism rings and formal group laws, and extend the case $A = C_2$ due to Hanke-Wiemeler (2018). We work in the framework of global homotop…
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.
Universal Embedding spaces for $G$-manifolds
本文为任意紧李群 G 构造了万有 G-流形 U_n(G)，使所有 n 维光滑 G-流形都能以平凡法丛嵌入其中且嵌入在等变合痕下唯一，并证明其上同调逆极限给出 G-流形的示性类类比，上同伦群等于等变配边群。
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…
Functoriality of the Klein-Williams Invariant and Universality Theory
本文证明了Klein-Williams不变量$\ell_G(f)$的函子性，总结了此类函子不变量的“万有性”理论并计算了万有不变量所在的群，最后探讨了其与广义等变Lefschetz不变量$\lambda_G(f)$的关系与比较。
Power operations preserve Thom classes in twisted equivariant Real K-theory
本文为拓扑栈的扭曲KR-理论构造幂运算，证明其保持Thom类，并得出扭曲Atiyah-Bott-Shapiro定向与幂运算交换的结论。
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formu…
We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex X. In the process, we identify the d^1 differential in terms of the coalgebra structure of H_*(X,\k), and the \kπ_1(X)-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov, on the mod p cohomology of cyclic p-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of X. It also yields computable upper boun…
We show that the center of a flat graded deformation of a standard Koszul algebra behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming the Koszul dual algebra. This is an algebraic version of a duality discovered by Goresky and MacPherson b…
Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable infinity-category, and we use this to show that universal examples of these objects are given by algebraic K-theory. More importantly, we introduce the unfurling of certain families of Waldhausen infinity-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic K-theory of such objects as spectra…
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$ for various symmetric pairs $(G,K)$. In type $A$, we realize the closures of $K=GL(p,\C) \times GL(q,\C)$-orbits on $GL(p+q,\C)/B$ as universal degeneracy loci for a vector bundle over a variety which is equipped with a single flag of subbundles and which splits as a direct sum of subbundles of ranks $p$ and $q$. The precise description of such a degeneracy locus relies upon knowing a set-theoretic descri…
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