Boris Chorny
We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result.
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We construct an explicit equivariant cellular decomposition of the $(4n-1)$-sphere with respect to binary polyhedral groups, and describe the associated cellular homology chain complex. As a corollary of the binary octahedral case, we deduce an $\mathfrak{S}_3$-equivariant decomposition of the flag manifold of $SL_3(\mathbb{R})$.
We show that several apparently unrelated formulas involving left or right Bousfield localizations in homotopy theory are induced by comparison maps associated with pairs of adjoint functors. Such comparison maps are used in the article to discuss the existence of functorial liftings of homotopical localizations and cellularizations to categories of algebras over monads acting on model categories, with emphasis on the cases of module spectra and algebras over simplicial operads. Some of our results hold for algebras up to homotopy as well; for example, if $T$ is the reduced monad associated wi…
In this note we present examples of localization functors (in the category of spaces) whose composition with certain cellularization functors is not idempotent, and vice versa.
We prove that, in the category of groups, the composition of a cellularization and a localization functor need not be idempotent. This provides a negative answer to a question of Emmanuel Dror Farjoun.
In this paper, we study the extent to which Bousfield and finite localizations relative to a thick subcategory of equivariant finite spectra preserve various kinds of highly structured multiplications. Along the way, we describe some basic, useful results for analyzing categories of acyclics in equivariant spectra, and we show that Bousfield localization with respect to an ordinary spectrum (viewed as an equivariant spectrum with trivial action) always preserves equivariant commutative ring spectra.
Localization of equivariant cohomology rings of real Grassmannians
本文通过局部化方法研究实格拉斯曼流形的等变上同调环，给出加法基并计算相关不变量，并利用等变形式性推广到普通上同调。
We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic convergence theorems for the Real Johnson-Wilson theories $E_{\mathbb{R}}(n)$ hold only after Borel completion. We establish analogous results for the $C_{2^n}$-equivariant Johnson-Wilson theories constructed by Beaudry, Hill, Shi, and Zeng. We show that induced localizations upgrade the available norms for an $N_\infty$-algebra, and we determine which new norms appear. Finally, we explore generalizations of our results on smashing localizations i…
Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions). In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces. This model category is not cofibrantly generated. We give conditions on a class of maps which…
Equivariant $K$-theory as a global localization
本研究建立斯奈思定理的等变全局精细化版本，通过预博特类的望远镜局部化，将等变K理论谱表示为特定的全局环谱局部化，拓展了复K理论的相关结果。
When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah--Bott--Berline--Vergne converts the integral of an equivariantly closed form to a finite sum over the fixed points, providing a powerful tool for computing integrals on a manifold. This article seeks to give an accessible exposition of the equivariant localization formula.
Good objects in the equivariant world
本文研究紧致李群作用下的等变局部化，建立了局部化与等变环路函子的交换规则，并利用Bredon上同调对好对象进行了分类。
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。
Let a torus $T$ act smoothly on a compact smooth manifold $M$. If the rational equivariant cohomology $H^*_T(M)$ is a free $H^*_T(pt)$-module, then according to the Chang-Skjelbred Lemma, it can be determined by the $1$-skeleton consisting of the $T$-fixed points and $1$-dimensional $T$-orbits of $M$. When $M$ is an even-dimensional, orientable manifold with 2-dimensional 1-skeleton, Goresky, Kottwitz and MacPherson gave a graphic description of the equivariant cohomology. In this paper, first we revisit the even-dimensional GKM theory and introduce a notion of GKM covering, then we consider t…
We give conditions on a monoidal model category M and on a set of maps C so that the Bousfield localization of M with respect to C preserves the structure of algebras over various operads. This problem was motivated by an example that demonstrates that, for the model category of equivariant spectra, preservation does not come for free, even for cofibrant operads. We discuss this example in detail and provide a general theorem regarding when localization preserves P-algebra structure for an arbitrary operad P. We characterize the localizations that respect monoidal structure and prove that all …
In this paper we compute homotopical bordism rings $MU^G_*$ for abelian compact Lie groups G, giving explicit generators and relations. The key constructions are operations on equivariant bordism which should play an important role in equivariant stable homotopy theory more generally. The main technique used is localization of the theory by inverting Euler classes. Applications to homotopy theory include analysis of the completion map from $MU^G_*$ to $MU^*(BG)$. Applications to geometry include classification up to cobordism of S^1 actions on stably complex four-manifolds with precisely three…
Virtual Localization in equivariant Witt cohomology
本文在 $\ ext{SL}_2$ 中环面正规化子作用且以 Witt 群层上同调替代 Chow 群的设定下，证明了 Graber-Pandharipande 虚拟局部化定理的类比。
Atiyah-Bott localization in equivariant Witt cohomology
本文在等变Witt上同调中为SL₂ⁿ和Nⁿ建立了Atiyah-Bott局部化定理及Bott留数公式的类似物，并分析了不同群作用下轨道类型与不动点分解的限制条件。
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。
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…
On the equivariant $KU_G$-local sphere for finite abelian groups
对于有限阿贝尔群 $G$，证明 $KU_G/p$-局部球面谱等价于 $p$-完备 $KO_{N_p}$-模谱的同伦不动点，并计算 $KU_G$-局部球面谱的 $\mathbb{Z}$-分次同伦 Mackey 函子，推广了 arXiv:2303.12271 的结果，最后通过比较 Bousfield 类证明 $KU_G/p$-局部球面谱等价于等变 Morava $K$-理论局部球面谱的楔和，并描述其 $RO(G)$-分次同伦 Mackey 函子。
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 develop a Chern-Weil theory for compact Lie group action whose generic stabilizers are finite in the framework of equivariant cohomology. This provides a method of changing an equivariant closed form within its cohomological class to a form more suitable to yield localization results. This work is motivated by our work on reproving wall crossing formulas in Seiberg-Witten theory, where the Lie group is the circle. As applications, we derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions on compact manifolds with boundary. One of the formulas is then used to yield a…
We develop a theory of equivariant factorization algebras on varieties with an action of a connected algebraic group $G$, extending the definitions of Francis-Gaitsgory [FG] and Beilinson-Drinfeld [BD1] to the equivariant setting. We define an equivariant analogue of factorization homology, valued in modules over $\text{H}^\bullet_G(\text{pt})$, and in the case $G=(\mathbb{C}^\times)^n$ we prove an equivariant localization theorem for factorization homology, analogous to the classical localization theorem [AtB]. We establish a relationship between $\mathbb{C}^\times$ equivariant factorization …
Atiyah's classical work on circular symmetry and stationary phase shows how the $\hat{A}$-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable ''antiholomorphic sector'' in the $\mathbb{C}/Λ$-equivariant cohomology of the conformal double loop space $\mathrm{Maps}(\mathbb{C}/Λ,X)$ of a rationally string manifold $X$ produces the Witten genus of $X$. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation o…
Equivariant Poincaré-Hopf theorem
本文利用局部化代数计算 Euler 特征算子的等变 K-同调类，结合 Witten 形变技术建立了 Poincaré-Hopf 定理的等变版本，推广了 Lück-Rosenberg 的经典结果。
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…
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。
For an $S^{1}$-manifold with boundary, we prove a localization formula applying to any equivariant cohomology theory satisfying a certain algebraic condition. We show how the localization result of Kalkman and a case of the quantization commutes with reduction theorem follow easily from the localization formula.
Localization in Khovanov homology
本文利用Burnside函子构造周期链环的等变Khovanov谱，通过不动点集识别，得到素数周期链环的奇偶Khovanov同调及环状滤过的秩不等式。
Localisation in Equivariant Cohomology
本文探讨等变上同调的理论基础与应用，重点阐述Atiyah-Bott局部化定理如何简化带李群作用的辛流形上的积分，并展示其在理论物理和几何学等领域的广泛影响。
For $X$ a smooth scheme acted on by a linear algebraic group $G$ and $p$ a prime, the equivariant Chow ring $CH^*_G(X)\otimes \mathbb{F}_p$ is an unstable algebra over the Steenrod algebra. We compute Lannes's $T$-functor applied to $CH^*_G(X)\otimes \mathbb{F}_p$. As an application, we compute the localization of $CH^*_G(X)\otimes \mathbb{F}_p$ away from $n$-nilpotent modules over the Steenrod algebra, affirming a conjecture of Totaro as a special case. The case when $X$ is a point and $n = 1$ generalizes and recovers an algebro-geometric version of Quillen's stratification theorem proved by …
Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut-Chern character.
We give an account of the theory of factorization spaces, categories, functors, and algebras, following the approach of [Ras1]. We apply these results to give geometric constructions of factorization $\mathbb{E}_n$ algebras describing mixed holomorphic-topological twists of supersymmetric gauge theories in low dimensions. We formulate and prove several recent predictions from the physics literature in this language: We recall the Coulomb branch construction of [BFN1] from this perspective. We prove a conjecture from [CosG] that the Coulomb branch factorization $\mathbb{E}_1$ algebra $\mathcal{…
A. Sikora has shown results which confirm the analogy between number fields and 3-manifolds. However, he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology.
I conjecture that index formulas for $K$-theory classes on the moduli of holomorphic $G$-bundles over a compact Riemann surface $Σ$ are controlled, in a precise way, by Frobenius algebra deformations of the Verlinde algebra of $G$. The Frobenius algebras in question are twisted $K$-theories of $G$, equivariant under the conjugation action, and the controlling device is the equivariant Gysin map along the "product of commutators" from $G^{2g}$ to $G$. The conjecture is compatible with naive virtual localization of holomorphic bundles, from $G$ to its maximal torus; this follows by localization …
The Hirzebruch $td_y(X)$ class of a complex manifold X is a formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class. The related $χ_y$-genus admits a generalization for singular complex algebraic varieties. The equivariant version of the Hirzebruch class can be developed as well. The general theory applied in the situation when a torus acts on a singular variety allows to apply powerful tools as the Localization Theorem of Atiyah and Segal for equivariant K-theory and Berline-Vergne formula for equivariant cohomology. We obtain a meaningful inva…
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