The $C_2$-equivariant $K(1)$-local sphere

Steinberg Summands in the free $\mathbb{F}_p$-module on the Equivariant Sphere Spectrum

Let $G$ be a finite $p$-group. The Eilenberg-Maclane spectrum of the constant Mackey functor $\underline{\mathbb{F}}_p$, denoted $H\underline{\mathbb{F}}_p$, is modeled by the free $\mathbb{F}_p$-module on the $G$-equivariant sphere spectrum. With this construction, one has a `word length' filtration $\{(H\underline{\mathbb{F}}_p)_n\}_{n\ge 1}$. Our main theorem is that the $k$-th layer $(H\underline{\mathbb{F}}_p)_{p^k}/(H\underline{\mathbb{F}}_p)_{p^{k-1}}$ is $p$-locally equivalent to the $k$-fold suspension of the Steinberg summand of the $G$-equivariant classifying space of $(\mathbb{Z}/p…

Equivariant principal bundles over spheres and cohomogeneity one manifolds

We classify SO(n)-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(Π, G)$-bundles over cohomogeneity one manifolds.

Fixed point Free Actions of Spheres and Equivariant maps

This paper generalizes the concept of index and co-index and some related results for free actions of G = S0 on a paracompact Hausdorff space which were introduced by Conner and Floyd. We define the index and co-index of a finitistic free G-space X, where G = Sd , d = 1 or 3 and prove that the index of X is not more than the mod 2 cohomology index of X. We observe that the index and co-index of a (2n + 1)-sphere (resp. (4n+3)-sphere) for the action of componentwise multiplication of G = S1 (resp. S3) is n. We also determine the orbit spaces of free actions of G = S3 on a finitistic space X wit…

Equivariant maps between representation spheres

Let $G$ be a compact Lie group. We prove that if $V$ and $W$ are orthogonal $G$-representations such that $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V) \to S(W)$ exists provided that $\dim V^H \leq \dim W^H$ for any closed subgroup $H\subseteq G$. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when $G$ is a torus.

Equivariant properties of symmetric products

The filtration on the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric produ…

Equivariant Equipartitions: Ham Sandwich Theorems for Finite Subgroups of Spheres

Equivariant "Ham Sandwich" Theorems are obtained for the finite subgroups G of the unit spheres S(F) in the classical algebras F = R, C, and H. Given any n F-valued mass distributions on F^n, it is shown that there exists a G-equivariant decomposition of F^n into |G| regular convex fundamental regions which "G-equipartition" each of the n measures, as realized by the vanishing of the "G-averages" of these regions' measures. Applications for real measures follow, among them that any n signed mass distributions on R^{(p-1)n} can be equipartitioned by a single regular p-fan when p a prime number.

Computational Topology of Equivariant Maps from Spheres to Complements of Arrangements

We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of calculating the obstructions for the existence of equivariant maps. A variety of techniques are introduced and discussed with the emphasis on concrete and explicit calculations. This eventually leads (Theorems 18 and 19) to an almost exhaustive analysis of when such maps do or do not exist in the particular case of interest.

Borel 与 genuine C_2-等变 Adams 谱序列

The Borel and genuine $C_2$-equivariant Adams spectral sequences

本文建立了截断实射影谱的经典 Adams 谱序列与 2-完备球面的 Borel 和 genuine C_2-等变 Adams 谱序列之间的联系，从而将 genuine 等变谱序列的计算归结为可计算的 Borel 等变谱序列。

从球面到复Stiefel流形的等变映射次数

On the degrees of equivariant maps from spheres to complex Stiefel manifolds

本文受相关学者研究启发，基于同伦论方法，利用多位学者的成果，确定了满足合适算术条件的从球面到复Stiefel流形的ℤ/m-等变映射的次数集合。

由球面上的向量场控制的 $RO(C_2)$-分次等变同调的Hurewicz定理

A Hurewicz Theorem for $RO(C_2)$-graded Equivariant Homology Governed by Vector Fields on Spheres

本文确定了 $C_2$-等变Eilenberg-MacLane谱的 $RO(C_2)$-分次Hurewicz像，并发现其与球面上向量场问题紧密相关，进而利用广义Leibniz规则和广义Mahowald技巧证明了真 $C_2$-等变Adams谱序列中存在任意长度的非零Adams微分。

Hamiltonian S^1-spaces with large equivariant pseudo-index

Let \((M,ω)\) be a compact symplectic manifold of dimension \(2n\) endowed with a Hamiltonian circle action with only isolated fixed points. Whenever \(M\) admits a toric \(1\)-skeleton \(\mathcal{S}\), which is a special collection of embedded \(2\)-spheres in \(M\), we define the notion of equivariant pseudo-index of \(\mathcal{S}\): this is the minimum of the evaluation of the first Chern class \(c_1\) on the spheres of \(\mathcal{S}\). This can be seen as the analog in this category of the notion of pseudo-index for complex Fano varieties. In this paper we provide upper bounds for the equi…

Geometric approach to stable homotopy groups of spheres, III. Abelian, cyclic and quaternionic structure for mappings with singularities

Collection of (equivariant) $\rm{PL}$-mappings admitting a relative abelian, cyclic, quaternionic, bicyclic, and quaternionic-cyclic structures are constructed.

关于有限阿贝尔群的等变 $KU_G$-局部球面

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。

对于循环群的点的Bredon等变上同调

The Bredon equivariant cohomology of a point for cyclic groups

本文研究了奇数阶循环群下点的Bredon等变上同调，通过代数方法将其转化为稳定球面同调群问题。

String homology of spheres and projective spaces

We study a spectral sequence that computes the (mod 2) S^1-equivariant homology of the free loop space LM of a manifold M (the "string homology" of M). Using it and knowledge of the string topology operations on the homology of LM, we compute the string homology of M when M is a sphere or a projective space.

Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture

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.

$C_3$-equivariant稳定同调群$π_{i,j}^{C_3}$的计算

$C_3$-equivariant stable stems

论文计算了$C_3$-equivariant稳定同调群$π_{i,j}^{C_3}$，针对茎小于25和权重范围-16到16的情况，并描述了几何固定点映射和底层映射。

Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum

We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. Our results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, our analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, we obtain an explicit description …

The $RO(\mathcal{K})$-graded Coefficients of $H\underline{A}$

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.

Equivariant Simplicial Cohomology With Local Coefficients and its Classification

We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of Bredon-Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients.

等变非阿贝尔庞加莱对偶性与Thom谱的等变因子同调

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。

Localization with respect to a class of maps II - Equivariant cellularization and its application

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.

$RO(C_2)$-graded $C_2$-equivariant 稳定茎的简化环

The reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems

研究$RO(C_2)$-graded $C_2$-equivariant 稳定茎的环结构，通过生成元和关系描述，确定其在去除所有 nilpotent 元素后的结构，并记录有理 $C_2$-equivariant 球面的同调群结构。

A nice acyclic matching on the nerve of the partition lattice

The author has already proven that the space $Δ(Π_n)/G$ is homotopy equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers $n\geq 3$ and all subgroups $G\subset S_1\times S_{n-1}$. We construct an $S_1\times S_{n-1}$-equivariant acyclic matching on $Δ(Π_n)$ together with a description of its critical simplices. This is also a more elementary approach to determining the number of spheres. We also develop new methods for Equivariant Discrete Morse Theory by adapting the Patchwork Theorem and poset maps with small fibers from Discrete Morse Theory.

The Hopf type theorem for equivariant local maps

We study otopy classes of equivariant local maps and prove the Hopf type theorem for such maps in the case of a real finite dimensional orthogonal representation of a compact Lie group.

Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant

In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a "10/8+4"-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known re…

Hyperplane mass partitions via relative equivariant obstruction theory

The Grünbaum-Hadwiger-Ramos hyperplane mass partition problem was introduced by Grünbaum (1960) in a special case and in general form by Ramos (1996). It asks for the "admissible" triples $(d,j,k)$ such that for any $j$ masses in $\mathbb{R}^d$ there are $k$ hyperplanes that cut each of the masses into $2^k$ equal parts. Ramos' conjecture is that the Avis-Ramos necessary lower bound condition $dk\ge j(2^k-1)$ is also sufficient. We develop a "join scheme" for this problem, such that non-existence of an $G_k$-equivariant map between spheres $(S^d)^{*k} \rightarrow S(W_k\oplus U_k^{\oplus j})$ t…

The geometry of the local cohomology filtration in equivariant bordism

Local cohomology techniques in equivariant homotopy theory, introduced by John Greenlees, may be applied to understand homology of classifying spaces through other equivariant data. In this paper we relate the local cohomology filtration to the families filtration. By doing so, we may identify geometry codified by the local cohomology filtration in the setting of equivariant bordism. The constructions which arise are naturally analyzed by localized K-theory machinery due to Atiyah and Segal, which we review.