Related papers: Rigidity and exotic models for $v_1$-local $G$-equ…

For any finite group G, we show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique equivariant model in the sense of Quillen model categories. This means that the suspension functor,…

We investigate a new case of rigidity in stable homotopy theory which is the rigidity of the $K(1)$-local stable homotopy category $\mathrm{Ho}(L_{K(1)}\mathrm{Sp})$ at $p=2$. In other words, we show that recovering higher homotopy…

We prove that if G is the circle group or a profinite group, then the all of the homotopical information of the category of rational G-spectra is captured by triangulated structure of the rational G-equivariant stable homotopy category.…

Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $KU_G/p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $KO_{N_p}$-module spectrum. Then we compute the…

Given a finite group $G$ and a commutative ring $G$-spectrum $R$, we study the separable commutative algebras in the category of compact $R$-modules. We isolate three conditions on the geometric fixed points of $R$ which ensure that every…

We prove that the category $\textbf{G-Cat}$ of small categories with $G$-action forms a model of unstable $G$-global homotopy theory for every discrete group $G$, generalizing Schwede's global model structure on $\textbf{Cat}$. As a…

Let $k$ be an arbitrary field of characteristic $p$ and let $G$ be a finite group. We investigate the representation type, derived representation type, and singularity category of the $k$-linear (cohomological) Mackey algebra. We classify…

The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. If $G$ is finite, describing them as homotopy colimits is convenient when applying…

Using Franke's methods we construct new examples of exotic equivalences. We show that for any symmetric ring spectrum $R$ whose graded homotopy ring $\pi_*R$ is concentrated in dimensions divisible by a natural number $N \geq 5$ and has…

We compute the homotopy Mackey functors of the $KU_G$-local equivariant sphere spectrum when $G$ is a finite $q$-group for an odd prime $q$, building on the degree zero case from arXiv:2204.03797.

If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG}…

This book introduces a new context for global homotopy theory, i.e., equivariant homotopy theory with universal symmetries. Many important equivariant theories naturally exist not just for a particular group, but in a uniform way for all…

If $(R,I)$ is a henselian pair with an action of a finite group $G$ and $n\ge 1$ is an integer coprime to $|G|$ and such that $n\cdot |G|\in R^*$, then the reduction map of mod-$n$ equivariant $K$-theory spectra \[…

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from…

In this paper we study the global structure of the stable homotopy theory of spectra. We establish criteria for when the homotopy theory associated to a given stable model category agrees with the classical stable homotopy theory of…

We first show that simply connected co-$H$-spaces and connected $H$-spaces can be uniquely decomposed into prime factors in the homotopy category of pointed $p$-local spaces of finite type, which is used to develop a $p$-local version of…

We study the closed group of homeomorphisms of the boundary of real hyperbolic space generated by a cocompact Kleinian group $G_1$ and a quasiconformal conjugate $h^{-1}G_2 h$ of a cocompact group $G_2$. We show that if the conjugacy $h$ is…

The problem of equivariant rigidity is the $\Gamma$-homeomorphism classification of $\Gamma$-actions on manifolds with compact quotient and with contractible fixed sets for all finite subgroups of $\Gamma$. In other words, this is the…

We develop a mechanism of "isotropy separation for compact objects" that explicitly describes an invertible $G$-spectrum through its collection of geometric fixed points and gluing data located in certain variants of the stable module…

Let p be an odd prime. Let G be a p-local finite group over the extraspecial p-group p_+^{1+2}. In this paper we study the cohomology and the stable splitting of their p-complete classifying space BG.