Related papers: Function spectra and continuous G-spectra

Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C.…

Let $p$ be a prime, let $KU_p$ be $p$-complete complex $K$-theory, and let $\mathbb{Z}_p^\times$ denote the group of units in the $p$-adic integers. The $p$-adic Adams operations induce an action of the profinite group $\mathbb{Z}_p^\times$…

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…

We give a new construction of the equivariant $K$-theory of group actions (cf. Barwick et al.), producing an infinite loop $G$-space for each Waldhausen category with $G$-action, for a finite group $G$. On the category $R(X)$ of retractive…

We introduce a family of twisted $K(n)$-local theories that behave analogous to twisted K-theory. Let $R_n= E_n^{hS\mathbb G_n}$, the homotopy fixed point spectrum under the action of the subgroup $S\mathbb G_n$ of the Morava stabilizer…

Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$-space. In this paper we analyze the homotopy type of the…

For a discrete group G, we represent the Bredon cohomology with local coefficients as the homotopy classes of maps in the category of equivaraint crossed complexes. Subsequently, we construct a naive parametrized G-spectrum, such that the…

The fixed point spectra of Morava E-theory $E_n$ under the action of finite subgroups of the Morava stabilizer group $\mathbb{G}_n$ and their K(n)-local Spanier--Whitehead duals can be used to approximate the K(n)-local sphere in certain…

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…

Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K(Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum Wh(*) is described. A suspended copy of the cokernel-of-J spectrum…

We show that Connes' B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a…

We construct the stable (representable) homotopy category of finite orbispectra, whose objects are formal desuspensions of finite orbi-CW-pairs by vector bundles and whose morphisms are stable homotopy classes of (representable) relative…

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 give an elementary proof of the proper homotopy invariance of the equivariant stable homotopy type of the configuration space $F(M,k)$ for a topological manifold $M$. Our technique is to compute the Spanier-Whitehead dual…

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.

Every homomorphism $\varphi: B(G) \rightarrow B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $\alpha: Y \rightarrow \Delta(B(G))$, where $Y$ is a set in the open coset…

We study the equivariant homotopy type of the poset of orthogonal decompositions of a finite-dimensional complex vector space. Suppose that n is a power of a prime p, and that D is an elementary abelian p-subgroup of U(n) acting on complex…

We prove that if $\rho: A(H) \to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $\alpha : G \to H$,…

We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove analogous conjectures in the stable homotopy category. We utilize simplicial schemes over an algebraically closed field of…

We prove that if a geodesically complete $\mathrm{CAT}(0)$ space $X$ admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of $X$ is less than $1$. Let $G$ be a finite connected graph, $\mu_1 (G)$ be the…