We introduce a version of algebraic $K$-theory for coefficient systems of rings which is valued in genuine $G$-spectra for a finite group $G$. We use this construction to build a genuine $G$-spectrum $K_G(\mathbb{Z}[\underline{π_1(X)}])$ associated to a $G$-space $X$, which provides a home for equivariant versions of classical invariants like the Wall finiteness obstruction and Whitehead torsion. We provide a comparison between our $K$-theory spectrum and the equivariant $A$-theory of Malkiewich--Merling via a genuine equivariant linearization map.
We construct a genuine $G$-equivariant extension of factorization homology for $G$ a finite group, assigning a genuine $G$-spectrum to a manifold with $G$-action. We show that $G$-factorization homology is compatible with Hill-Hopkins-Ravenel norms and satisfies equivariant $\otimes$-excision. Following Ayala-Francis we prove an axiomatic characterization of genuine $G$-factorization homology. Applications include a description of real topological Hochschild homology and relative topological Hochschild homology of $C_n$-rings using genuine $G$-factorization homology.
We build new algebraic structures, which we call genuine equivariant operads, which can be thought of as a hybrid between equivariant operads and coefficient systems. We then prove an Elmendorf-Piacenza type theorem stating that equivariant operads, with their graph model structure, are equivalent to genuine equivariant operads, with their projective model structure. As an application, we build explicit models for the $N_{\infty}$-operads of Blumberg and Hill.
In this paper, we build on the work from our previous paper (arXiv:2002.01556) to show that periodic rational $G$-equivariant topological $K$-theory has a unique genuine-commutative ring structure for $G$ a finite abelian group. This means that every genuine-commutative ring spectrum whose homotopy groups are those of $KU_{\mathbb{Q},G}$ is weakly equivalent, as a genuine-commutative ring spectrum, to $KU_{\mathbb{Q},G}$. In contrast, the connective rational equivariant $K$-theory spectrum does not have this type of uniqueness of genuine-commutative ring structure.
Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homolo…
We use geometric fixed points to describe the homotopy theory of genuine equivariant commutative ring spectra after inverting the group order. The main innovation is the use of the extra structure provided by the Hill-Hopkins-Ravenel norms in the form of additional norm maps on geometric fixed point diagrams, which turns out to be computationally managable.
We construct a generalization of the operadic nerve, providing a translation between the equivariant simplicially enriched operadic world to the parametrized $\infty$-categorical perspective. This naturally factors through genuine equivariant operads, a model for "equivariant operads with norms up to homotopy". We introduce the notion of an op-fibration of genuine equivariant operads, extending Grothendieck op-fibrations, and characterize fibrant operads as the image of genuine equivariant symmetric monoidal categories. Moreover, we show that under the operadic nerve, this image is sent to $G$…
There is a lack of knowledge about the topological invariants of non-linear $d$-dimensional systems with a periodic potential. We study these systems through a classification of the linearized NLS/GP equation around their soliton solutions. Stability conditions under linearized (mode) adiabatic evolution can be interpreted topologically and we can use equivariant $\mathit{K}$-theory and cohomology for their classification. On a lattice with crystallographic point group $P$, modes around stable, $P$-symmetric solitons are coarsely classified by the groups $\bar{\mathit{K}}_{P}^{0,τ}(\mathbb{T}^…
We establish a novel approach to computing $G$-equivariant cohomology for a finite group $G$, and demonstrate it in the case that $G = C_{p^n}$. For any commutative ring spectrum $R$, we prove a symmetric monoidal reconstruction theorem for genuine $G$-$R$-modules, which records them in terms of their geometric fixedpoints as well as gluing maps involving their Tate cohomologies. This reconstruction theorem follows from a symmetric monoidal stratification (in the sense of \cite{AMR-strat}); here we identify the gluing functors of this stratification in terms of Tate cohomology. Passing from ge…
We define a genuine $\mathbb{Z}/2$-equivariant real algebraic $K$-theory spectrum $KR(A)$, for every genuine $\mathbb{Z}/2$-equivariant spectrum $A$ equipped with a compatible multiplicative structure. This construction extends the real $K$-theory of Hesselholt-Madsen for discrete rings and the Hermitian $K$-theory of Burghelea-Fiedorowicz for simplicial rings. We construct a natural trace map of $\mathbb{Z}/2$-spectra $tr\colon KR(A)\to THR(A)$ to the real topological Hochschild homology spectrum, which extends the $K$-theoretic trace of Bökstedt-Hsiang-Madsen. The trace provides a splitting …
University of Virginia（弗吉尼亚大学）；University of Pennsylvania（宾夕法尼亚大学）；Michigan State University（密歇根州立大学）；Reed College（里德学院）；University of Minnesota（明尼苏达大学）
We set up operadic foundations for equivariant iterated loop space theory. We start by building up from a discussion of the approximation theorem and recognition principle for V-fold loop G-spaces to several avatars of a recognition principle for infinite loop G-spaces. We then explain what genuine permutative G-categories are and, more generally, what E_{\infty} G-categories are, giving examples showing how they arise. As an application, we prove the equivariant Barratt-Priddy-Quillen theorem as a statement about genuine G-spectra and use it to give a new, categorical, proof of the tom Dieck …
We introduce a global equivariant refinement of algebraic K-theory; here `global equivariant' refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global $Ω$-spectrum that keeps track of genuine $G$-equivariant infinite loop spaces, for all finite groups $G$. The resulting global algebraic K-theory spectrum is a rigid way of packaging the representation K-theory, or `Swan K-theory' into one highly structured object.
For a finite group G, we introduce the complete suboperad $Q_G$ of the categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but $Q_G$ is finitely generated and is a genuine $E_\infty$ G-operad (i.e., it is $N_\infty$ and includes all norms). For G cyclic of order 2 or 3, we determine presentations of the object operad of $Q_G$ and conclude with a discussion of algebras over $Q_G$, which we call biased permutative equivariant categories.
A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action induced by a $G$-map of $G$-rings is not equivariant. We define a version of equivariant algebraic $K$-theory which encodes a group action on the input in a functorial way to produce a $genuine$ algebraic $K$-theory $G$-spectrum for a finite group $G$. The main technical work lies in studying coherent actions on the input category. A payoff of our approach i…
In this note we compare the fracture squares from genuine equivariant stable homotopy theory and the fracture squares which appear in the Goodwillie tower for the norm functor.
For a global field, local field, or finite field $k$ with infinite Galois group, we show that there can not exist a functor from the Morel--Voevodsky $\mathbb{A}^1$-homotopy category of schemes over $k$ to a genuine Galois equivariant homotopy category satisfying a list of hypotheses one might expect from a genuine equivariant category and an étale realization functor. For example, these hypotheses are satisfied by genuine $\mathbb{Z}/2$-spaces and the $\mathbb{R}$-realization functor constructed by Morel--Voevodsky. This result does not contradict the existence of étale realization functors t…
For a "genuine" equivariant commutative ring spectrum $R$, $π_0(R)$ admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on $R$ arising from the existence of multiplicative norm maps. Motivated by the surprising fact that Bousfield localization can destroy some of the norm maps, in previous work we studied equivariant commutative ring structures parametrized by $N_\infty$ operads. In a precise sense, these interpolate between "naive" and "genuine" equivariant ring structures. In this paper, we describe the algebraic analogue of $N_\infty$…
This dissertation comprises three collections of results, all united by a common theme. The theme is the study of categories via algebraic techniques, considering categories themselves as algebraic objects. This algebraic approach to category theory is central to noncommutative algebraic geometry, as realized by recent advances in the study of noncommutative motives. We have success proving algebraic results in the general setting of symmetric monoidal and semiring $\infty$-categories, which categorify abelian groups and rings, respectively. For example, we prove that modules over the semiring…
Topological cyclic homology is a refinement of Connes--Tsygan's cyclic homology which was introduced by Bökstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and …
