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We report significant progress toward establishing an obstruction theory of equivariant Thom spectra with multiplicative structures arising from maps of $E_V$ spaces. Focusing on the Fujii-Landweber Real bordism spectrum, we explain an argument to show that a homotopy ring Real orientation of an $E_{\rho}$-spectrum satisfying strong equivariant evenness and mild additional commutativity condition lifts to an $E_{\rho}$ map.First, we address the foundational issues. We discuss model categories for equivariant $C_2$-spectra indexed on Real inner product spaces and comparisons among them. We explain which foundations are needed for our project and describe the parts that would be original even non-equivariantly.Next, we construct equivalences between several $E_V$ operads, including little disks and Steiner operads. We show that algebras over $E_{V \oplus \Rbb}$ operads can be strictified to monoids in the category of $E_V$-algebras.To compute obstruction groups for maps of $E_V$ algebras, we analyze the spectra of the derived indecomposibles of augmented $E_V$-algebras. We determine that such spectra are $V$-fold desuspensions of a lift of the May delooping machine to the augmented algebras.Next, we discuss the induced map on cohomology for a map between spaces whose $RO(C_2)$-graded cohomology carries obstruction classes for homotopy ring and $E_\rho$ ring maps out of the Real bordism spectrum.Finally, we review the strategies for the proofs of the goal results of our project, explaining how to use the technique of climbing the slice tower of the target to construct highly structured orientations out of Real bordism.

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory, the global homotopy theory of Schwede (2018). Specifically, for a global ring spectrum R, we consider which classes of ring homomorphisms ηe W πe eR ! Se can be realised by a map ηW R ! S in the category of global R– modules, and what multiplicative structures can be placed on S. If ηe witnesses Se as a projective πe eR–module, then such an η exists as a map between homotopy commutative global R–algebras. If ηe is in addition étale or S0 is a Q–algebra, then η can be upgraded to a map of E1–global R–algebras or a map of G1–R–algebras, respectively. Various global spectra and E1–global ring spectra are then obtained from classical homotopy-theoretic and algebraic constructions, with a controllable global homotopy type.

We extend the Cisinski-Moerdijk-Weiss theory of $\infty$-operads to the equivariant setting to obtain a notion of $G$-$\infty$-operads that encode "equivariant operads with norm maps" up to homotopy. At the root of this work is the identification of a suitable category of $G$-trees together with a notion of $G$-inner horns capable of encoding the compositions of norm maps. Additionally, we follow Blumberg and Hill by constructing suitable variants associated to each of the indexing systems featured in their work., Final version to appear in Algebraic & Geometric Topology

We introduce the analogues of the notions of complete Segal space and of Segal category in the context of equivariant operads with norm maps, and build model categories with these as the fibrant objects. We then show that these model categories are Quillen equivalent to each other and to the model category for [math] – [math] –operads built in a previous paper. ¶ Moreover, we establish variants of these results for the Blumberg–Hill indexing systems. ¶ In an appendix, we discuss Reedy categories in the equivariant context.

Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in $\mathscr{F}^{\mathrm{Nil}}$ (which we call $\mathscr{F}$-nilpotent) can be reconstructed from its underlying $H$-spectra as $H$ varies over $\mathscr{F}$. A similar result holds for calculating $G$-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition $E\in \mathscr{F}^{\mathrm{Nil}}$ implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin and Brauer type induction theorems for $G$-equivariant $E$-homology and cohomology, and generalizations of Quillen's $\mathcal{F}_p$-isomorphism theorem when $E$ is a homotopy commutative $G$-ring spectrum. We show that the subcategory $\mathscr{F}^{\mathrm{Nil}}$ contains many $G$-spectra of interest for relatively small families $\mathscr{F}$. These include $G$-equivariant real and complex $K$-theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, any $L_n$-local spectrum, the classical bordism theories, connective real $K$-theory, and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family such that these results hold., Comment: 63 pages. Many edits and some simplifications. Final version, to appear in Geometry and Topology

Let $k$ be a field with a real embedding. We compare the motivic slice filtration of a motivic spectrum over $Spec(k)$ with the $C_2$-equivariant slice filtration of its equivariant Betti realization, giving conditions under which realization induces an equivalence between the associated slice towers. In particular, we show that, up to reindexing, the towers agree for all spectra obtained from localized quotients of $MGL$ and $MR$, and for motivic Landweber exact spectra and their realizations. As a consequence, we deduce that equivariant spectra obtained from localized quotients of $MR$ are even in the sense of Hill--Meier, and give a computation of the slice spectral sequence converging to $\pi_{*,*}BP\langle n \rangle/2$ for $1 \le n \le \infty$., Comment: Version to appear in Algebraic & Geometric Topology

In this paper, we prove a conjecture of Blumberg and Hill regarding the existence of $N_\infty$-operads associated to given sequences $\mathcal{F} = (\mathcal{F}_n)_{n \in \mathbb{N}}$ of families of subgroups of $G\times \Sigma_n$. For every such sequence, we construct a model structure on the category of $G$-operads, and we use these model structures to define $E_\infty^{\mathcal{F}}$-operads, generalizing the notion of an $N_\infty$-operad, and to prove the Blumberg-Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these $E_\infty^{\mathcal{F}}$-operads, obtaining some new results as well for $N_\infty$-operads., Comment: This version has been accepted to Algebraic & Geometric Topology

We introduce and explore multiple categorical frameworks which allow for a general investigation of the homotopy theory of equivariant operads. We extend the Cisinski-Moerdijk-Weiss theory of dendroidal sets by suitably generalizing the category of trees in order to record equivariant composition information. This formalism of $G$-trees is then applied to a rebuilt free operad monad in order to endow the category of $G$-operads in $\V$ with an $\F$-semi-model structure for any weak indexing system $\F$ and fairly general model categories $\V$; as a consequence, we prove that all indexing systems can be realized as $N_\infty$-operads, confirming a conjecture of Blumberg-Hill. Also using $G$-trees, we define appropriate notions of inner $G$-horns and $G$-$\infty$-operads. Inspired by the internal algebra of $G$-trees and $G$-$\infty$-operads, we extend $G$-operads to the new algebraic notion of genuine equivariant operads, which allow us to record the equivariance of our operadic compositions, while removing rigidity conditions on fixed points without relaxing the strictness of the composition laws. Lastly show that there is a natural homotopy strictification functor, sending $G$-$\infty$-operads to an associated genuine $G$-operad.

We review how core structures of string/M-theory emerge as higher structures in super homotopy theory; namely from systematic analysis of the brane bouquet of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion-subgroups in brane charges and focuses on tangent spaces of super space-time. Already at this level, super homotopy theory discovers all super $p$-brane species, their intersection laws, their M/IIA-, T- and S-duality relations, their black brane avatars at ADE-singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D-branes it recovers twisted K-theory, rationally, but for the M-branes it gives cohomotopy cohomology theory. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M-theory in super homotopy theory.

A solution to Linckelmannʼs gluing problem in all fusion systems of blocks yields an alternative formulation of Alperinʼs weight conjecture. Using Bredon equivariant cohomology techniques we solve the gluing problem in fusion systems (of blocks) which are isomorphic to the fusion systems of the symmetric groups or the alternating groups or GLd(q) or SLd(q) at the defining characteristic.

The main thrust of this paper is to generalize certain aspects of equivariant Nielsen fixed point theory to coincidences, and to extend both equivariant fixed point, and coincidence point theory so as to include certain types of disconnected fixed point sets. These extensions allow us to exhibit certain important new examples, which enable us to utilize equivariant Nielsen theory in an essential way. In particular in these examples (unlike previous ones), the minimum number of coincidence points cannot be accurately estimated by other existing, or simpler, Nielsen theories. Also new is the inclusion of examples which illustrate the non-abelian side of the story. Finally, we fill a number of gaps and make corrections from earlier expositions in equivariant Nielsen theory.

Call a map f : A→B of O(2)-spaces a weak equivalence if f H :A H →B H is a weak homotopy equivalence for all finite cyclic and dihedral subgroups H⊆O(2). We show that the resulting homotopy category is equivalent to the homotopy category of dihedral sets with suitable weak equivalences. There is a similar relation between Pin(2)-spaces and quaternionic sets.

Let G be a compact Lie group and V be a G -representation. We define V -dimensional equivariant Eilenberg-MacLane spaces and show that their elementary properties imply a Seifert-van Kampen theorem and a suspension theorem for the V th homotopy groups of G -spaces. Our equivariant suspension theorem is radically different from those that have appeared previously. Rather than asserting, under certain very restrictive hypotheses, that the suspension map is an isomorphism (or epimorphism), our theorem describes, under milder hypotheses, the precise extent to which this map fails to be injective and surjective.

This paper constructs an equivariant homotopy spectral sequence for any finite group G, any finite dimensional representation V, and two suitably connected G-CW complexes X and Y. The spectral sequence converges to the collection of equivariant homotopy groups of the wedge of X and Y, while the E2 term depends only on the equivariant homotopy groups of X and of Y, along with primary homotopy operations. The edge homomorphism of the spectral sequence is actually an isomorphism in a range, which is the equivariant van Kampen theorem of L.G. Lewis Jr. When G is the trivial group, the spectral sequence reduces to that of C.R. Stover.

The homotopy limit problem for Karoubiʼs Hermitian K-theory (Karoubi, 1980) [26] was posed by Thomason (1983) [44] . There is a canonical map from algebraic Hermitian K-theory to the Z / 2 -homotopy fixed points of algebraic K-theory. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. In this paper, we solve this problem completely for fields of characteristic 0 (Theorems 16, 20). We show that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd 2 ( F [ i ] ) ∞ , but not in general.

Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum $MU_{\mathbb{R}}$. In particular, we construct an equivariant version of the B\"okstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green functors, first introduced by Blumberg, Gerhardt, Hill, and Lawson., Comment: Minor revision. Accepted version