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In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory. [ABSTRACT FROM AUTHOR]

Following ideas of Lurie, we give a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain, in particular, equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori. [ABSTRACT FROM AUTHOR]

We compute the homotopy Mackey functors of the K U G -local equivariant sphere spectrum when G is a finite q-group for an odd prime q, building on the degree zero case due to Bonventre and the third and fifth authors. [ABSTRACT FROM AUTHOR]

Let C 2 denote the cyclic group of order 2. We compute the R O (C 2) -graded cohomology of all C 2 -surfaces with constant integral coefficients. We show when the action is nonfree, the answer depends only on the genus, the orientability of the underlying surface, the number of isolated fixed points, the number of fixed circles with trivial normal bundles, and the number of fixed circles with nontrivial normal bundles. When the action on the surface is free, we show the answer depends only on the genus, the orientability of the underlying surface, whether or not the action preserves the orientation, and one other invariant. [ABSTRACT FROM AUTHOR]

Let Q denote the cyclic group of order two. Using the Tate diagram we compute the RO(Q)-graded coefficients of Eilenberg–MacLane Q-spectra and describe their structure as modules over the coefficients of the Eilenberg–MacLane spectrum of the Burnside Mackey functor. If the underlying Mackey functor is a Green functor, we also obtain the multiplicative structure on the RO(Q)-graded coefficients. [ABSTRACT FROM AUTHOR]

We construct a 2-category version of tom Dieck's equivariant fundamental groupoid for representable orbifolds and show that the discrete fundamental groupoid is Morita invariant; hence it is an orbifold invariant for representable orbifolds. [ABSTRACT FROM AUTHOR]

Let G be a finite group with 2-Sylow subgroup of order less than or equal to 16. For such a G, we prove a quantified version of Quillen's uniform Fp-isomorphism theorem, which holds uniformly for all G-spaces. We do this by bounding from above the exponent of Borel equivariant F2-cohomology, as introduced by Mathew--Naumann--Noel, with respect to the family of elementary abelian 2-subgroups. [ABSTRACT FROM AUTHOR]

We present upper and lower bounds for symmetrized topological complexity T C Σ (X) in the sense of Basabe–González–Rudyak–Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of the symmetric square S P 2 (X). We also show that symmetrized topological complexity coincides with its monoidal version, where the path from a point to itself is required to be constant. Using these results, we calculate the symmetrized topological complexity of all odd spheres. [ABSTRACT FROM AUTHOR]

We extend Schwede’s work on the unstable global homotopy theory of orthogonal spaces and L-spaces to the category of ∗-modules (i.e., unstable S-modules). We prove a theorem which transports model structures and their properties from L-spaces to ∗-modules and show that the resulting global model structure for ∗-modules is monoidally Quillen equivalent to that of orthogonal spaces. As a consequence, there are induced Quillen equivalences between the associated model categories of monoids, which identify equivalent models for the global homotopy theory of A∞-spaces. [ABSTRACT FROM AUTHOR]

We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [ X / G ] , where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic homotopy theory: the (unstable) purity and gluing theorems of Morel–Voevodsky and the (stable) ambidexterity theorem of Ayoub. Our proof of the latter is different than Ayoub's and is of interest even when G is trivial. Using these results, we construct a formalism of six operations for equivariant motivic spectra, and we deduce that any cohomology theory for G -schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology. [ABSTRACT FROM AUTHOR]

We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological presheaves indexed by the orbit category of a fixed topological group G and the category of G-spaces can be endowed with Quillen equivalent model category structures.We prove an analogous result for any cofibrantly generated model category and discrete group G, under certain conditions on the fixed point functors of the subgroups of G. These conditions hold in many examples, though not in the category of chain complexes, where we nevertheless establish and generalize to collections an equivariant Whitehead Theorem à la Kropholler and Wall for the normalized chain complexes of simplicial G-sets. [ABSTRACT FROM AUTHOR]

We define a model category structure on a slice category of simplicial spaces, called the "Segal group action" structure, whose fibrant-cofibrant objects may be viewed as representing spaces X with an action of a fixed Segal group (i.e. a group-like, reduced Segal space). We show that this model structure is Quillen equivalent to the projective model structure on G-spaces, SBG, where G is a simplicial group corresponding to the Segal group. One advantage of this model is that if we start with an ordinary group action X ε SBG and apply a weakly monoidal functor of spaces L : S → S (such as localization or completion) on each simplicial degree of its associated Segal group action, we get a new Segal group action of LG on LX which can then be rigidified via the above-mentioned Quillen equivalence. [ABSTRACT FROM AUTHOR]

In this paper we give a model for equivariant (∞, 1)-categories. We modify an approach of Shimakawa for equivariant Γ-spaces to the setting of simplicial spaces. We then adapt Rezk’s Segal and completeness conditions to fit with this setting. [ABSTRACT FROM AUTHOR]

Given a finite group G acting on a ring R , Merling constructed an equivariant algebraic K -theory G -spectrum, and work of Malkiewich and Merling, as well as work of Barwick, provides an interpretation of this construction as a spectral Mackey functor. This construction is powerful, but highly categorical; as a result the Mackey functors comprising the homotopy are not obvious from the construction and have therefore not yet been calculated. In this work, we provide an interpretation of the homotopy Mackey functors of equivariant algebraic K -theory in terms of a purely algebraic construction. In particular, we construct Mackey functors out of the n th algebraic K -groups of group rings whose multiplication is twisted by the group action. Restrictions and transfers for these functors admit a tractable algebraic description in that they arise from restriction and extension of scalars along homomorphisms of twisted group rings. In the case where the group action is trivial, our construction recovers work of Dress and Kuku from the 1980's which constructs Mackey functors out of the algebraic K -theory of group rings. We develop many families of examples of Mackey functors, both new and old, including K -theory of endomorphism rings, the K -theory of fixed subrings of Galois extensions, and (topological) Hochschild homology of twisted group rings. [ABSTRACT FROM AUTHOR]

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 M U R. In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green functors, first introduced by Blumberg, Gerhardt, Hill, and Lawson. [ABSTRACT FROM AUTHOR]

For any finite group $$G$$ , we construct a spectral sequence for computing the Bredon cohomology of a $$G$$ -CW complex $$\mathbf{X }$$ , starting with the cohomology of $$\mathbf{X }^{H}/\bigcup _{K>H}\mathbf{X }^{K}$$ with suitable local coefficients, for various $$H\le G$$ . [ABSTRACT FROM AUTHOR]

Abstract: The homotopy limit problem for Karoubiʼs Hermitian K-theory (Karoubi, 1980) was posed by Thomason (1983) . There is a canonical map from algebraic Hermitian K-theory to the -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 , but not in general. [Copyright &y& Elsevier]

Abstract: 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 or at the defining characteristic. [Copyright &y& Elsevier]

We construct a map from the suspension G -spectrum Σ G ∞ M of a smooth compact G -manifold to the equivariant A -theory spectrum A G (M) , and we show that its fiber is, on fixed points, a wedge of stable h -cobordism spectra. This map is constructed as a map of spectral Mackey functors, which is compatible with tom Dieck style splitting formulas on fixed points. In order to reconcile different definitions of the suspension G -spectrum as a spectral Mackey functor, we present a new perspective on spectral Mackey functors, viewing them as multifunctors on indexing categories for "rings on many objects" and modules over such. This perspective should be of independent interest. [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]

This paper describes several model structures on the categories of ex-spaces and ex-G-spaces when G is a compact Lie group. Two of these are of particular interest in that they have expected applications to the study of transfer maps and to parametrized spectra. These two structures are shown to coincide on the collection of Hurewicz fibrations, and an indication is also given, mainly via examples, of how they differ. The last two sections of this paper are mostly expository; they set forth the model category techniques needed to prove the main theorems. [Copyright &y& Elsevier]

We give an algebro-geometric interpretation of C 2 -equivariant stable homotopy theory by means of the b -topology introduced by Claus Scheiderer in his study of 2-torsion phenomena in étale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a ∞-topos X with G -action a presentable stable ∞-category Sp G (X) , which recovers the ∞-category Sp G of genuine G -spectra when X is the terminal G -∞-topos. Given a scheme X with 1 2 ∈ O X , our construction then specializes to produce an ∞-category Sp b C 2 (X) of " b -sheaves with transfers" as b -sheaves of spectra on the small étale site of X equipped with certain transfers along the extension X [ i ] → X ; if X is the spectrum of a real closed field, then Sp b C 2 (X) recovers Sp C 2 . On a large class of schemes, we prove that, after p -completion, our construction assembles into a premotivic functor satisfying the full six functors formalism. We then introduce the b -variant SH b (X) of the ∞-category SH (X) of motivic spectra over X (in the sense of Morel-Voevodsky), and produce a natural equivalence of ∞-categories SH b (X) p ∧ ≃ Sp b C 2 (X) p ∧ through amalgamating the étale and real étale motivic rigidity theorems of Tom Bachmann. This involves a purely algebro-geometric construction of the C 2 -Tate construction, which may be of independent interest. Finally, as applications, we deduce a " b -rigidity" theorem, use the Segal conjecture to show étale descent of the 2-complete b -motivic sphere spectrum, and construct a parametrized version of the C 2 -Betti realization functor of Heller-Ormsby. [ABSTRACT FROM AUTHOR]

We build new algebraic structures, which we call genuine equivariant operads and which can be thought of as a hybrid between 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 ∞ -operads of Blumberg and Hill. [ABSTRACT FROM AUTHOR]

There are fundamental open problems in the precise global nature of RR-field tadpole cancellation conditions in string theory. Moreover, the non-perturbative lift as M5/MO5-anomaly cancellation in M-theory had been based on indirect plausibility arguments, lacking a microscopic underpinning in M-brane charge quantization. We provide a framework for answering these questions, crucial not only for mathematical consistency but also for phenomenological accuracy of string theory, by formulating the M-theory C-field on flat M-orientifolds in the generalized cohomology theory called Equivariant Cohomotopy. This builds on our previous results for smooth but curved spacetimes, showing in that setting that charge quantization in twisted Cohomotopy rigorously implies a list of expected anomaly cancellation conditions. Here we further expand this list by proving that brane charge quantization in unstable equivariant Cohomotopy implies the anomaly cancellation conditions for M-branes and D-branes on flat orbi-orientifolds. For this we (a) use an unstable refinement of the equivariant Hopf-tom Dieck theorem to derive local/twisted tadpole cancellation, and in addition (b) the lift to super-differential cohomology to establish global/untwisted tadpole cancellation. Throughout, we use (c) the unstable Pontrjagin–Thom theorem to identify the brane/O-plane configurations encoded in equivariant Cohomotopy and (d) the Boardman homomorphism to equivariant K-theory to identify Chan–Paton representations of D-brane charge. We find that unstable equivariant Cohomotopy, but not its image in equivariant K-theory, distinguishes D-brane charge from the finite set of types of O-plane charges. [ABSTRACT FROM AUTHOR]

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. The emergence of core structures in string and M‐theory as higher structures in super homotopy theory is reviewed. In particular, super homotopy theory discovers all super p‐branes, their intersection laws, and their dualities including T‐ and S‐duality. An outlook as how to formalise microscopic M‐theory in super homotopy theory is given. [ABSTRACT FROM AUTHOR]