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In this paper we study the simplicial complex induced by the poset of Brauer pairs ordered by inclusion for the family of finite reductive groups. In the defining characteristic case the homotopy type of this simplicial complex coincides with that of the Tits building thanks to a well-known result of Quillen. On the other hand, in the non-defining characteristic case, we show that the simplicial complex of Brauer pairs is homotopy equivalent to a simplicial complex determined by generalised Harish-Chandra theory. This extends earlier results of the author on the Brown complex and makes use of the theory of connected subpairs and twisted block induction developed by Cabanes and Enguehard. [ABSTRACT FROM AUTHOR]

Fix a finite group G and an n-dimensional orthogonal G-representation V. We define the equivariant factorization homology of a V-framed smooth G-manifold with coefficients in an E V -algebra using a two-sided bar construction, generalizing (Andrade, From manifolds to invariants of E n -algebras. PhD thesis, Massachusetts Institute of Technology, 2010; Kupers and Miller, Math Ann 370(1–2):209–269, 2018). This construction uses minimal categorical background and aims for maximal concreteness, allowing convenient proofs of key properties, including invariance of equivariant factorization homology under change of tangential structures. Using a geometrically-seen scanning map, we prove an equivariant version (eNPD) of the nonabelian Poincaré duality theorem due to several authors. The eNPD states that the scanning map gives a G-equivalence from the equivariant factorization homology to mapping spaces out the one-point compactification of the G-manifolds, when the coefficients are G-connected. For non-G-connected coefficients, when the G-manifolds have suitable copies of R in them, the scanning map gives group completions. This generalizes the recognition principle for V-fold loop spaces in Guillou and May (Algebr Geom Topol 17(6):3259–3339, 2017). [ABSTRACT FROM AUTHOR]

We describe the slices of positive integral suspensions of the equivariant Eilenberg–MacLane spectrum H F 2 ̲ for the constant Mackey functor over the Klein four-group C 2 × C 2 . [ABSTRACT FROM AUTHOR]

Arone and Lesh (J Reine Angew Math 604:73–136, 2007; Fund Math 207(1):29–70, 2010) constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg–MacLane spectrum for the integers. In this paper we consider (global) equivariant generalizations of these filtrations and another closely related class of filtrations, the modified rank filtrations of the K-theory spectra themselves. We lift Arone and Lesh's description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups. It turns out that these representation ring filtrations are considerably easier to express in a global equivariant context than over a fixed compact Lie group. Furthermore, they have formal similarities to the filtration on Burnside rings induced by the symmetric products of spheres, which was computed by Schwede (J Am Math Soc 30(3):673–711, 2017). [ABSTRACT FROM AUTHOR]

This paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory. At the same time, we provide a new perspective by interpreting it as the homotopy theory of 'spaces with an action of the universal compact Lie group'. The upshot is a novel way to construct and study genuine cohomology theories on stacks, orbifolds, and orbispaces, defined from stable global homotopy types represented by orthogonal spectra. The universal compact Lie group (which is neither compact nor a Lie group) is a well-known object, namely the topological monoid L of linear isometric self-embeddings of R ∞ . The underlying space of L is contractible, and the homotopy theory of L -spaces with respect to underlying weak equivalences is just another model for the homotopy theory of spaces. However, the monoid L contains copies of all compact Lie groups in a specific way, and we define global equivalences of L -spaces by testing on corresponding fixed points. We establish a global model structure on the category of L -spaces and prove it to be Quillen equivalent to the global model category of orthogonal spaces, and to the category of orbispaces, i.e., presheaves of spaces on the global orbit category. [ABSTRACT FROM AUTHOR]

In this paper, we compute the R O (C pq) -graded cohomology of C pq -orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong and Lewis for the group C p . The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group C p was proved by Lewis. We demonstrate that certain complex projective spaces and complex Grassmannians satisfy the freeness theorem. [ABSTRACT FROM AUTHOR]