In this paper we show that the category of symmetric spectra can be used to model global equivariant homotopy theory of finite groups.

Abstract. In this paper we develop the basic homotopy theory of G-symmetric spectra (that is, symmetric spectra with a G-action) for a finite group G, as a model for equivariant stable homotopy with respect to a G-set universe. This model lies in between Mandell's equivariant symmetric spectra and the G-orthogonal spectra of Mandell and May and is Quillen equivalent to the two. We further discuss equivariant semistability, construct model structures on module, algebra and commutative algebra categories and describe the homotopical properties of the multiplicative norm in this context. IntroductionStable equivariant homotopy theory has seen various applications both in equivariant and non-equivariant topology. Many of these applications make use of constructions which require or are simplified by a highly structured model of equivariant spectra with a point-set level smash product. For example, model structures on module or (commutative) algebra categories allow one to employ algebraic techniques in order to obtain new equivariant spectra with desired properties, fixed points of commutative equivariant ring spectra form non-equivariant E ∞ -ring spectra and their homotopy groups inherit the structure of a Tambara functor. Another example is the construction of the spectrum level multiplicative norm introduced in [HHR16], which plays a central role in the solution of the Kervaire invariant one problem.The first highly structured models for stable equivariant homotopy theory were the S Gmodules and G-orthogonal spectra of [MM02] and the equivariant symmetric spectra of [Man04]. While the first two can be formed over arbitrary compact Lie groups, the latter only works for finite groups. In turn it has the advantage that it can also be based on simplicial sets, whereas the other two need topological spaces.In this paper we develop a further model we call G-symmetric spectra. It also requires the group to be finite and can be based both on simplicial sets and topological spaces. Conceptually, it lies in between Mandell's equivariant symmetric spectra and G-orthogonal spectra. In order to describe the difference, we have to recall the definitions. G-orthogonal spectra are formed with respect to a G-representation universe which determines what kinds of representation spheres S V become invertible in the homotopy category. Then, roughly speaking, a G-orthogonal spectrum X consists of a family of based G-spaces X(V ) for every finite dimensional subrepresentation V of the chosen universe, together with structure maps of the form X(V ) ∧ S W → X(V ⊕ W ). In addition, every X(V ) possesses an action of the orthogonal group O(V ) which is suitably compatible with the G-action and the structure maps. G-orthogonal spectra formed with respect to different universes are connected by so-called change of universe functors. It was noticed in [MM02, V, Thm. 1.5] that on the point-set level these are always equivalences of categories (though the derived functors are not). In other words, the underlying category does not really depen...

For a finite abelian group A, we determine the Balmer spectrum of Sp ω A , the compact objects in genuine A-spectra. This generalizes the case A = Z/pZ due to Balmer and Sanders [BS17], by establishing (a corrected version of) their logp-conjecture for abelian groups. We also work out the consequences for the chromatic type of fixed-points and establish a generalization of Kuhn's blue-shift theorem for Tate-constructions [Kuh04].

We study the Balmer spectrum of the category of finite G-spectra for a compact Lie group G, extending the work for finite G by Strickland, Balmer-Sanders, Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton and others. We give a description of the underlying set of the spectrum and show that the Balmer topology is completely determined by the inclusions between the prime ideals and the topology on the space of closed subgroups of G. Using this, we obtain a complete description of this topology for all abelian compact Lie groups and consequently a complete classification of thick tensor-ideals. For general compact Lie groups we obtain such a classification away from a finite set of primes p. Contents2010 Mathematics Subject Classification 55P42, 55P91

In [AL07] and [AL10], Arone and Lesh 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 of 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 globally 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 Schwede computed in [Sch14].