Equivariant Structure on Smash Powers

Brun, Morten; Dundas, Bjørn Ian; Stolz, Martin

doi:10.48550/arxiv.1604.05939

Cited by 10 publications

(16 citation statements)

References 20 publications

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“…The next theorem shows that smashing with a G-flat orthogonal G-spectrum preserves π * -isomorphisms. This result is due to Stolz Since Stolz' thesis [163] is not published, the notation and level of generality in [32] is different from ours, and the characterization of flat objects in terms of latching maps is not explicitly mentioned in [32] nor [163], we spell out the argument.…”

Section: Productsmentioning

confidence: 87%

“…i.e., the value of |X| at an inner product space V is the realization of the simplicial space [n] → X n (V), as discussed in Construction A. 32. By Proposition A.35, the realization can be formed in the ambient category of all topological spaces, and the result is automatically compactly generated.…”

Section: Global Model Structure For Orthogonal Spacesmentioning

confidence: 99%

“…We do not discuss model category structures for orthogonal G-spectra; the interested reader can find different ones in the memoir of Mandell and May [108], in the thesis of Stolz [163], the article by Brun, Dundas and Stolz [32] and (for finite groups) in the paper of Hill, Hopkins and Ravenel [77].…”

Section: Equivariant Stable Homotopy Theorymentioning

confidence: 99%

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Global homotopy theory

Schwede

2018

Preprint

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Equivariant stable homotopy theory has a long tradition, starting from geometrically motivated questions about symmetries of manifolds. The homotopy theoretic foundations of the subject were laid by tom Dieck, Segal and May and their students and collaborators in the 70's, and during the last decades equivariant stable homotopy theory has been very useful to solve computational and conceptual problems in algebraic topology, geometric topology and algebraic K-theory. Various important equivariant theories naturally exist not just for a particular group, but in a uniform way for all groups in a specific class. Prominent examples of this are equivariant stable homotopy, equivariant K-theory or equivariant bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., the adjective 'global' refers to simultaneous and compatible actions of all compact Lie groups.This book introduces a new context for global homotopy theory. Various ways to provide a home for global stable homotopy types have previously been explored in [100, Ch. II], [68, Sec. 5], [18] and [19]. We use a different approach: we work with the well-known category of orthogonal spectra, but use a much finer notion of equivalence, the global equivalences, than what is traditionally considered. The basic underlying observation is that an orthogonal spectrum gives rise to an orthogonal G-spectrum for every compact Lie group G, and the fact that all these individual equivariant objects come from one orthogonal spectrum implicitly encodes strong compatibility conditions as the group G varies. An orthogonal spectrum thus has G-equivariant homotopy groups for every compact Lie group, and a global equivalence is a morphism of orthogonal spectra that induces isomorphisms for all equivariant homotopy groups for all compact Lie groups (based on 'complete G-universes', compare Definition 4.1.3).The structure on the equivariant homotopy groups of an orthogonal spectrum gives an idea of the information encoded in a global homotopy type in our sense: the equivariant homotopy groups π G k (X) are contravariantly funcv vi Preface torial for continuous group homomorphisms ('restriction maps'), and they are covariantly functorial for inclusions of closed subgroups ('transfer maps'). The restriction and transfer maps enjoy various transitivity properties and interact via a double coset formula. This kind of algebraic structure has been studied before under different names, e.g., 'global Mackey functor', 'inflation functor', . . . . From a purely algebraic perspective, there are various parameters here than one can vary, namely the class of groups to which a value is assigned and the classes of homomorphisms to which restriction maps respectively transfer maps are assigned, and lots of variations have been explored. However, the decision to work with orthogonal spectra and equivariant homotopy groups on complete universes dictates a canonical choice: we prove in Theorem 4.2.6 that the algebra of natural operations between the equivariant homotopy groups of ...

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Section: Productsmentioning

confidence: 87%

Section: Global Model Structure For Orthogonal Spacesmentioning

confidence: 99%

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Global homotopy theory

Schwede

2018

Preprint

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“…We mention that all the known classical models for G-equivariant stable homtopy theory based on spaces (and not on simplicial sets) are cofibrantly generated, proper G-equivariant stable model categories. These include as already mentioned Sp G [MM02], the S-model structure (flat model structure, see [Sto11,Theorem 2.3.27] and also [BDS16]), the model category of S G -modules [MM02, IV.2], the model category of G-equivariant continuous functors [Blu06] and the model categories of G-equivariant topological symmetric spectra in the sense of [Man04] and [Hau17].…”

Section: Background and Conventionsmentioning

confidence: 99%

Rigidity and exotic models for $v_1$-local $G$-equivariant stable homotopy theory

Patchkoria¹,

Roitzheim²

2017

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We prove that the v1-local G-equivariant stable homotopy category for G a finite group has a unique G-equivariant model at p = 2. This means that at the prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p ≥ 5 equivariant rigidity fails in general.

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“…Building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory [20], the topological Hochschild homology of a ring spectrum R can be viewed as the norm N S 1 e R (see [3] and [10]). This viewpoint leads to several natural generalizations.…”

Section: Introductionmentioning

confidence: 99%

Computational tools for twisted topological Hochschild homology of equivariant spectra

Adamyk¹,

Gerhardt²,

Hess³

et al. 2020

Preprint

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Twisted topological Hochschild homology of Cn-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.

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Equivariant Structure on Smash Powers

Cited by 10 publications

References 20 publications

Global homotopy theory

Global homotopy theory

Rigidity and exotic models for $v_1$-local $G$-equivariant stable homotopy theory

Computational tools for twisted topological Hochschild homology of equivariant spectra

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