Filtrations of global equivariant K-theory

Hausmann, Markus; Ostermayr, Dominik

doi:10.48550/arxiv.1510.04011

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“…On the other hand, Remark 6.3.38 shows that the map is not always surjective, as the class tr S U( 2) N (1) in π S U (2) 0 (ku) is not in the image. In [74], Hausmann and Ostermayr give a complete calculation of π 0 (ku) as a global functor. The strategy of [74] is to identify the global homotopy types of the subquotients of the rank filtration and deduce from it a presentation π 0 (ku) by generators and relations.…”

Section: This Map Coincides With the Compositementioning

confidence: 99%

“…In [74], Hausmann and Ostermayr give a complete calculation of π 0 (ku) as a global functor. The strategy of [74] is to identify the global homotopy types of the subquotients of the rank filtration and deduce from it a presentation π 0 (ku) by generators and relations. The final answer is that π 0 (ku) is generated as a global functor by the classes…”

Section: This Map Coincides With the Compositementioning

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: This Map Coincides With the Compositementioning

confidence: 99%

Section: This Map Coincides With the Compositementioning

confidence: 99%

Global homotopy theory

Schwede

2018

Preprint

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“…Globally defined Mackey functors have emerged in the last two decades as a naturally occuring algebraic structure with applications in group cohomology and representation theory [Web93], [Web00], [Bou06], [Bou07], [Web10]. Global functors are a particular version (called inflation functors in [Web93]) appearing in equivariant topology, where the 'global' perspective has also proven to be illuminating in the study of equivariant homotopy types [Sch17], [HO15], [Hau18]. They are defined more generally for compact Lie groups, but we will restrict attention to finite groups in the following.…”

Section: Introductionmentioning

confidence: 99%

Rational extensions of the representation ring global functor and a splitting of global equivariant K-theory

Wimmer

2018

Bull. London Math. Soc.

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We identify the group of homomorphisms HomGF (F, RU Q ) in the abelian category GF of (fin)global functors to the rationalization of the unitary representation ring functor RU Q . The result is a certain inverse limit over the values of F at cyclic groups and we deduce that the higher Ext-groups Ext n GF (F, RU Q ) have to vanish for n 2. This leads to a rational splitting of the (fin)global equivariant K-theory spectrum KU into a sum of Eilenberg-MacLane spectra. Interpreted in terms of cohomology theories, it means that the equivariant Chern character is compatible with restrictions along all group homomorphisms.

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Filtrations of global equivariant K-theory

Cited by 2 publications

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Global homotopy theory

Global homotopy theory

Rational extensions of the representation ring global functor and a splitting of global equivariant K-theory

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