Proper Equivariant Stable Homotopy Theory

Dieter Degrijse, Markus Hausmann, Wolfgang Lück, Irakli Patchkoria, Stefan Schwede

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This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthm¨uller isomorphisms, and the resulting equivariant cohomology theories support the analog of an RO(G)- grading.
Our model for genuine proper G-equivariant stable homotopy theory is the category of orthogonal G-spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G. This class of π∗-isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G-spectrum represents an equivariant cohomology theory on the category of G-spaces. These represented cohomology theories are designed to only depend on the ‘proper G-homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G-sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G-actions has a finite G-CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G-CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G-vector bundles. Via this description, we can identify the previously defined G-cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G-spectra.
Original languageEnglish
JournalMemoirs of the American Mathematical Society
Publication statusAccepted/In press - 13 Oct 2020

Bibliographical note

All five authors were supported by the Hausdorff Center for Mathematics at the University of Bonn (DFG GZ 2047/1, project ID 390685813) and by the Centre for Symmetry and Deformation at the University of Copenhagen (CPH-SYM-DNRF92); we would like to thank these two institutions for their hospitality, support and the stimulating atmosphere. Hausmann, Patchkoria and Schwede were partially supported by the DFG Priority Programme 1786 ‘Homotopy Theory and Algebraic Geometry’. Work on this monograph was funded by the ERC Advanced Grant ‘KL2MG-interactions’ of L¨uck (Grant ID 662400), granted by the European Research Council. Patchkoria was supported by the Shota Rustaveli National Science Foundation Grant 217-614. Patchkoria and Schwede would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme ‘Homotopy harnessing higher structures’, when work on this paper was undertaken (EPSRC grant number EP/R014604/1). We would also like to thank Bob Oliver and Søren Galatius for helpful conversations on topics related to this project, and the anonymous referee for his or her careful reading and the many useful comments.


  • Lie group
  • equivariant homotopy theory
  • proper action


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