Project description

This project proposal is concerned with several basic questions about global equivariant homotopy theory, where the adjective `global' refers to simultaneous and compatible actions of all compact Lie groups. The framework is the model for global stable homotopy theory based on the category of orthogonal spectra, using a finer notion of equivalence than is usually considered, namely the global equivalences. The basic underlying observation is that every 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. Prominent examples of global theories that can be modeled in this framework are equivariant stable homotopy, equivariant K-theory or equivariant bordism. Specific aims of this project are to develop and study:
  • global algebraic K-theory;
  • global units and global orientations;
  • a global generalization of Snaith's theorem that constructs equivariant Ktheory from~C Pinfty by inverting a suitable Bott class;
  • a global obstruction theory for ultra-commutative multiplications.

Related publications

Published articles

Stefan Schwede, Global homotopy theory New Mathematical Monographs 34. Cambridge University Press, Cambridge, 2018. xvi+828 pp. www.cambridge.org/9781108425810.

Christian Wimmer, Rational extensions of the representation ring global functor and a splitting of global equivariant $K$-theory, Bulletin of the London Mathematical Society 50 (2018), 863-873 doi.org/10.1112/blms.12189

Stefan Schwede, Orbispaces, orthogonal spaces, and the universal compact Lie group, Mathematische Zeitschrift 294 (2020), 71-107 doi.org/10.1007/s00209-019-02265-1

Markus Hausmann, Symmetric products and subgroup lattices, Geometry & Topology 22 (2018), no. 3, 1547–1591.

Markus Hausmann, Symmetric spectra model global homotopy theory of finite groups, Algebr. Geom. Topol. 19 (2019), no. 3, 1413–1452. doi.org/10.2140/agt.2019.19.1413

Markus Hausmann, Dominik Ostermayr, Filtrations of global equivariant K-theory, Mathematische Zeitschrift 295 (2020), 161--210 doi.org/10.1007/s00209-019-02338-1

Stefan Schwede, Equivariant properties of symmetric products, Journal of the American Mathematical Society 30 (2017), 673-711.

Stefan Schwede, Global algebraic K-theory, Journal of Topology 15 (2022), 1325-1454 DOI: 10.1112/topo.12241

Stefan Schwede, Global stable splittings of Stiefel manifolds, Documenta Mathematica 27 (2022), 789-845 DOI: 10.25537/dm.2022v27.789-845

Stefan Schwede, Splittings of global Mackey functors and regularity of equivariant Euler classes, Proceedings of the London Mathematical Society 125 (2022), 258-276 DOI: 10.1112/plms.12446

Preprints

Michael Stahlhauer, \(G_\infty\)-ring spectra and Moore spectra for \(\beta\)-rings. To appear, Algebraic & Geometric Topology. arXiv:2007.14304

Christian Wimmer, Rational global homotopy theory and geometric fixed points. PhD thesis, Universität Bonn, 2017