geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
The possible actions of well-behaved topological groups (such as compact Lie groups) on topological n-spheres display various interesting patterns in their classification.
This entry is meant to eventually list and discuss some of these. For the moment it mainly just collects some references.
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Given an continuous action of the circle group on the topological 4-sphere, its fixed point space is of one of two types:
either it is the 0-sphere $S^0 \hookrightarrow S^4$
or it has the rational homotopy type of an even-dimensional sphere.
(Félix-Oprea-Tanré 08, Example 7.39)
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Discussion of free group actions on spheres by finite groups includes
C. T. C. Wall, Free actions of finite groups on spheres, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 (pdf)
Alejandro Adem, Constructing and deconstructing group actions (arXiv:0212280)
Discussion of circle group-actions on spheres includes
The subgroups of SO(8) which act freely on $S^7$ have been classified in
and lifted to actions of Spin(8) in
Further discussion of these actions of $Spin(8)$ on the 7-sphere is in
Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)
Paul de Medeiros, José Figueroa-O'Farrill, Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, Euclid)
where they are related to the black M2-brane BPS-solutions of 11-dimensional supergravity at ADE-singularities.