The cohomology ring of subspaces of universal \(S^1\)-space with finite orbit types (Q388023)

In the classical case of universal bundles, a topological group acts freely on a contractible space. The orbit space is the classifying space. In the paper of \textit{R. S. Palais} [The classification of \(G\)-spaces, Mem. Am. Math. Soc. 36, 72 p. (1960; Zbl 0119.38403)], this is extended (for compact groups, with some further restrictions) to construct universal \(G\)-spaces, with varying orbit-types. The authors call these universal, isovariant extensors.NEWLINENEWLINEThe general existence problem, where orbits vary, was solved in [\textit{S. M. Ageev}, Sb. Math. 203, No. 6, 769--797 (2012); translation from Mat. Sb. 203, No. 6, 3-34 (2012; Zbl 1257.54038)] and [Izv. Math. 76, No. 5, 857--880 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 5, 3-28 (2012; Zbl 1266.54081)].NEWLINENEWLINEFor such spaces, the orbit bundle holds much information, like characteristic classes in this general setting. The main theorem of this paper calculates the rational cohomology of the orbit space (\({\mathcal F}\) orbit bundle) when the group is \(G=\mathrm{SO}(2)\).