Document Zbl 0626.55012 - zbMATH Open

Consider a closed normal subgroup \(\Pi\) of a topological group \(\Gamma\) with quotient group G. If Y is a \(\Gamma\)-space such that \(\Pi\) acts freely on it then the projection \(Y\to Y/\Pi\) is a G-map and it is called a principal \((G,\Pi)\)-bundle. There exists a universal (G,\(\Pi)\)-bundle E(G,\(\Pi)\to B(G,\Pi).\)
The authors shows that \(H^*_ G(B(G,\Pi)) = H^*(B\Gamma)\) where \(H^*_ G(-) = H^*(EG\times_{G}-)\) is the Borel cohomology. In the special case when \(\Gamma =G\times \Pi\) he observes that characteristic classes of a (G,\(\Pi)\)-bundle over a G-space X with values in the Borel cohomology are defined by the \(H^*(BG)\)-module structure on \(H^*_ G(X)\) and the ordinary characteristic classes of the \(\Pi\)-bundle \(EG\times_ GY\to EG\times_ GX\).

MSC:

55R40	Homology of classifying spaces and characteristic classes in algebraic topology
55N91	Equivariant homology and cohomology in algebraic topology
57S99	Topological transformation groups
55N25	Homology with local coefficients, equivariant cohomology

References:

[1]	J.F. Adams, J.-P. Haeberly, S. Jackowski, and J.P. May, A generalization of the Segal conjecture, Topology, to appear.; J.F. Adams, J.-P. Haeberly, S. Jackowski, and J.P. May, A generalization of the Segal conjecture, Topology, to appear.
[2]	J.F. Adams, J.-P. Haeberly, S. Jackowski, and J.P. May, A generalization of the Atiyah-Segal completion theorem, Topology, to appear.; J.F. Adams, J.-P. Haeberly, S. Jackowski, and J.P. May, A generalization of the Atiyah-Segal completion theorem, Topology, to appear. · Zbl 0657.55007
[3]	Elmendorf, A., Systems of fixed point sets, Trans. Amer. Math. Soc., 277, 275-284 (1983) · Zbl 0521.57027
[4]	Lashof, R. K., Equivariant bundles, Illinois J. Math., 26, 257-271 (1982) · Zbl 0458.55009
[5]	R.K. Lashof and J.P. May, Generalized equivariant bundles, to appear.; R.K. Lashof and J.P. May, Generalized equivariant bundles, to appear. · Zbl 1542.55005
[6]	Lashof, R. K.; May, J. P.; Segal, G. B., Equivariant bundles with Abelian structural group, Contemporary Math., 19, 167-176 (1983) · Zbl 0526.55020
[7]	Lewis, G.; May, J. P.; Steinberger, M., Equivariant stable homotopy theory, (Lecture Notes in Math., 1213 (1986), Springer: Springer Berlin), (with contributions by J.E. McClure) · Zbl 0611.55001

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.