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Computations in \(C_{pq}\)-Bredon cohomology. (English) Zbl 1435.55003
Bredon cohomology came into being as the natural generalization of ordinary cohomology to the category of \(G\)-spaces. While Borel’s equivariant cohomology theory for \(G\)-spaces is used more often in the literature due to availability of non-equivariant techniques, Bredon cohomology has turned out to be difficult to compute even for simple spaces like spheres with a linear \(G\)-action.
“We compute the \(RO(C_{pq})\)-graded cohomology of \(C_{pq}\)-orbits. We deduce that in all the cases the Bredon cohomology groups are a function of the fixed-point dimensions of the underlying virtual representations. Further, when thought of as a Mackey functor, the same independence result holds in almost all cases. This generalizes earlier computations of Stong [unpublished, cited and reproduced in J. L. Caruso, Math. Proc. Camb. Philos. Soc. 126, No. 3, 521–541 (1999; Zbl 0933.55009)] and L. G. Lewis jun. [Lect. Notes Math. 1361, 53–122 (1988; Zbl 0669.57024)] for the cyclic group \(C_p\). The computations of cohomology of orbits are used to prove a freeness theorem. The analogous result for the group \(C_p\) was proved by Lewis. We demonstrate that certain complex projective spaces and complex Grassmannians satisfy the freeness theorem.”
The paper is well-written with details spelled out clearly.
MSC:
55N91 Equivariant homology and cohomology in algebraic topology
55P91 Equivariant homotopy theory in algebraic topology
57S17 Finite transformation groups
14M15 Grassmannians, Schubert varieties, flag manifolds
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References:
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