Computations in \(C_{pq}\)-Bredon cohomology.

*(English)*Zbl 1435.55003Bredon 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 This generalizes earlier computations of Stong [unpublished, cited and reproduced in J. L. Caruso, Math.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.

“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 This generalizes earlier computations of Stong [unpublished, cited and reproduced in J. L. Caruso, Math.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.

Reviewer: Mahender Singh (Manauli)

##### 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 |

##### References:

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