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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 This generalizes earlier computations of Stong [unpublished, cited and reproduced in J. L. Caruso, Math.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.
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
Full Text: DOI arXiv
[1] Bredon, G.E.: Equivariant Cohomology Theories, Lecture Notes in Mathematics, vol. 34. Springer, Berlin (1967) · Zbl 0162.27202
[2] Caruso, J.L.: Operations in equivariant \[{\bf Z}{/}p\] Z/p-cohomology. Math. Proc. Camb. Philos. Soc. 126, 521-541 (1999) · Zbl 0933.55009
[3] Dress, A.W.M.: Contributions to the theory of induced representations. In: Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math, vol. 342, pp. 183-240. Springer, Berlin (1973)
[4] Ferland, K.K.: On the RO(G)-graded equivariant ordinary cohomology of generalized G-cell complexes for G = Z/p, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.). Syracuse University (1999)
[5] Ferland, K.K., Lewis, L.G., Jr.: The \[R{\rm O}(G)\] RO(G)-graded equivariant ordinary homology of \[G\] G-cell complexes with even-dimensional cells for \[G={\mathbb{Z}}/p\] G=Z/p, Memoirs of the American Mathematical Society, vol. 167, pp. viii+129 (2004) · Zbl 1052.55007
[6] Greenlees, J.P.C., May, J.P.: Equivariant Stable Homotopy Theory, in Handbook of Algebraic Topology, pp. 277-323. North-Holland, Amsterdam (1995) · Zbl 0866.55013
[7] Hill, M.A., Hopkins, M.J., Ravenel, D.C.: On the nonexistence of elements of Kervaire invariant one. Ann. Math. (2) 184, 1-262 (2016) · Zbl 1366.55007
[8] Lewis Jr., L.G.: The \[R{\rm O}(G)\] RO(G)-graded equivariant ordinary cohomology of complex projective spaces with linear \[{\bf Z}/p\] Z/p actions, in algebraic topology and transformation groups (Göttingen, vol. 1361 of Lecture Notes in Math. 1988, pp. 53-122. Springer, Berlin (1987)
[9] Lewis Jr., L.G.: The equivariant Hurewicz map. Trans. Am. Math. Soc. 329, 433-472 (1992) · Zbl 0769.54042
[10] Lewis, L.G., Jr.: The category of Mackey functors for a compact Lie group, in Group representations: cohomology, group actions and topology. In: Seattle, WA: vol. 63 of Proc. Sympos. Pure Math., Amer. Math. Soc. Providence, RI, vol. 1998, pp. 301-354 (1996)
[11] Lewis, L.G. Jr., May, J.P., Steinberger, M., McClure, J.E.: Equivariant stable homotopy theory. With contributions by J. E. McClure. Lecture Notes in Mathematics, vol. 1213, x+538 pp, Springer-Verlag, Berlin (1986)
[12] Lewis Jr., L.G.: The theory of green functors. Mimeographed notes (1981)
[13] May, J.P.: Equivariant homotopy and cohomology theory, vol. 91 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner
[14] Wasserman, A.G.: Equivariant differential topology. Topology 8, 127-150 (1969) · Zbl 0215.24702
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