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A generalization of Smith theory. (English) Zbl 0635.57020
Let G be a finite p-group and let X be a finite dimensional G-CW-complex with finite dimensional mod p-cohomology. Let SX denote the subcomplex of singular points X and let \(FX:=X/SX\). The author found some inequalities which involve dimensions of the mod p-cohomology groups of X, SX and FX/G. His inequalities are sharper than those which can be obtained from the classical P. A. Smith theory. The proofs are based on an interpretation of the (ordinary) cohomology groups of X, SX and FX/G as Bredon equivariant cohomology groups of X with coefficients in suitable systems.
Reviewer: S.Jackowski

MSC:
57S17 Finite transformation groups
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
55N25 Homology with local coefficients, equivariant cohomology
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[1] Glen E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin-New York, 1967. · Zbl 0162.27202
[2] E. E. Floyd, On periodic maps and the Euler characteristics of associated spaces, Trans. Amer. Math. Soc. 72 (1952), 138 – 147. · Zbl 0046.16603
[3] Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. Bertram Huppert and Norman Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 243, Springer-Verlag, Berlin-New York, 1982.
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