The classifying space of a permutation representation

Blowers, James V.

doi:10.1090/S0002-9947-1977-0442931-4

The classifying space of a permutation representation
HTML articles powered by AMS MathViewer

by James V. Blowers PDF

Trans. Amer. Math. Soc. 227 (1977), 345-355 Request permission

Abstract:

In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. An example of this classifying space is given by a generalization of the infinite join construction that defines the standard example of a classifying space of a group. In a previous paper of the author, the join of two permutation representations was defined, and it was shown that the cohomology ring of the join was trivial. In this paper the classifying space of the join of two permutation representations is shown to be the topological join of the two classifying spaces and from this the triviality of the cup-product is derived topologically.

References

James V. Blowers, The cohomology rings of certain finite permutation representations, Proc. Amer. Math. Soc. 45 (1974), 157–163. MR 379690, DOI 10.1090/S0002-9939-1974-0379690-2
Glen E. Bredon, Equivariant cohomology theories, Lecture Notes in Mathematics, No. 34, Springer-Verlag, Berlin-New York, 1967. MR 0214062, DOI 10.1007/BFb0082690
Ronald Brown, Elements of modern topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1968. MR 0227979
George E. Cooke and Ross L. Finney, Homology of cell complexes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1967. Based on lectures by Norman E. Steenrod. MR 0219059, DOI 10.1515/9781400877751
Samuel Eilenberg and J. C. Moore, Foundations of relative homological algebra, Mem. Amer. Math. Soc. 55 (1965), 39. MR 178036
Morton E. Harris, Generalized group cohomology, Fund. Math. 65 (1969), 269–288. MR 252481, DOI 10.4064/fm-65-3-269-288
G. Hochschild, Relative homological algebra, Trans. Amer. Math. Soc. 82 (1956), 246–269. MR 80654, DOI 10.1090/S0002-9947-1956-0080654-0
John Milnor, Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430–436. MR 77932, DOI 10.2307/1970012
Ernst Snapper, Cohomology of permutation representations. I. Spectral sequences, J. Math. Mech. 13 (1964), 133–161. MR 0157998
N. E. Steenrod, Cohomology operations, Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. MR 0145525