Some general theorems on the cohomology of classifying spaces of compact Lie groups

Feshbach, Mark

doi:10.1090/S0002-9947-1981-0597866-4

Some general theorems on the cohomology of classifying spaces of compact Lie groups
HTML articles powered by AMS MathViewer

by Mark Feshbach

Trans. Amer. Math. Soc. 264 (1981), 49-58

DOI: https://doi.org/10.1090/S0002-9947-1981-0597866-4

PDF | Request permission

Abstract:

This paper is divided into two parts. The first part proves a number of general theorems on the cohomology of the classifying spaces of compact Lie groups. These theorems are proved by transfer methods, relying heavily on the double coset theorem [F$_{1}$]. Several of these results are well known while others are quite new. For the most part the proofs of the theorems are independent of each other and are quite short. Nevertheless they are true in great generality. Several are proven for arbitrary compact Lie groups and arbitrary cohomology theories. Perhaps the most interesting of the new results relates the cohomology of the classifying space of an arbitrary compact Lie group with that of the normalizer of a maximal torus. The second part of the paper generalizes many theorems to certain equivariant cohomology theories. Some of these theorems appear in [F$_{2}$].

References

J. C. Becker and D. H. Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976), no. 2, 107–133. MR 436137
G. Brumfiel and I. Madsen, Evaluation of the transfer and the universal surgery classes, Invent. Math. 32 (1976), no. 2, 133–169. MR 413099, DOI 10.1007/BF01389959
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Albrecht Dold, The fixed point transfer of fibre-preserving maps, Math. Z. 148 (1976), no. 3, 215–244. MR 433440, DOI 10.1007/BF01214520
Albrecht Dold, Transfert des points fixes d’une famille continue d’applications, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1291–1293 (French). MR 348734
Albrecht Dold, The fixed point index of fibre-preserving maps, Invent. Math. 25 (1974), 281–297. MR 380776, DOI 10.1007/BF01389731
Eldon Dyer, Cohomology theories, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0268883
Leonard Evens, On the Chern classes of representations of finite groups, Trans. Amer. Math. Soc. 115 (1965), 180–193. MR 212099, DOI 10.1090/S0002-9947-1965-0212099-X
Mark Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. 251 (1979), 139–169. MR 531973, DOI 10.1090/S0002-9947-1979-0531973-8
Mark Feshbach, The transfer and compact Lie groups, Bull. Amer. Math. Soc. 83 (1977), no. 3, 372–374. MR 440548, DOI 10.1090/S0002-9904-1977-14274-2
Mark Feshbach, The transfer and characteristic classes, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 657, Springer, Berlin, 1978, pp. 156–162. MR 513546
H. Hopf and H. Samelson, Ein Satz über die Wirkungsräume geschlossener Liescher Gruppen, Comment. Math. Helv. 13 (1941), 240–251 (German). MR 6546, DOI 10.1007/BF01378063
Wu-yi Hsiang, Cohomology theory of topological transformation groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York-Heidelberg, 1975. MR 0423384
Robert M. Switzer, Algebraic topology—homotopy and homology, Die Grundlehren der mathematischen Wissenschaften, Band 212, Springer-Verlag, New York-Heidelberg, 1975. MR 0385836