A geometric interpretation of a classical group cohomology obstruction

Hill, R. O.

doi:10.1090/S0002-9939-1976-0391089-3

A geometric interpretation of a classical group cohomology obstruction
HTML articles powered by AMS MathViewer

by R. O. Hill PDF

Proc. Amer. Math. Soc. 54 (1976), 405-412 Request permission

Abstract:

For a non-Abelian group $G$, we show that the obstruction to the existence of an extension of $G$ by $\Pi$ that induces $\phi :\Pi \to$ Out $G$ is also the $k$-invariant of the classifying space for $K(G,1)$-bundles.

References

M. G. Barratt, V. K. A. M. Gugenheim, and J. C. Moore, On semisimplicial fibre-bundles, Amer. J. Math. 81 (1959), 639–657. MR 111028, DOI 10.2307/2372920
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
Samuel Eilenberg, Homology of spaces with operators. I, Trans. Amer. Math. Soc. 61 (1947), 378–417; errata, 62, 548 (1947). MR 21313, DOI 10.1090/S0002-9947-1947-0021313-4

Cohomology theory in abstract groups

48

8

9

Samuel Gitler, Cohomology operations with local coefficients, Amer. J. Math. 85 (1963), 156–188. MR 158398, DOI 10.2307/2373208
Daniel H. Gottlieb, On fibre spaces and the evaluation map, Ann. of Math. (2) 87 (1968), 42–55. MR 221508, DOI 10.2307/1970593
R. O. Hill Jr., On characteristic classes of groups and bundles of $K(\Pi ,\,1)$’s, Proc. Amer. Math. Soc. 40 (1973), 597–603. MR 319192, DOI 10.1090/S0002-9939-1973-0319192-1
James F. McClendon, Obstruction theory in fiber spaces, Math. Z. 120 (1971), 1–17. MR 296943, DOI 10.1007/BF01109713

Homology

28

J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892

Semi-principal bundles and stable nonorientable obstruction theory

Paul Olum, On mappings into spaces in which certain homotopy groups vanish, Ann. of Math. (2) 57 (1953), 561–574. MR 54250, DOI 10.2307/1969737
Paul Olum, Factorizations and induced homomorphisms, Advances in Math. 3 (1969), 72–100. MR 238314, DOI 10.1016/0001-8708(69)90003-6
Jerrold Siegel, Higher order cohomology operations in local coefficient theory, Amer. J. Math. 89 (1967), 909–931. MR 225319, DOI 10.2307/2373410
Jerrold Siegel, $k$-invariants in local coefficient theory, Proc. Amer. Math. Soc. 29 (1971), 169–174. MR 307224, DOI 10.1090/S0002-9939-1971-0307224-4
C. A. Robinson, Moore-Postnikov systems for non-simple fibrations, Illinois J. Math. 16 (1972), 234–242. MR 298664

Additional Information

Journal: Proc. Amer. Math. Soc. 54 (1976), 405-412
DOI: https://doi.org/10.1090/S0002-9939-1976-0391089-3
MathSciNet review: 0391089