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 S. Eilenberg and S. Mac Lane, Cohomology theory in abstract groups. I, II, Ann. of Math. (2) 48 (1947), 51-78, 326-341. MR 8, 367; 9, 7.
- 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 S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #122.
- 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 F. Nussbaum, Thesis, Northwestern University, 1970. —, Semi-principal bundles and stable nonorientable obstruction theory (to appear).
- 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
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 405-412
- DOI: https://doi.org/10.1090/S0002-9939-1976-0391089-3
- MathSciNet review: 0391089