Classifying spaces and homotopy sets of axes of pairings

Ishiguro, Kenshi

doi:10.1090/S0002-9939-96-03497-1

Classifying spaces and homotopy sets of axes of pairings
HTML articles powered by AMS MathViewer

by Kenshi Ishiguro PDF

Proc. Amer. Math. Soc. 124 (1996), 3897-3903 Request permission

Abstract:

We consider the maps between classifying spaces of the form $BK \times BL \rightarrow BG$. The main theorem shows that if the restriction map on $BL$ is a weak epimorphism, then the restriction on $BK$ should factor through the classifying spaces of the center of the compact Lie group $G$. An application implies that $BG$ is an H–space (Hopf space) if and only if $G$ is abelian.

References

Jaume Aguadé and Larry Smith, On the mod$\,p$ torus theorem of John Hubbuck, Math. Z. 191 (1986), no. 2, 325–326. MR 818677, DOI 10.1007/BF01164037
Giora Dula and Daniel H. Gottlieb, Splitting off $H$-spaces and Conner-Raymond splitting theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 2, 321–334. MR 1071426
W. G. Dwyer and G. Mislin, On the homotopy type of the components of $\textrm {map}_*(BS^3, BS^3)$, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 82–89. MR 928824, DOI 10.1007/BFb0083001
W.G. Dwyer and C. Wilkerson, The center of a $p$–compact group, Preprint.
W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 106–119. MR 928826, DOI 10.1007/BFb0083003
Yutaka Hemmi, The projective plane of an $H$-pairing, J. Pure Appl. Algebra 75 (1991), no. 3, 277–296. MR 1137841, DOI 10.1016/0022-4049(91)90137-Q
Kenshi Ishiguro, A $p$-local splitting of $B\textrm {U}(n)$, Proc. Amer. Math. Soc. 95 (1985), no. 2, 307–311. MR 801344, DOI 10.1090/S0002-9939-1985-0801344-3
Kenshi Ishiguro and Dietrich Notbohm, Fibrations of classifying spaces, Trans. Amer. Math. Soc. 343 (1994), no. 1, 391–415. MR 1231336, DOI 10.1090/S0002-9947-1994-1231336-4
Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
S. Jackowski, J.E. McClure and B. Oliver, Self homotopy equivalences of classifying spaces of compact connected Lie groups, Preprint.
R. K. Lashof, J. P. May, and G. B. Segal, Equivariant bundles with abelian structural group, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 167–176. MR 711050, DOI 10.1090/conm/019/711050
James P. Lin, A cohomological proof of the torus theorem, Math. Z. 190 (1985), no. 4, 469–476. MR 808914, DOI 10.1007/BF01214746
Guido Mislin and Jacques Thévenaz, The $Z^*$-theorem for compact Lie groups, Math. Ann. 291 (1991), no. 1, 103–111. MR 1125010, DOI 10.1007/BF01445193
Dietrich Notbohm, Maps between classifying spaces, Math. Z. 207 (1991), no. 1, 153–168. MR 1106820, DOI 10.1007/BF02571382
Dietrich Notbohm, Maps between classifying spaces and applications, J. Pure Appl. Algebra 89 (1993), no. 3, 273–294. MR 1242722, DOI 10.1016/0022-4049(93)90057-Z
Nobuyuki Oda, The homotopy set of the axes of pairings, Canad. J. Math. 42 (1990), no. 5, 856–868. MR 1080999, DOI 10.4153/CJM-1990-044-3
Nobuyuki Oda, Localization of the homotopy set of the axes of pairings, Adams Memorial Symposium on Algebraic Topology, 1 (Manchester, 1990) London Math. Soc. Lecture Note Ser., vol. 175, Cambridge Univ. Press, Cambridge, 1992, pp. 163–177. MR 1170577, DOI 10.1017/CBO9780511526305.012