Self-maps of flag manifolds

Glover, Henry H.; Homer, William D.

doi:10.1090/S0002-9947-1981-0626481-9

Self-maps of flag manifolds
HTML articles powered by AMS MathViewer

by Henry H. Glover and William D. Homer PDF

Trans. Amer. Math. Soc. 267 (1981), 423-434 Request permission

Abstract:

Rationally, a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology. Two algebraic results for cohomology endomorphisms then serve (a) to determine those flag manifolds which have (nontrivial) self-maps that factor through a complex projective space, and (b) for a special class of flag manifolds, to classify the self-maps of their rationalizations up to homotopy.

References

J. F. Adams and Z. Mahmud, Maps between classifying spaces, Inv. Math. 35 (1976), 1–41. MR 0423352, DOI 10.1007/BF01390132
Paul F. Baum, On the cohomology of homogeneous spaces, Topology 7 (1968), 15–38. MR 219085, DOI 10.1016/0040-9383(86)90012-1
A. K. Bousfield and V. K. A. M. Gugenheim, On $\textrm {PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 425956, DOI 10.1090/memo/0179
Armand Borel, Topics in the homology theory of fibre bundles, Lecture Notes in Mathematics, No. 36, Springer-Verlag, Berlin-New York, 1967. Lectures given at the University of Chicago, 1954; Notes by Edward Halpern. MR 0221507
A. Borel and J. De Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200–221 (French). MR 32659, DOI 10.1007/BF02565599

Automorphisms of the cohomology ring of finite Grassmann manifolds

John Ewing and Arunas Liulevicius, Homotopy rigidity of linear actions on homogeneous spaces, J. Pure Appl. Algebra 18 (1980), no. 3, 259–267. MR 593617, DOI 10.1016/0022-4049(80)90003-1
Eric M. Friedlander, Maps between localized homogeneous spaces, Topology 16 (1977), no. 3, 205–216. MR 501047, DOI 10.1016/0040-9383(77)90001-5
Henry Glover and Bill Homer, Endomorphisms of the cohomology ring of finite Grassmann manifolds, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 657, Springer, Berlin, 1978, pp. 170–193. MR 513548

Fixed points on flag manifolds

Cohomology endomorphisms of flag manifolds

Henry Glover and Bill Homer, Immersing manifolds and $2$-equivalence, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 657, Springer, Berlin, 1978, pp. 194–197. MR 513549
Henry H. Glover, William D. Homer, and Robert E. Stong, Splitting the tangent bundle of projective space, Indiana Univ. Math. J. 31 (1982), no. 2, 161–166. MR 648168, DOI 10.1512/iumj.1982.31.31015
Peter Hilton, Guido Mislin, and Joe Roitberg, Localization of nilpotent groups and spaces, North-Holland Mathematics Studies, No. 15, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0478146
Arunas Liulevicius, Homotopy rigidity of linear actions: characters tell all, Bull. Amer. Math. Soc. 84 (1978), no. 2, 213–221. MR 475124, DOI 10.1090/S0002-9904-1978-14457-7
Arunas Liulevicius, Line bundles, cohomology automorphisms, and homotopy rigidity of linear actions, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 658, Springer, Berlin-New York, 1978, pp. 224–233. MR 513578
Arunas Liulevicius, Flag manifolds and homotopy rigidity of linear actions, Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977) Lecture Notes in Math., vol. 673, Springer, Berlin-New York, 1978, pp. 254–261. MR 517097

The fixed point property for Grassmann manifolds

Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078