Self-maps of flag manifolds

Glover, Henry H.; Homer, William D.

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

Self-maps of flag manifolds

Authors: Henry H. Glover and William D. Homer
Journal: Trans. Amer. Math. Soc. 267 (1981), 423-434
MSC: Primary 55P62; Secondary 14M17, 57T15
DOI: https://doi.org/10.1090/S0002-9947-1981-0626481-9
MathSciNet review: 626481
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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.

[AM] J. F. Adams and Z. Mahmud, Maps between classifying spaces, Invent. Math. 35 (1976), 1-41. MR 0423352 (54:11331)
[Ba] P. F. Baum, On the cohomology of homogeneous spaces, Topology 7 (1968), 15-38. MR 0219085 (36:2168)
[BG] A. K. Bousfield and V. K. A. M. Gugenheim, On PL de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. No. 179 (1976). MR 0425956 (54:13906)
[B] A. Borel, Topics in the homology theory of fiber bundles, Lecture Notes in Math., vol. 36, Springer-Verlag, Berlin and New York, 1967. MR 0221507 (36:4559)
[BS] A. Borel and J. de Siebenthal, Les sous-groupes fermes de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200-221. MR 0032659 (11:326d)
[Br] S. Brewster, Automorphisms of the cohomology ring of finite Grassmann manifolds, Ph. D. Dissertation, Ohio State Univ., Columbus, 1978.
[EL] J. Ewing and A. Liulevicius, Homotopy rigidity of linear actions on friendly homogeneous spaces, J. Pure Appl. Algebra 18 (1980), 259-267. MR 593617 (82e:57020)
[Fr] E. Friedlander, Maps between localized homogeneous spaces, Topology 16 (1977), 205-216. MR 0501047 (58:18511)
[GH1] H. Glover and W. Homer, Endomorphisms of the cohomology ring of finite Grassmann manifolds (Proc. Northwestern Univ. Conf. Geometric Applications of Homotopy Theory), Lecture Notes in Math., vol. 657, Springer-Verlag, Berlin and New York, 1978, pp. 170-193. MR 513548 (80e:55003)
[GH2] -, Fixed points on flag manifolds (preprint).
[GH3] -, Cohomology endomorphisms of flag manifolds. I, II, 1978, (Preliminary report).
[GHM] H. Glover, W. Homer and G. Mislin, Immersions in manifolds of positive weights, Algebraic Topology (Proc. Conf., Vancouver, 1977), Lecture Notes in Math., vol. 673, Springer-Verlag, Berlin and New York, pp. 88-92. MR 517085 (82c:57017b)
[GHS] H. Glover, W. Homer and R. Stong, Splitting the tangent bundle of projective space, Indiana Univ. Math. J. (to appear). MR 648168 (83f:57016)
[HMR] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, Math. Studies No. 15, North-Holland, Amsterdam, 1975. MR 0478146 (57:17635)
[L1] A. Liulevicius, Homotopy rigidity of linear actions: characters tell all, Bull. Amer. Math. Soc. 84 (1978), 213-221. MR 475124 (81f:57040)
[L2] -, Line bundles, cohomology automorphisms, and homotopy rigidity of linear actions, (Proc. Northwestern Univ. Conf. Geometric Applications of Homotopy Theory), Lecture Notes in Math., vol. 658, Springer-Verlag, Berlin and New York, 1978, pp. 224-233. MR 513578 (84a:57041)
[L3] -, Flag manifolds and homotopy rigidity of linear actions, Algebraic Topology (Proc. Conf., Vancouver, 1977), Lecture Notes in Math., vol. 673, Springer-Verlag, Berlin and New York, 1978, pp. 254-261. MR 517097 (84a:57042)
[O] L. O'Neill, The fixed point property for Grassmann manifolds, Ph.D. Dissertation, Ohio State Univ., Columbus, 1974.
[S] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269-332. MR 0646078 (58:31119)