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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
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, Inv. Math.
35 (1976), 1–41. MR 0423352
(54 #11331)
- [Ba]
Paul
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. 8
(1976), no. 179, ix+94. MR 0425956
(54 #13906)
- [B]
Armand
Borel, Topics in the homology theory of fibre bundles,
Lectures given at the University of Chicago, vol. 1954,
Springer-Verlag, Berlin-New York, 1967. MR 0221507
(36 #4559)
- [BS]
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 0032659
(11,326d)
- [Br]
S. Brewster, Automorphisms of the cohomology ring of finite Grassmann manifolds, Ph. D. Dissertation, Ohio State Univ., Columbus, 1978.
- [EL]
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 (82e:57020), http://dx.doi.org/10.1016/0022-4049(80)90003-1
- [Fr]
Eric
M. Friedlander, Maps between localized homogeneous spaces,
Topology 16 (1977), no. 3, 205–216. MR 0501047
(58 #18511)
- [GH1]
Henry
Glover and Bill
Homer, Endomorphisms of the cohomology ring of finite Grassmann
manifolds, Geometric applications of homotopy theory (Proc. Conf.,
Evanston, Ill., 1977), I, Lecture Notes in Math., vol. 657, Springer,
Berlin, 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]
Henry
Glover, Bill
Homer, and Guido
Mislin, Immersions in manifolds of positive weights, Algebraic
topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977)
Lecture Notes in Math., vol. 673, Springer, Berlin, 1978,
pp. 88–92. MR 517085
(82c:57017b)
- [GHS]
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 (83f:57016), http://dx.doi.org/10.1512/iumj.1982.31.31015
- [HMR]
Peter
Hilton, Guido
Mislin, and Joe
Roitberg, Localization of nilpotent groups and spaces,
North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier
Publishing Co., Inc., New York, 1975. North-Holland Mathematics Studies,
No. 15; Notas de Matemática, No. 55. [Notes on Mathematics, No. 55].
MR
0478146 (57 #17635)
- [L1]
Arunas
Liulevicius, Homotopy rigidity of linear actions:
characters tell all, Bull. Amer. Math. Soc.
84 (1978), no. 2,
213–221. MR
475124 (81f:57040), http://dx.doi.org/10.1090/S0002-9904-1978-14457-7
- [L2]
Arunas
Liulevicius, Line bundles, cohomology automorphisms, and homotopy
rigidity of linear actions, Geometric applications of homotopy theory
(Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math.,
vol. 658, Springer, Berlin-New York, 1978, pp. 224–233. MR 513578
(84a:57041)
- [L3]
Arunas
Liulevicius, Flag manifolds and homotopy rigidity of linear
actions, Lecture Notes in Math., vol. 673, Springer, Berlin-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]
Dennis
Sullivan, Infinitesimal computations in topology, Inst. Hautes
Études Sci. Publ. Math. 47 (1977), 269–331
(1978). MR
0646078 (58 #31119)
- [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)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1981-0626481-9
PII:
S 0002-9947(1981)0626481-9
Keywords:
Flag manifold,
homogeneous space,
rational homotopy,
self-map
Article copyright:
© Copyright 1981
American Mathematical Society
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