Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Applications of bundle map theory


Author: Daniel Henry Gottlieb
Journal: Trans. Amer. Math. Soc. 171 (1972), 23-50
MSC: Primary 55F10
DOI: https://doi.org/10.1090/S0002-9947-1972-0309111-X
MathSciNet review: 0309111
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper observes that the space of principal bundle maps into the universal bundle is contractible. This fact is added to I. M. James' Bundle map theory which is slightly generalized here. Then applications yield new results about actions on manifolds, the evaluation map, evaluation subgroups of classifying spaces of topological groups, vector bundle injections, the Wang exact sequence, and $ H$-spaces.


References [Enhancements On Off] (What's this?)

  • [1] Guy Allaud, On the classification of fibre spaces, Math. Z. 92 (1966), 110-125. MR 32 #6462. MR 0189035 (32:6462)
  • [2] A. Borel, Topics in the homology theory of fibre bundles, Lecture Notes in Math., no. 36, Springer-Verlag, Berlin and New York, 1967. MR 36 #4559. MR 0221507 (36:4559)
  • [3] R. F. Brown, On the Lefshetz number and the Euler class, Trans. Amer. Math. Soc. 118 (1965), 174-179. MR 30 #3481. MR 0173268 (30:3481)
  • [4] Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239-281. MR 20 #3542. MR 0097062 (20:3542)
  • [5] James Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [6] E. Fadell, Generalized normal bundles for locally-flat imbeddings, Trans. Amer. Math. Soc. 114 (1965), 488-513. MR 31 #4037. MR 0179795 (31:4037)
  • [7] Daniel H. Gottlieb, On fibre spaces and the evaluation map, Ann. of Math. (2) 87 (1968), 42-55. MR 36 #4560. MR 0221508 (36:4560)
  • [8] -, Evaluation subgroups of homotopy groups, Amer. J. Math. 91 (1969), 729-755. MR 0275424 (43:1181)
  • [9] -, The evaluation map and homology, Michigan Math. J. (to appear). MR 0343261 (49:8005)
  • [10] I. M. James, On the space of bundle maps, Topology 2 (1963), 45-59. MR 33 #6630. MR 0198472 (33:6630)
  • [11] John Milnor, Construction of universal bundles. II, Ann. of Math. (2) 63 (1956), 430-436. MR 17, 1120. MR 0077932 (17:1120a)
  • [12] -, Lectures on characteristic classes, Princeton University, Princeton, N. J., 1957 (mimeographed).
  • [13] A. Phillips, Submersions of open manifolds, Topology 6 (1967), 171-296. MR 34 #8420. MR 0208611 (34:8420)
  • [14] E. Spanier, Infinite symmetric products, function spaces and duality, Ann. of Math. (2) 69 (1959), 142-198; erratum ibid., 733 (see §13). MR 21 #3851. MR 0105106 (21:3851)
  • [15] -, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
  • [16] N. E. Steenrod, Topology of fibre bundles, Princeton Math. Series, vol. 14, Princeton Univ. Press, Princeton, N. J., 1951, 2nd printing, 1957. MR 12, 522. MR 0039258 (12:522b)
  • [17] -, Milgram's classifying space of a topological group, Topology 7 (1968), 349-368. MR 38 #1675. MR 0233353 (38:1675)
  • [18] T. E. Stewart, Lifting group actions in fibre bundles, Ann. of Math. (2) 74 (1961), 192-198. MR 23 #A3798. MR 0126502 (23:A3798)
  • [19] René Thom, L'homologie des espaces fonctionnels, Colloque de Topologie Algébrique (Louvain, 1956), Georges Thone, Liége; Masson, Paris, 1957. MR 19, 669. MR 0089408 (19:669h)
  • [20] S. Weingram, On the incompressibility of certain maps, Ann. of Math (2) 93 (1971), 476-485. MR 0301735 (46:890)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55F10

Retrieve articles in all journals with MSC: 55F10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0309111-X
Keywords: Fibre bundle map, evaluation subgroup, characteristic classes of manifolds, evaluation map, Wang exact sequence, $ H$-spaces
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society