Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the cohomology of real Grassmanians
HTML articles powered by AMS MathViewer

by Howard L. Hiller PDF
Trans. Amer. Math. Soc. 257 (1980), 521-533 Request permission

Abstract:

Let ${G_k}({\textbf {R}^{n + k}})$ denote the grassman manifold of k-planes in real $(n + k)$-space and ${w_1} \in {H^1}({G_k}({\textbf {R}^{n + k}}); {\textbf {Z}_2})$ the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height of ${w_1}$ in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.
References
    I. N. Berstein, I. M. Gelfand and S. I. Gelfand, Schubert cells and cohomology of the spaces $G/P$, Russian Math. Surveys (1973), 1-26.
  • Israel Berstein, On the Lusternik-Schnirelmann category of Grassmannians, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 129–134. MR 400212, DOI 10.1017/S0305004100052142
  • I. Berstein and T. Ganea, The category of a map and of a cohomology class, Fund. Math. 50 (1961/62), 265–279. MR 139168, DOI 10.4064/fm-50-3-265-279
  • Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207 (French). MR 51508, DOI 10.2307/1969728
  • Shiing-shen Chern, On the multiplication in the characteristic ring of a sphere bundle, Ann. of Math. (2) 49 (1948), 362–372. MR 24127, DOI 10.2307/1969285
  • Charles Ehresmann, Sur la topologie de certains espaces homogènes, Ann. of Math. (2) 35 (1934), no. 2, 396–443 (French). MR 1503170, DOI 10.2307/1968440
  • Tudor Ganea, Some problems on numerical homotopy invariants, Symposium on Algebraic Topology (Battelle Seattle Res. Center, Seattle Wash., 1971) Lecture Notes in Math., Vol. 249, Springer, Berlin, 1971, pp. 23–30. MR 0339147
  • F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
  • L. Lusternik and L. Schnirelmann, Methodes topologiques dans les problemes variationnels, Hermann, Paris, 1934.
  • V. Oproiu, Some non-embedding theorems for the Grassmann manifolds $G_{2,n}$ and $G_{3,n}$, Proc. Edinburgh Math. Soc. (2) 20 (1976/77), no. 3, 177–185. MR 445530, DOI 10.1017/S0013091500026249
  • H. Schubert, Kalkül der abzählenden Geometrie, Teubner, Leipzig, 1879.
  • Paul A. Schweitzer, Secondary cohomology operations induced by the diagonal mapping, Topology 3 (1965), 337–355. MR 182969, DOI 10.1016/0040-9383(65)90002-9
  • Wilhelm Singhof, On the Lusternik-Schnirelmann category of Lie groups, Math. Z. 145 (1975), no. 2, 111–116. MR 391075, DOI 10.1007/BF01214775
  • Richard P. Stanley, Some combinatorial aspects of the Schubert calculus, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977, pp. 217–251. MR 0465880
  • N. E. Steenrod, Cohomology operations, Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. MR 0145525
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 14M15, 55R40, 57T15
  • Retrieve articles in all journals with MSC: 14M15, 55R40, 57T15
Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 257 (1980), 521-533
  • MSC: Primary 14M15; Secondary 55R40, 57T15
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0552272-2
  • MathSciNet review: 552272