On the cohomology of real Grassmanians
HTML articles powered by AMS MathViewer
- by Howard L. Hiller
- Trans. Amer. Math. Soc. 257 (1980), 521-533
- DOI: https://doi.org/10.1090/S0002-9947-1980-0552272-2
- PDF | 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
Bibliographic 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