Rusin, Tomáš Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds. (English) Zbl 1474.57011 Arch. Math., Brno 54, No. 5, 313-329 (2018). J. Korbaš in [Bull. Belg. Math. Soc. - Simon Stevin 17, No. 1, 69–81 (2010; Zbl 1194.57032)] presented the value of the cup-length cup\(\left(\tilde{G}_{2^{t}-1,3}\right)=2^{t}-3\) of the oriented Grasmann manifold, introducing the notion of the characteristic rank, which was generalized by A. C. Naolekar and A. S. Thakur [Math. Slovaca 64, No. 6, 1525–1540 (2014; Zbl 1349.57008)] to the characteristic rank of a vector bundle.In this paper, the author presents a theorem providing a lower bound for the characteristic rank of the canonical \(k\)-plane bundle \(\tilde{\gamma}_{n,k}\) over \(\tilde{G}_{n,k}\).Also, a theorem is given providing an uniform upper bound of cup\(\left(\tilde{G}_{{n}_{k,t}}+a,k\right)\) where \[n_{k,t}=k\cdot 2^{t-1}-\frac{2^{t}-x_{k,t}}{k-1}-1\] and \(x_{k,t}\) is the smallest positive integer such that \(2^{t}-x_{k,t}\) is divisible by \(k-1\). Reviewer: Alice Kimie Miwa Libardi (São Paulo) Cited in 1 Document MSC: 57T15 Homology and cohomology of homogeneous spaces of Lie groups 57R20 Characteristic classes and numbers in differential topology 55R25 Sphere bundles and vector bundles in algebraic topology Keywords:cup-length; Grassmann manifold; characteristic rank; Stiefel-Whitney class Citations:Zbl 1194.57032; Zbl 1349.57008 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Borel, A., La cohomologie mod \(2\) de certains espaces homogènes, Comment. Math. Helv. 27 (1953), 165-197 · Zbl 0052.40301 · doi:10.1007/BF02564561 [2] Fukaya, T., Gröbner bases of oriented Grassmann manifolds, Homology, Homotopy Appl. 10 (2008), 195-209 · Zbl 1156.57029 · doi:10.4310/HHA.2008.v10.n2.a10 [3] Jaworowski, J., An additive basis for the cohomology of real Grassmannians, Lecture Notes in Math., vol. 1474, Springer, Berlin, 1991, Algebraic topology Poznań 1989, 231-234 · Zbl 0733.57025 · doi:10.1007/BFb0084749 [4] Korbaš, J., The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 69-81 · Zbl 1194.57032 [5] Korbaš, J., The characteristic rank and cup-length in oriented Grassmann manifolds, Osaka J. Math. 52 (2015), 1163-1172 · Zbl 1333.57040 [6] Korbaš, J.; Rusin, T., A note on the \(\mathbb{Z}_2\)-cohomology algebra of oriented Grassmann manifolds, Rend. Circ. Mat. Palermo (2) 65 (2016), 507-517 · Zbl 1357.57065 · doi:10.1007/s12215-016-0249-7 [7] Korbaš, J.; Rusin, T., On the cohomology of oriented Grassmann manifolds, Homology, Homotopy, Appl. 18 (2) (2016), 71-84 · Zbl 1357.57060 · doi:10.4310/HHA.2016.v18.n2.a4 [8] Milnor, J.; Stasheff, J., Characteristic Classes, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974 · Zbl 0298.57008 [9] Naolekar, A. C.; Thakur, A. S., Note on the characteristic rank of vector bundles, Math. Slovaca 64 (2014), 1525-1540 · Zbl 1349.57008 · doi:10.2478/s12175-014-0289-4 [10] Petrović, Z. Z.; Prvulović, B. I.; Radovanović, M., Characteristic rank of canonical vector bundles over oriented Grassmann manifolds \(\widetilde{G}_{n,3}\), Topology Appl. 230 (2017), 114-121 · Zbl 1376.57029 [11] Rusin, T., A note on the characteristic rank of oriented Grassmann manifolds, Topology Appl. 216 (2017), 48-58 · Zbl 1357.57062 · doi:10.1016/j.topol.2016.11.009 [12] Stong, R. E., Semicharacteristics of oriented Grassmannians, J. Pure Appl. Algebra 33 (1984), 97-103 · Zbl 0573.57007 · doi:10.1016/0022-4049(84)90029-X This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.