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Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds. (English) Zbl 1474.57011

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 \(k-1\).

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
Full Text: DOI

References:

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