Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds

Tomáš Rusin

Address: Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Mlynská Dolina, 842 48 Bratislava, Slovakia

E-mail: tomas.rusin@fmph.uniba.sk

Abstract: We estimate the characteristic rank of the canonical $k$–plane bundle over the oriented Grassmann manifold $\widetilde{G}_{n,k}$. We then use it to compute uniform upper bounds for the $\mathbb{Z}_2$–cup-length of $\widetilde{G}_{n,k}$ for $n$ belonging to certain intervals.

AMSclassification: primary 57T15; secondary 57R20, 55R25.

Keywords: cup-length, Grassmann manifold, characteristic rank, Stiefel-Whitney class.

DOI: 10.5817/AM2018-5-313