A note on the cohomology ring of the oriented Grassmann manifolds $\widetilde{G}_{n,4}$

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 use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold $\widetilde{G}_{n,4}$ to compute the generators of the $\mathbb{Z}_2$–cohomology groups $H^j(\widetilde{G}_{n,4})$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of $\widetilde{G}_{n,3}$ we conjecture some predictions.

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

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

DOI: 10.5817/AM2019-5-319