On integral cohomology algebra of some oriented Grassmann manifolds (Q6186997)

The oriented Grassmann manifold \(\tilde{G}_{n,k}\) is defined as a space of oriented \(k\)-dimensional subspaces of \(\mathbb R^n\). Recently, much work has been done in understanding the \(\mathbb Z_2\) cohomology algebra of oriented Grassmann manifolds. For \(k = 2\) it was determined in [\textit{S. Basu} and \textit{P. Chakraborty}, J. Homotopy Relat. Struct. 15, No. 1, 27--60 (2020; Zbl 1441.57027)], and in [\textit{J. Korbaš} and \textit{T. Rusin}, Rend. Circ. Mat. Palermo (2) 65, No. 3, 507--517 (2016; Zbl 1357.57065)] the authors obtain an almost complete description of this algebra for \(k = 3\) and \(n\) close to a power of two. On the other hand, it seems that very little is known about the integral cohomology of \(\tilde{G}_{n,k}\), especially about the algebra structure. In their recent work [Indag. Math., New Ser. 32, No. 3, 579--597 (2021; Zbl 1467.57022)], \textit{M. Kalafat} and \textit{E. Yalçınkaya} have managed to describe the integral cohomology algebra of \(\tilde{G}_{6,3}\) by using the special Lagrangian locus of \(\tilde{G}_{6,3}\). \par In this paper, using the Leray-Serre spectral sequence, the author gives the complete structure of the integral cohomology algebra of \(\tilde{G}_{n,3}\) for \(n = 8\) and \(n = 10\). She also gives a few general facts about the integral cohomology algebra of \(\tilde{G}_{n,3}\).