Gr?bner bases of oriented Grassmann manifolds

摘要

For $n=2^{m+1}-4$, $mgeq 2$, we determine the cup-length of $H^*(widetilde{G}_{n,3}; mathbb{Z}/2)$ by finding a Gr?bner basis associated with a certain subring, where $widetilde{G}_{n,3}$ is the oriented Grassmann manifold $extit{SO}(n+3)/extit{SO}(n) imes extit{SO}(3)$. As an application, we provide not only a lower but also an upper bound for the LS-category of $widetilde{G}_{n,3}$. We also study the immersion problem of $widetilde{G}_{n,3}$.