Gröbner bases of oriented Grassmann manifolds

For $n=2^{m+1}-4$, $m\geq 2$, we determine the cup-length of $H^*(\widetilde{G}_{n,3}; \mathbb{Z}/2)$ by finding a Gröbner basis associated with a certain subring, where $\widetilde{G}_{n,3}$ is the oriented Grassmann manifold $\textit{SO}(n+3)/\textit{SO}(n) \times \textit{SO}(3)$. 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 $\widetilde{G}_{n,3}$.

Keywords

cup-length, LS-category, Gröbner bases, immersion

2010 Mathematics Subject Classification

Primary 55M30. Secondary 13P10, 57T15.

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Published 1 January 2008