Gröbner bases of oriented Grassmann manifolds (Q951137)

Let \(\widetilde{G}_{n,3} = SO(n+3)/SO(n) \times SO(3)\) denote the oriented Grassmann manifold. The author studies \(H^*(\widetilde{G}_{n,3};\mathbb{Z}/2)\) focusing on the calculation of the cup-length. As in \textit{J. Korbaš} [Topology Appl. 153, No.~15, 2976--2986 (2006; Zbl 1099.55001)], the approach is via the double covering \(p_n \colon \widetilde{G}_{n,3} \to G_{n,3} \) where \(G_{n,3}\) is the oriented Grassmanian. Classical results imply \(\mathrm{Im}p_n^* = \mathbb{Z}/2[ w_2, w_3]/J\) where the \(w_i\) correspond to Stiefel-Whitney classes and \(J\) is an ideal. Further, \( \mathrm{cup}_{\mathbb{Z}/2}(\widetilde{G}_{n, 3}) = \mathrm{cup}(\mathrm{Im}p_n^*).\) The author identifies a Gröbner basis for the ideal \(J\) which facilitates the calculations of cup-length. When \(n =2^{m+1}-4\) for \(m \geq 2,\) the author's results are quite explicit. In this case, \[ \mathrm{cup}_{\mathbb{Z}/2}(\widetilde{G}_{n, 3}) = n+1. \] This result, in turn, gives an estimate for the L.S. category: \[ n+1 \leq \mathrm{cat}(\widetilde{G}_{n, 3}) < \tfrac{3}{2}n \] and \(\mathrm{cat}(\widetilde{G}_{4, 3}) = 5.\) Finally, the author applies the cup-length result above to study the immersion problem for \(\widetilde{G}_{n, 3}\): When \(n =2^{m+1}-4\) for \(m \geq 3\), \(\widetilde{G}_{n, 3}\) immerses in \(\mathbb{R}^{6n-3}\) but not in \(\mathbb{R}^{3n+8}.\) Further, \(G_{4,3}\) immerses in \(\mathbb{R}^{24}\) but not in \(\mathbb{R}^{17}\)