A note on the characteristic rank of oriented Grassmann manifolds
Let \(\tilde{\gamma}_{n,3}\) denote the canonical 3-plane bundle over the oriented Grassmann manifold \(\tilde{G}_{n, 3}=\mathrm{SO}(n)/(\mathrm{SO}(3)\times\mathrm{SO}(n-3))\). By the characteristic rank of \(\tilde{\gamma}_{n,3}\) is meant the largest integer \(q\) such that each element of \(H^j(\tilde{G}_{n, 3}; \mathbb{Z}_2)\) with \(j \leq q\) can be expressed as a polynomial in the Stiefel-Whitney classes \(w_i(\tilde{\gamma}_{n,3})\) of \(\tilde{\gamma}_{n,3}\). We denote it by \(\text{charrank}(\tilde{\gamma}_{n,3})\). Then the main result of this paper states that if \(t\geq 3\) and \(2^t+2^{t-1} \leq n \leq 2^{t+1}-4\), then \(\text{charrank}(\tilde{\gamma}_{n,3})\geq 2^{t+1}-5\). With the help of this result the author determines the \(\mathbb{Z}_2\)-cup-length of \(\tilde{G}_{2^t+2^{t-1}+i, 3}\) for \(i=1, 2\); in fact it is proved that \(\text{cup}(\tilde{G}_{2^t+2^{t-1}+1, 3})=2^t+2^{t-2}\) and \(\text{cup}(\tilde{G}_{2^t+2^{t-1}+2, 3})=2^t+2^{t-2}+1\). This allows us to verify the corresponding claims of \textit{T. Fukaya}'s conjecture [Homology Homotopy Appl. 10, No. 2, 195--209 (2008; Zbl 1156.57029)]. The proof of the main result is based on the result due to \textit{J. Korbaš} and the author concerning the cohomology of oriented Grassmann manifolds [Homology Homotopy Appl. 18, No. 2, 71--84 (2016; Zbl 1357.57060)]. In accordance with their approach it is shown that \(w_1 : H^{n-3+x}(G_{n, 3}; \mathbb{Z}_2)\to H^{n-2+x}(G_{n, 3}; \mathbb{Z}_2)\) is injective for all \(x\) such that \(0 \leq x \leq 2^{t+1}-n-2\) where \(w_1\) is defined by \(w_1(x)=w_1(\gamma_{n, 3})\smile x\) (cup product), \(\gamma_{n, 3}\) denoting the canonical 3-plane bundle over \(G_{n, 3}\). This ensures that the desired inequality holds.
scientific article; zbMATH DE number 6668171
scientific article