On the cohomology of oriented Grassmann manifolds
The \(\mathbb{Z}_2\)-cohomology \(H^*(G_{n, k})\) of the Grassmann manifold \(G_{n,k}= O(n)/(O(k) \times O(n - k))\) \((k\leq n-k)\) is isomorphic to the quotient algebra of \(\mathbb{Z}_2[w_1, \dots, w_k]\) by the ideal generated by \(k\) homogeneous components of \((1+w_1+ \cdots +w_k)^{-1}\) in dimensions \(n-k+1, \dots, n\) where \(w_i\) denotes the \(i\)th Stiefel-Whitney class of the canonical \(k\)-plane bundle \(\gamma_{n, k}\) over \(G_{n, k}\). Let \(w_1 : H^j (G_{n, k})\to H^{j+1}(G_{n, k})\) be the homomorphism induced by the cup-product with \(w_1\). Let \(\tilde{G}_{n, k}=\mathrm{SO}(n)/(\mathrm{SO}(k)\times\mathrm{SO}(n-k))\) and \(p : \tilde{G}_{n, k}\to {G}_{n, k}\) be the canonical covering projection. Then we know that \(\text{Ker}(w_1) \cong H^*(\tilde{G}_{n, k})/\text{Im}(p^*)\). Besides, since \(p^*\gamma_{n, k}=\tilde{\gamma}_{n, k}\), we have \(p^*w_i=\tilde{w_i}\) where \(\tilde{\gamma}_{n, k}\) denotes the canonical \(k\)-plane bundle over \(\tilde{G}_{n, k}\) and \(\tilde{w}_i\) is its \(i\)th Stiefel-Whitney class. So we see that \(\text{Im}(p^*)\) consists only of elements which can be expressed as polynomials in \(\tilde{w}_1, \dots, \tilde{w}_k\). This tells us that analyzing \(\text{Ker}(w_1)\) we can get more detailed information and understanding about \(H^*(\tilde{G}_{n, k})\). In this paper the authors develop new tools for studying this kernel. Let us write \(\alpha_j(\tilde{G}_{n, k})\) for the dimension of \(\text{Ker}(w_1) \subset H^j (G_{n, k})\). By \(\text{charrank}(\tilde{\gamma}_{n, k})\) we mean the greatest integer \(q\) such that \(\alpha_i(\tilde{G}_{n, k})=0\) \((0\leq i\leq q)\) and call it the characteristic rank of \(\tilde{\gamma}_{n, k}\). The authors examine this invariant using an additive basis (called here standard basis) of \(H^j(G_{n, k})\) introduced by \textit{J. W. Milnor} and \textit{J. D. Stasheff} [Characteristic classes. Princeton, N.J.: Princeton University Press and University of Tokyo Press (1974; Zbl 0298.57008)]. For example, it is proved that \textit{if \(1\leq x \leq n-k\), then all the elements in the standard basis for \(H^{n-k-x}(G_{n, k})\) are regular, thus we have \(\alpha_{n-k-x}(\tilde{G}_{n, k})=0\)} (definition omitted). This recovers the result due to the first author [Topology Appl. 153, No. 15, 2976--2986 (2006; Zbl 1099.55001)], which says that if \(w_1\) is injective for \(0\leq j \leq n-k-1\), then \(\text{charrank}(\tilde{\gamma}_{n, k})\geq n-k-1\). For one more example, we have that \textit{if \(x \leq n-k-1\) and there are \(t\) linearly independent elements in the set \(N_x(G_{n, k})\), then \(\alpha_{n-k+x}(\tilde{G}_{n, k})\leq p(\{1, 2, \dots, k-1\}, x)-t\)}. With these results, the authors obtain new results on bounds of \(\text{charrank}(\tilde{\gamma}_{n, k})\). Applying these new ones, the authors succeed in determining the exact values of \(\text{charrank}(\tilde{\gamma}_{n, 3})\) and \(\text{cup}(\tilde{\gamma}_{n, 3})\) for infinitely many \(n\) of the form \(2^q+1\), \(2^q+2\) and \(2^q+3\).
scientific article; zbMATH DE number 6676562