On cup products in some manifolds (Q1971664)

The authors consider smooth fiber bundles \(p:E \to B\) whose fiber is \(G_{n,k}\), the Grassmann manifold of \(k\)-dimensional linear subspaces of \(\mathbb{R}^n\). As \(G_{n,k}\) and \(G_{n,n-k}\) are diffeomorphic the authors restrict themselves to the case where \(2 \leq 2k \leq n\); moreover they assume \(E\) to be a closed connected manifold. In their main theorem they give various lower bounds for \(\text{cup}(E; \mathbb{Z}/2)\), the cup-length of \(E\) with respect to \(\mathbb{Z}/2\)-cohomology. These lower bounds typically depend on arithmetic properties of \(n\) and \(k\). E.g. the authors show that if each power of \(2\) dividing \(n\) also divides \(k\) then \(\text{cup}(E; \mathbb{Z}/2)\geq \text{cup} (B;\mathbb{Z}/2)\), or that if \(n\) is odd then \(\text{cup}(E;\mathbb{Z}/2) \geq \text{cup} (G_{n,k};\mathbb{Z}/2) + \text{cup}(B;\mathbb{Z}/2)\). Using results of \textit{H. L. Hiller} [Trans. Am. Math. Soc. 257, 521-533 (1980; Zbl 0462.57021)] and \textit{R. E. Stong} [Topology Appl. 13, 103-113 (1982; Zbl 0469.55005)] on the \(\mathbb{Z}/2\)-cohomology ring of \(G_{n,k}\) the lower bounds are made more explicit for certain \(n\) and \(k\).