On the topological complexity of Grassmann manifolds (Q2057894)

Let \(X\) be a path connected topological space and \(PX\) the space of paths in \(X\). The topological complexity of \(X\), denoted by \(TC(X)\), is the (reduced) Schwarz genus of the fibration \(\pi:PX\rightarrow X\times X\) given by \(\pi(\omega)=(\omega(0),\omega(1))\). The most notable lower bounds for \(TC(X)\) are the \textit{\(K\)-zero-divisor cup-lengths} of \(X\), denoted by \(\mathrm{zcl}_K(X)\), for various fields \(K\). This is the maximal number of elements of positive degree in \(\mathrm{ker}(\cup:H^*(X;K)\otimes H^*(X;K)\rightarrow H^*(X;K))\) with nonzero product in the algebra \(H^*(X;K)\otimes H^*(X;K)\) (where \(\cup\) is the cohomology cup product). Also, a well-known upper bound for topological complexity of an \(r\)-connected polyhedron \(X\) is \(2\dim(X)/(r+1)\). In this paper the author proves that for any quaternionic flag manifold its topological complexity equals the half of its real dimension. It is shown that the \(\mathbb Q\)-zero-divisor cup-length of such a manifold (a lower bound for the topological complexity) coincides with the above mentioned upper bound. The other class of manifolds considered in the paper are the Grassmann manifolds \(\widetilde G_{n,k}\) of oriented \(k\)-planes in \(\mathbb R^n\). The author presents a complete calculation of \(\mathrm{zcl}_\mathbb Q(\widetilde G_{n,k})\) (for \(3\leq k\leq [n/2]\)), and in the process completes the work of \textit{J. Korbaš} [Topology Appl. 153, No. 15, 2976--2986 (2006; Zbl 1099.55001)] on computing the \(\mathbb Q\)-cup-length of these manifolds. In some low-dimensional cases the \(\mathbb Z_2\)-zero-divisor cup-length of \(\widetilde G_{n,k}\) is calculated, which turns out to be a better lower bound for \(TC(\widetilde G_{n,k})\) than \(\mathrm{zcl}_\mathbb Q(\widetilde G_{n,k})\).