Topological complexity of oriented Grassmann manifolds

We study the topological complexity of the Grassmann manifolds Gn,3 of oriented $3$-dimensional vector subspaces in Rⁿ. By a result of Farber, for any field K, the topological complexity of a space X is greater than K(X), where K(X) is the K-zero-divisor cup-length of X. In this paper we examine Z₂( Gn,3). Some lower and upper bounds for this invariant are obtained for all integers n≥6. For infinitely many of them the exact value of Z₂( Gn,3) is computed, and in the rest of the cases these bounds differ by 1. We thus establish lower bounds for the topological complexity of Grassmannians Gn,3.