Topological Complexity of Oriented Grassmann Manifolds

Abstract

We study the topological complexity of the Grassmann manifolds (Formula presented) 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 zcl<inf>K</inf> (X), where zcl<inf>K</inf> (X) is the K-zero-divisor cup-length of X. In this paper we examine zcl<inf>ℤ2</inf> (^˜G<inf>n,3</inf>). Some lower and upper bounds for this invariant are obtained for all integers n ≥ 6. For infinitely many of them the exact value of zcl<inf>ℤ2</inf> ((Formula presented)) 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 (Formula presented).