Higher Topological Complexities of Real Grassmannians and Semi-complete Real Flag Manifolds

Abstract

Topological complexity and its higher analogs naturally appear in motion planning in robotics. In this paper, we consider the problem of finding higher topological complexities (TC h) of the real Grassmann manifold Gk(Rn) of k-dimensional subspaces in Rn and semi-complete real flag manifold F(1 k, m) (here 1 k means that 1 appears k times). We use cohomology methods to prove some general bounds on the h-th zero-divisor cup-length (zcl h), and then use them to obtain the exact values of TCh(G2(R2s+1)) for h⩾ 2 s+1- 1 , and TC h(F(1 k, 2 s- k+ 1)) for h⩾ k⩾ 3. Additionally, we determine zcl h(G2(Rn)) for h⩾ 2 s+1- 1 (where 2 s< n⩽ 2 s+1), and resolve two questions from González et al. (Homol Homotopy Appl 82(2):359–375, 2016).