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

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 G_k(ℝ^n) of k -dimensional subspaces in ℝ^n 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 TC_h(G_2(ℝ^2^s+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(G_2(ℝ^n)) for h⩾ 2^s+1-1 (where 2^s更多