Topological complexity of real Grassmannians

Abstract

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds G k (ℝ n ) to compute zero-divisor cup-length and estimate topological complexity of motion planning for k-linear subspaces in ℝ n . In addition, we obtain results about monotonicity of Lusternik–Schnirelmann category and topological complexity of G k (ℝ n ) as a function of n.

On the topological complexity and zero-divisor cup-length of real Grassmannians

“…We will compare our results with the following upper bound from [9] (this result is a consequence of a general result from [3, theorem 1]).…”

“…In this paper we reconsider this problem, and as an outcome correct and improve several results from [9]. As in [9], we use the cohomology method to obtain our results.…”

“…Unfortunately, there is a problem with the proof of the main lemma of that paper (lemma 4.4) and the consequential results on the topological complexity (theorems 4.5, 4.8 and 4.12); see [10]. In this paper we reconsider this problem, and as an outcome correct and improve several results from [9]. As in [9], we use the cohomology method to obtain our results.…”

“…In both cases x ⊗ y + y ⊗ x = 0, which implies p = 0. Also, we recall some results from [9] that will be used in our calculations.…”

“…Throughout the paper we use the same notation as in this section. Finally, let us say a few words on lemma 4.4 from [9] and our strategy that bypasses the application of this lemma. In lemma 4.4 from [9] the author assumes that u 1 , .…”