Projective product coverings and sequential motion planning algorithms in real projective spaces

TL;DR

This paper characterizes the higher topological complexity of even-dimensional real projective spaces using equivariant maps from Davis' projective product spaces and computes exact values for large s.

Contribution

It generalizes existing work on topological complexity by relating it to equivariant maps and computes exact values for certain cases.

Findings

Higher topological complexity $TC_s$ characterized via equivariant maps.

Exact $TC_s$ values computed for even-dimensional real projective spaces.

Generalization of previous work relating $TC_2$ to immersion dimension.

Abstract

For positive integers $m$ and $s$, let $\mathbf{m}_s$ stand for the $s$-th tuple $(m,\ldots,m)$. We show that, for large enough $s$, the higher topological complexity $TC_s$ of an even dimensional real projective space $RP^m$ is characterized as the smallest positive integer $k=k(m,s)$ for which there is a $(\mathbb{Z}_2)^{s-1}$-equivariant map from Davis' projective product space $P_{\mathbf{m}_s}$ to the $(k+1)$-th join-power $((\mathbb{Z}_2)^{s-1})^{\ast(k+1)}$. This is a (partial) generalization of Farber-Tabachnikov-Yuzvinsky's work relating $TC_2$ to the immersion dimension of real projective spaces. In addition, we compute the exact value of $TC_s(RP^m)$ for $m$ even and $s$ large enough.