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

Abstract

For positive integers $m$ and $s$, let ${\mathbf{m}}_s$ stand for the $s$-th tuple $(m,\dots ,m)$. We show that, for large enough $s$, the higher topological complexity $T{C}_s$ of an even dimensional real projective space $R{P}^m$ 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 $T{C}_2$ to the immersion dimension of real projective spaces. In addition, we compute the exact value of $T{C}_s(R{P}^m)$ for $m$ even and $s$ large enough.

arXiv comment11 pages

References in corpus (1)

• The stability of the higher topological complexity of real projective spaces: an approach to their immersion dimension

Cited by in corpus (1)

• The stability of the higher topological complexity of real projective spaces: an approach to their immersion dimension