Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension

Abstract

Thes-th higher topological complexity{\operatorname{TC}_{s}(X)}of a spaceXcan be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when{X=\operatorname{\mathbb{R}P}^{m}}, the real projective space of dimensionm. In particular, we describe a number{r(m)}, which depends on the structure of zeros and ones in the binary expansion ofm, and with the property that{0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)}for{s\geq r(m)}, where{\delta_{s}(m)=(0,1,0)}for{m\equiv(0,1,2)\bmod 4}. Such an estimation for{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}appears to be closely related to the determination of the Euclidean immersion dimension of{\operatorname{\mathbb{R}P}^{m}}. We illustrate the phenomenon in the case{m=3\cdot 2^{a}}. In addition, we show that, for large enoughsand evenm,{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}is characterized as the smallest positive integer{t=t(m,s)}for which there is a suitable equivariant map from Davis’ projective product space{\mathrm{P}_{\mathbf{m}_{s}}}to the{(t+1)}-st join-power{((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating{\operatorname{TC}_{2}}to the immersion dimension of real projective spaces.

Bounds for higher topological complexity of real projective space implied by BP

“…Jesus González ( [2]) has particular interest in estimates for TC k pP 3¨2 e q. We shall prove the interesting fact that our Theorems 1.1 and 1.4 improve significantly on the cohomological lower bound for TC 3 pP 3¨2 e q, but not for TC k pP 3¨2 e q when k ą 3.…”

“…In this section, we prove Theorems 1.1 and 1.4. The first step, Theorem 2.1, follows suggestions of Jesus González, and is similar to work in [2]. We are very grateful to González for these suggestions.…”

“…Recall from [2] or [6] that zcl k pP n q is the lower bound for TC k pP n q implied by mod-2 cohomology. It is an analogue of Theorem 2.1, except that classes are in grading 1 rather than grading 2.…”

A lower bound for higher topological complexity of real projective space

“…In this section, we examine some special cases of our results (in Propositions 3.1 and 3.5) and make comparisons with some work in [1]. The numbers z 3 pnq " zcl 3 pP n q are 1 less than a sequence which was listed by the author as A290649 at [3] in August 2017.…”

“…In [1,Thm 5.7], it is shown that our g k pnq in Theorem 1.6 is a decreasing function of k, and achieves a stable value of 2 νpn`1q´1 for sufficiently large k. They defined spnq to be the minimal value of k such that g k pnq " 2 νpn`1q´1 . We obtain a formula for the precise value of spnq in our next result.…”