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

Abstract

Abstract The s-th higher topological complexity TC s ⁡ ( X ) {\operatorname{TC}_{s}(X)} of a space X can 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 = ℝ ⁢ P m {X=\operatorname{\mathbb{R}P}^{m}} , the real projective space of dimension m. In particular, we describe a number r ⁢ ( m ) {r(m)} , which depends on the structure of zeros and ones in the binary expansion of m, and with the property that 0 ≤ s ⁢ m - TC s ⁡ ( ℝ ⁢ P m ) ≤ δ s ⁢ ( m ) {0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)} for s ≥ r ⁢ ( m ) {s\geq r(m)} , where δ s ⁢ ( m ) = ( 0 , 1 , 0 ) {\delta_{s}(m)=(0,1,0)} for m ≡ ( 0 , 1 , 2 ) mod 4 {m\equiv(0,1,2)\bmod 4} . Such an estimation for TC s ⁡ ( ℝ ⁢ P m ) {\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} appears to be closely related to the determination of the Euclidean immersion dimension of ℝ ⁢ P m {\operatorname{\mathbb{R}P}^{m}} . We illustrate the phenomenon in the case m = 3 ⋅ 2 a {m=3\cdot 2^{a}} . In addition, we show that, for large enough s and even m, TC s ⁡ ( ℝ ⁢ P m ) {\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})} is characterized as the smallest positive integer t = t ⁢ ( m , s ) {t=t(m,s)} for which there is a suitable equivariant map from Davis’ projective product space P 𝐦 s {\mathrm{P}_{\mathbf{m}_{s}}} to the ( t + 1 ) {(t+1)} -st join-power ( ( ℤ 2 ) s - 1 ) ∗ ( t + 1 ) {((\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 TC 2 {\operatorname{TC}_{2}} to the immersion dimension of real projective spaces.