Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension
Natalia Cadavid-Aguilar, Jesús González, Darwin Gutiérrez, Aldo Guzmán-Sáenz, Adriana Lara · Forum Mathematicum · 2017
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