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

The s-th higher topological complexity TC_s ( X ) 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 = RP^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 ( RP^m ) ≤ δ s ( m ) for s ≥ r ( m ), where δ_s (m) = ( 0 , 1 , 0 ) s(m)=(0,1,0) for m = ( 0 , 1 , 2 ) mod 4. Such an estimation for TC_s ( RP^m ) TC_s(RP^m) appears to be closely related to the determination of the Euclidean immersion dimension of RP^m . We illustrate the phenomenon in the case m = 3 .2 a. In addition, we show that, for large enough s and even m, TC s ( RP^m ) TC_s 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_s to the ( t + 1 ) (t+1)-st join-power ( (2) s-1 )( t + 1 ) ((Z₂)^s-1(t+1) . This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating TC₂ to the immersion dimension of real projective spaces.