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The  s -th higher topological complexityTC  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 whenX  =    ℝ  ⁢  P   m{X=\operatorname{\mathbb{R}P}^{m}}   , the real projective space of dimension  m . In particular, we describe a numberr  ⁢   (  m  ){r(m)}   , which depends on the structure of zeros and ones in the binary expansion of  m , and with the property that0  ≤    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)}    fors  ≥   r  ⁢   (  m  ){s\geq r(m)}   , whereδ  s   ⁢   (  m  )    =   (  0  ,  1  ,  0  ){\delta_{s}(m)=(0,1,0)}    form  ≡    (  0  ,  1  ,  2  )   mod  4{m\equiv(0,1,2)\bmod 4}   . Such an estimation forTC  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 ofℝ  ⁢  P   m{\operatorname{\mathbb{R}P}^{m}}   . We illustrate the phenomenon in the casem  =   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 integert  =   t  ⁢   (  m  ,  s  ){t=t(m,s)}    for which there is a suitable equivariant map from Davis’ projective product spacePs{\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 relatingTC  2{\operatorname{TC}_{2}}    to the immersion dimension of real projective spaces.

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