The stability of the higher topological complexity of real projective spaces: an approach to their immersion dimension

The $s$-th higher topological complexity of a space $X$, $TC_s(X)$, 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 $r(m)$, which depends on the structure of zeros and ones in the binary expansion of $m$, and with the property that $TC_s(RP^m)$ is given by $sm$ with an error of at most one provided $s \geq r(m)$ and $m \not\equiv 3 \bmod 4$ (the error vanishes for even $m$). The latter fact appears to be closely related to the estimation of the Euclidean immersion dimension of $RP^m$. We illustrate the phenomenon in the case $m=3 \cdot 2^a$.