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 $X$, $T{C}_s(X)$, 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 $X=R{P}^m$, the real projective space of dimension $m\mathrm{.}$ 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 $T{C}_s(R{P}^m)$ is given by $sm$ with an error of at most one provided $s\ge r(m)$ and $m \not\equiv 3 \bmod 4$(theerrorvanishesforeven$m$). The latter fact appears to be closely related to the estimation of the Euclidean immersion dimension of $R{P}^m$. We illustrate the phenomenon in the case $m=3\cdot 2^a$.