Thes-th higher topological complexity{\operatorname{TC}_{s}(X)}of a spaceXcan 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=\operatorname{\mathbb{R}P}^{m}}, the real projective space of dimensionm. In particular, we describe a number{r(m)}, which depends on the structure of zeros and ones in the binary expansion ofm, and with the property that{0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)}for{s\geq r(m)}, where{\delta_{s}(m)=(0,1,0)}for{m\equiv(0,1,2)\bmod 4}. Such an estimation for{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}appears to be closely related to the determination of the Euclidean immersion dimension of{\operatorname{\mathbb{R}P}^{m}}. We illustrate the phenomenon in the case{m=3\cdot 2^{a}}. In addition, we show that, for large enoughsand evenm,{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}is characterized as the smallest positive integer{t=t(m,s)}for which there is a suitable equivariant map from Davis’ projective product space{\mathrm{P}_{\mathbf{m}_{s}}}to the{(t+1)}-st join-power{((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating{\operatorname{TC}_{2}}to the immersion dimension of real projective spaces.

Abstract. Let R be a commutative ring containing 1/2. We compute the R-cohomology ring of the configuration space Conf(RP m , k) of k ordered points in the m-dimensional real projective space RP m . The method uses the observation that the orbit configuration space of k ordered points in the m-dimensional sphere (with respect to the antipodal action) is a 2 k -fold covering of Conf(RP m , k). This implies that, for odd m, the Leray spectral sequence for the inclusion Conf(RP m , k) ⊂ (RP m ) k collapses after its first non-trivial differential, just as it does when RP m is replaced by a complex projective variety. The method also allows us to handle the R-cohomology ring of the configuration space of k ordered points in the punctured manifold RP m − ⋆. Lastly, we compute the LusternikSchnirelmann category and all of the higher topological complexities of some of the auxiliary orbit configuration spaces.MSC 2010: Primary 55R80, 55T10, 55M30.

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