An upper bound for the rational topological complexity of a family of elliptic spaces
Said Hamoun, Youssef Ramı, Lucile Vandembroucq · Proceedings of the American Mathematical Society · 2024
In this work, we show that, for any simply-connected elliptic space S S admitting a pure minimal Sullivan model with a differential of constant length, we have T C 0 ( S ) ≤ 2 c a t 0 ( S ) + χ π ( S ) TC_0(S)\leq 2cat_0(S)+\chi _{\pi }(S) where T C 0 ( S ) TC_0(S) and c a t 0 ( S ) cat_0(S) denote respectively the rational topological complexity and Lusternik-Schnirelmann category, and χ π ( S ) \chi _{\pi }(S) is the homotopy characteristic of S S . This is obtained by establishing a structure theorem showing that such a space can be rationally realized as the total space of a fibration over an F 0 F_0 -space with fibre a product of odd-dimensional spheres.