Le genre de Mislin des espaces rationnellement équivalents à un produit de deux sphères de dimensions différentes. (The Mislin genus of spaces rationally equivalent to a product of two spheres of different dimensions.) (Q1601085)

Using, as algebraic tools, the exact sequences of Zabrodsky which express the genus set of a nilpotent space \(X\) of finite type, in terms of its self-maps, when \(X\) has the rational homotopy type of a co-\(H\)-space or an \(H\)-space, the author proves the main result of this paper. He establishes the existence of a Zabrodsky exact sequence and expresses the genus of CW-complexes with three cells rationally equivalent to the product of two spheres of different dimensions, \(S^k\times S^n\), \(n>k\geq 2\). He deduces, from results of Morisugi-Oshima, the genus of some spherical bundles. Explicit examples show that these methods cannot be generalized to the class of all simply connected finite CW-complexes.