Relative Gottlieb groups of mapping spaces and their rational cohomology

Abstract

Let f:X →Y be a map of simply connected CW-complexes of finite type. Put maxπ_★(Y)⊗Q = max{ i | π_i(Y)⊗Q≠0 }. In this paper we compute the relative Gottlieb groups of f when X is an F₀-space and Y is a product of odd spheres. Also, under reasonable hypothesis, we determine these groups when X is a product of odd spheres and Y is an F₀-space. As a consequence, we show that the rationalized G-sequence associated to f splits into a short exact sequence. Finally, we prove that the rational cohomology of map(X,Y;f) is infinite dimensional whenever maxπ_★(Y)⊗Q is even.