A rational splitting of a based mapping space

Let F∗(X,Y) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y . Consider a CW complex of the form X∪αe and a space Y whose connectivity exceeds the dimension of the adjunction space. Using a Quillen–Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map α : Sk → X is greater than the Whitehead length WL(Y) of Y , then F∗(X ∪α ek+1, Y) has the rational homotopy type of the product space F∗(X, Y)× Ωk+1Y . This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex X are greater than WL(Y) and the connectivity of Y is greater than or equal to dimX , then the mapping space F∗(X,Y) can be decomposed rationally as the product of iterated loop spaces.