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 cup_{alpha}e^{k+1} 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 alpha: S^k --> X is greater than the Whitehead length WL(Y) of Y, then F_*(X cup_{alpha}e^{k+1}, Y) has the rational homotopy type of the product space F_*(X, Y) times Omega^{k+1}Y. 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 dim X, then the mapping space F_*(X, Y) can be decomposed rationally as the product of iterated loop spaces.