A rational splitting of a based mapping space

Let ${\mathcal F}_*(X, Y)$ 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 $\alpha : S^k \to X$ is greater than the Whitehead length WL$(Y)$ of $Y$, then ${\mathcal F}_*(X\cup_{\alpha}e^{k+1}, Y)$ has the rational homotopy type of the product space ${\mathcal F}_*(X, Y)\times \Omega^{k+1}Y$. 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 ${\mathcal F}_*(X, Y)$ can be decomposed rationally as the product of iterated loop spaces.