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.