Rank of the fundamental group of any component of a function space

We compute the rank of the fundamental group of any connected component of the space $\mathrm{map}(X, Y)$ for $X$ and $Y$ connected, nilpotent CW complexes of finite type with $X$ finite. For the component corresponding to a general homotopy class $f \colon X \to Y$, we give a formula directly computable from the Sullivan model for $f$. For the component of the constant map, our formula retrieves a known expression for the rank in terms of classical invariants of $X$ and $Y$. When both $X$ and $Y$ are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of $\mathrm{map}(X, Y)$ explicitly in terms of the homomorphism induced by $f$ on rational cohomology.