In this note we give an algorithm to determine the rational homotopy type of the free and pointed mapping spaces $\map(F(\br^m,k),\bs^n)$ and $\map^*(F(\br^m,k),\bs^n)$. An explicit description of these spaces is given for $k=3$. The general case for $n$ odd is also presented as an immediate consequence of the rational version of a classical result of Thom.
BUIJS, URTZI; GARVIN, ANTONIO; and MURILLO, ANICETO
"Rational maps from Euclidean configuration spaces to spheres,"
Turkish Journal of Mathematics: Vol. 43:
5, Article 17.
Available at: https://journals.tubitak.gov.tr/math/vol43/iss5/17