Rational maps from Euclidean Configuration Spaces to Spheres

Abstract

In this note we give an algorithm to determine the rational homotopy type of the free and pointed mapping spaces $map(F({\mathbb{R}}^m,k),S^n)$ and $ma{p}^{\ast }(F({\mathbb{R}}^m,k),S^n)$. 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.