Rational maps from Euclidean Configuration Spaces to Spheres

TL;DR

This paper provides an algorithm to determine the rational homotopy types of mapping spaces from Euclidean configuration spaces to spheres, with explicit descriptions for specific cases and general results for odd-dimensional spheres.

Contribution

It introduces a new algorithm for computing the rational homotopy types of these mapping spaces and offers explicit descriptions for certain configurations.

Findings

Explicit description for k=3 configuration spaces

Algorithm for rational homotopy type determination

Results for odd-dimensional 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 $ map^*(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.