Symmetric Joins and Weighted Barycenters

Given a space X, we study the homotopy type of ${\mathcal B}_n(X)$ the space obtained as "the union of all (n-1)-simplexes spanned by points in X". This is a space encountered in non-linear analysis under the name of "space of barycenters" or in differential geometry in the case n=2 as the "space of chords". We first relate this space to a more familiar symmetric join construction and then determine its stable homotopy type in terms of the symmetric products on the suspension of X. This leads to a complete understanding of the homology of ${\mathcal B}_n(X)$ as a functor of X, and to an expression for its Euler characteristic given in terms of that of $X$. A sharp connectivity theorem is also established. Finally barycenter spaces of spheres are studied in details.