Formal Barycenter Spaces with Weights: The Euler Characteristic

We compute the Euler characteristic with compact supports $\chi_c$ of the formal barycenter spaces with weights of a finite CW complex, connected or not. This reduces to the topological Euler characteristic $\chi$ when the weights of the singular points are less than one. As foresighted by A. Malchiodi, our formula is related to the Leray-Schauder degree for mean field equations on a compact Riemann surface obtained by C.C. Chen and C.S. Lin.