We compute the Euler characteristic with compact supports $\chi_c$ of the formal barycenter spaces with weights of some locally compact spaces, 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 Andrea 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.
"Formal barycenter spaces with weights: the Euler characteristic." Topol. Methods Nonlinear Anal. 53 (2) 801 - 823, 2019. https://doi.org/10.12775/TMNA.2019.019