Formal Barycenter Spaces with Weights: The Euler Characteristic

TL;DR

This paper calculates the Euler characteristic with compact supports for weighted formal barycenter spaces on finite CW complexes, linking it to topological Euler characteristics and mean field equations on Riemann surfaces.

Contribution

It provides a formula for the Euler characteristic with weights, extending known results and connecting to the Leray-Schauder degree in mean field equations.

Findings

Derived a formula for the Euler characteristic with weights.

Connected the weighted Euler characteristic to topological Euler characteristic.

Linked the results to mean field equations on Riemann surfaces.

Abstract

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.