Eigenvalues and signature of quadratic forms associated with finite topological spaces

Abstract

Given any finite topological space X and a field K, we associate a quadratic space (QX,VX), consisting of a vector space VXover Kand a quadratic form QX:VX×VX→K, to X. The eigenvalues and signature of QXare topological invariants of X. We study their relations with X. From this, we obtain restrictions to check whether a finite topological space can be embedded into another one. Additionally, we compute these invariants for minimal finite models of spheres and other families of finite spaces.