Eigenvalues and signature of quadratic forms associated with finite topological spaces

Given any finite topological space X and a field K , we associate a quadratic space ( Q X , V X ) , consisting of a vector space V X over K and a quadratic form Q X : V X × V X → K , to X . The eigenvalues and signature of Q X are 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.