Dilute homologies and fixed elements in a complex of multi-ary relations

Abstract

Having an abstract simplicial complex, its homology groups, the Euler-Poincare theorem about the link between the Euler characteristics of the complex and ranks of these groups, we construct in a purely combinatorial way dilute homologies, prove the related Euler-Poincare-Hopf theorem, and deduce the formula for fixed simplices with respect to the indicated isomorphism. We also give concrete examples of such fixed elements.