Local and global coincidence homology classes

TL;DR

This paper explores local and global coincidence homology classes for differentiable maps, providing combinatorial insights, explicit formulas, and a generalization of the Lefschetz coincidence point formula.

Contribution

It introduces a combinatorial approach to coincidence homology classes, clarifies their relation with existing cohomology classes, and generalizes classical formulas.

Findings

Explicit expressions for local homology classes

Relations between local and global coincidence classes

Generalization of the Lefschetz coincidence point formula

Abstract

For two differentiable maps between two manifolds of possibly different dimensions, the local and global coincidence homology classes are introduced and studied by Bisi- Bracci-Izawa-Suwa (2016) in the framework of Cech-de Rham cohomology. We take up the problem from the combinatorial viewpoint and give some finer results, in particular for the local classes. As to the global class, we clarify the relation with the cohomology coincidence class as studied by Biasi-Libardi-Monis (2015). In fact they introduced such a class in the context of several maps and we also consider this case. In particular we define the local homology class and give some explicit expressions. These all together lead to a generalization of the classical Lefschetz coincidence point formula.