Conversion between chains of maps and chains of surfaces; application to the computation of incidence graphs homology

Abstract

Many combinatorial cellular structures have been defined in order to represent the topology of subdivided geometric objects. Two main classes can be distinguished. According to the terminology of [8], one is related to incidence graphs and the other to ordered models. Both classes have their own specificities and their use is relevant in different contexts. It is thus important to create bridges between them. So we define here chains of surfaces (a subclass of incidence graphs) and chains of maps without multi-incidence (a subclass of ordered models), which are able to represent the topology of subdivided objects, whose cells have " manifold-like " properties. We show their equivalence by providing conversion operations. As a consequence, it is hence possible to directly apply on each model results obtained on the other. We extend here classical results related to homology computation obtained for incidence graphs corresponding to regular CW −complexes and recent results about combinatorial cell complexes where cells are not necessarily homeomorphic to balls