In this paper, we deal with the problem of the computation of the homology of a finite simplicial complex after an “elementary simplicial perturbation” process such as the inclusion or elimination of a maximal simplex or an edge contraction. To this aim we compute an algebraic topological model that is an special chain homotopy equivalence connecting the simplicial complex with its homology (wo...

This paper contains a survey of the A-theory of simplicial complexes and graphs, a combinatorial homotopy theory developed recently.The initialmotivation arises from the use of simplicial complexes as models for a variety of complex systems and their dynamics. This theory diverges from classical homotopy theory in several crucial aspects. It is related to prior work in matroid theory, graph the...

In the field of Intelligent Information Processing, mechanized reasoning systems provide a chance of increasing the reliability of software systems, namely Computer Algebra Systems. This paper is devoted to a concrete case of this topic. The notion of simplicial complex, see [7], is the most elementary method to settle a connection between common “general” topology and homological algebra. The ...

The family of Buchsbaum simplicial posets generalizes the family of simplicial cell manifolds. The h′vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced Betti numbers of its geometric realization. Novik and Swartz showed that the h′-vector of a Buchsbaum simplicial poset satisfies certain simple inequalities. In t...

This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simpli...

We study the Fräıssé limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fräıssé limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal exampl...