Random simplicial complexes, duality and the critical dimension

In this paper, we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behavior of the Betti numbers in the lower model is characterized by the notion of critical dimension, which was introduced by Costa and Farber in [Large random simplicial complexes III: The critical dimension, J. Knot Theory Ramifications 26 (2017) 1740010]: random simplicial complexes in the lower model are homologically approximated by a wedge of spheres of dimension equal the critical dimension. In this paper, we study the Betti numbers in the upper model and introduce new notions of critical dimension and spread. We prove that (under certain conditions) an upper random simplicial complex is homologically approximated by a wedge of spheres of the critical dimension.

Betti splitting from a topological point of view

A Betti splitting [Formula: see text] of a monomial ideal [Formula: see text] ensures the recovery of the graded Betti numbers of [Formula: see text] starting from those of [Formula: see text] and [Formula: see text]. In this paper, we introduce an analogous notion for simplicial complexes, using Alexander duality, proving that it is equivalent to a recursive splitting condition on links of some vertices. We provide results ensuring the existence of a Betti splitting for a simplicial complex [Formula: see text], relating it to topological properties of [Formula: see text]. Among other things, we prove that orientability for a manifold without boundary is equivalent to the admission of a Betti splitting induced by the removal of a single facet. Taking advantage of our topological approach, we provide the first example of a monomial ideal which admits Betti splittings in all characteristics but with characteristic-dependent resolution. Moreover, we introduce new numerical descriptors for simplicial complexes and topological spaces, useful to deal with questions concerning the existence of Betti splitting.

Alexander duality for monomial ideals associated with isotone maps between posets

For a pair [Formula: see text] of finite posets the generators of the ideal [Formula: see text] correspond bijectively to the isotone maps from [Formula: see text] to [Formula: see text]. In this note we determine all pairs [Formula: see text] for which the Alexander dual of [Formula: see text] coincides with [Formula: see text], up to a switch of the indices.