Betti splitting from a topological point of view

A Betti splitting 𝐼=𝐽+𝐾 of a monomial ideal 𝐼 ensures the recovery of the graded Betti numbers of 𝐼 starting from those of 𝐽,𝐾 and 𝐽∩𝐾. 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 Δ, relating it to topological properties of Δ. 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.