# Betti splitting from a topological point of view

## TL;DR

This paper explores Betti splitting in simplicial complexes through a topological lens, linking algebraic properties to topological features and providing new examples and concepts like splitting probability.

## Contribution

It introduces a topological perspective on Betti splitting, relating it to properties like orientability and characteristic dependence, and defines the novel concept of splitting probability.

## Key findings

- Betti splitting for simplicial complexes is equivalent to a recursive condition on links.
- Orientability of a manifold without boundary is characterized by a Betti splitting from a single facet removal.
- First example of Betti splitting with characteristic-dependent resolution.

## Abstract

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 $J \cap K$. In this paper, we introduce this condition for simplicial complexes, and, by using Alexander duality, we prove that it is equivalent to a recursive splitting conditions on links of some vertices. The adopted point of view enables for relating the existence of a Betti splitting for a simplicial complex $\Delta$ to the topological properties of $\Delta$. Among other results, we prove that orientability for a manifold without boundary is equivalent to admit a Betti splitting induced by the removal of a single facet. Taking advantage of this topological approach, we provide the first example in literature admitting Betti splitting but with characteristic-dependent resolution. Moreover, we introduce the notion of splitting probability, useful to deal with results concerning existence of Betti splitting.