Lattice cohomology and subspace arrangements: the topological and analytic cases

Abstract

In this paper we consider the (topological) lattice cohomology \(\mathbb {H}^*\) of a surface singularity with rational homology sphere link. In particular, we will be studying two sets of (topological) invariants related to it: the weight function \(\upchi \) that induces the cohomology and the topological subspace arrangement \(T(\ell ,I)\) at each lattice point \(\ell \) — the latter of which is the weaker of the two. We shall prove that the two are in fact equivalent by establishing an algorithm to compute \(\upchi \) from the subspace arrangement. Replacing the topological arrangements with the analytic, we get another formula — one that connects them with the analytic lattice cohomology introduced and studied in our earlier papers. In fact, this connection served as the original motivation for the definition of the latter. Aside from the historical interest, this parallel also provides us with tools to study more easily the connection between the two cohomologies.