Weakly Equivariant Classification of Small Covers over a Product of Simplicies

Given a dimension function \(\omega\), we define a notion of an \(\omega\)-vector weighted digraph and an \(\omega\)-equivalence between them. Then we establish a bijection between the weakly \((\mathbb{Z}/2)^n\)-equivariant homeomorphism classes of small covers over \(\Delta^{n_1}\times\cdots \times \Delta^{n_k}\) and the set of \(\omega\)-equivalence classes of \(\omega\)-vector weighted digraphs with \(k\)-labeled vertices. As an example, we obtain a formula for the number of weakly \((\mathbb{Z}/2)^n\)-equivariant homeomorphism classes of small covers over a product of three simplices.