For an \((n-1)\)-dimensional simplicial complex \(K\) with \(m\) vertices, the moment-angle complex \(\mathcal{Z}_K\) admits a canonical action of the \(m\)-dimensional torus \(T^m\). The Buchstaber number \(s_K\) is the maximal integer \(r\) for which there exists a subtorus \(H\) of rank \(r\) acting freely on \(\mathcal{Z}_K\). It is known that \(1 \leq s_K \leq m-n\). If \(s_K\) is maximal, i.e., \(s_K = m-n\), the quotient \(\mathcal{Z}_K / H\) is related to many important mathematical objects such as toric manifolds or quasitoric manifolds.

In this talk, I will introduce an inductive method to study \(K\) that admits a maximal Buchstaber number and share some examples where the toric wedge induction method has been used to address various unsolved problems with toric manifolds that have a Picard number of 4 or less.

This research is a joint work with Hyeontae Jang and Mathieu Vallee.