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.