Maximal Buchstaber Numbers and Toric Wedge Induction | BIMSA Topology Seminar

Abstract

For an \((n-1)\)-dimensional simplicial complex \(K) with \(ml) vertices, the moment-anglecomplex \(mathcalZ} Kl) admits a canonical action of the \(ml)-dimensional torus \(T^ml). TheBuchstaber number \(s Kl) is the maximal integer \(rl) for which there exists a subtorus \(H) of rank\(rly) acting freely on \(lmathcal{Z} Ky). lt is known that \(1 Vleg s K Vleg m-nl). lf \(s kl) is maximali.e.., \s K = m-nl), the guotient \(mathcalZ} K/ Hl is related to many important mathematicalobjects such as toric manifolds or quasitoric manifolds.

In this talk, we introduce an effective method known as toric wedge induction to study \(k)admitting a maximal Buchstaber number and to prove certain properties of (quasi)toric manifoldsthat have a specific Picard number. This method is inspired by the research on classifying toricmanifolds conducted by the speaker and Hanchul Park, which uses a process called the wedgeoperation in combinatorics. We will share some examples where the toric wedge induction methodhas been used to address various unresolved issues with toric manifolds that have a Picardnumber of 4 or less.