The topology of an algebraic variety is a central subject in algebraic geometry. Instead of a variety, we consider the topology of a pair (X,D)
which is a variety X with a divisor D, but in the coarsest level. More precisely, we study the dual complex defined as the combinatorial datum
characterizing how the components of D intersect with each other. We will discuss how to use the minimal model program (MMP) to investigate it.
As one concrete application, we will explain how close the dual complex of a log Calabi-Yau pair (X,D) is to a finite quotient of a sphere.