UC Berkeley

We consider the notion of expander degree, defined for certain smooth 4-orbifolds M with boundary, defined by Bamler and Chen. The expander degree is a topological invariant of M, and importantly, when the expander degree of a 4-manifold M is nonzero, there exists a gradient expanding soliton asymptotic to any nonnegatively curved cone metric. This fact is significant in the resolution of conical singularities in 4-dimensional Ricci flow by gradient expanding solitons.We begin by reviewing Ricci flow and properties of Ricci solitons. Then, we give an introduction to the theory of the expander degree. Subsequently, we define a version of the expander degree in a cohomogeneity one setting and calculate it for certain topologies. In doing so, we describe a relation between the two notions of expander degree and compare the general degree theory with the cohomogeneity one setting.