UC Riverside

Given a finite group G, an equivariant map between spaces with group actions is a map that preserves the G-action. We can keep track of the equivariant maps between the quotient sets of G using the orbit category. An isovariant map is an equivariant map that preserves isotropy groups. Yeakel defines the link orbit category, which records the isovariant maps between the quotient sets of G.The subdivision of the orbit category is a stratified version of the orbit category that explicitly encodes paths through the orbit category. The classifying space functor, B, assigns to each category a topological space and interprets functors as continuous maps, and a functor is a weak equivalence if its image under B is a homotopy equivalence.In this thesis, for finite abelian groups, I construct a weak equivalence from the subdivision of the orbit category to the link orbit category using Quillen's Theorem A.