# Abe homotopy groups of homogeneous spaces with applications to topological phases of matter

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## Abstract

Describing physical systems can be complicated. In order to reduce the complexity of certain models, one will sometimes restrict attention to the so called order parameter field. The order parameter field can take many different forms depending on the physical system to be modeled. We can look at the topological characteristics of the order parameter space to model what has become known as topological phases of these systems. The aspect of physical systems we will focus on here is topological excitations, also known as topological defects. These defects are often classified by the homotopy groups of the associated order parameter manifold M, with the nth homotopy group of M, [pi][subscript]n(M), describing the n-dimensional defects. These homotopy groups give all the possible defects that can occur in a given dimension, as well as how the defects can interact with each other. Furthermore, the action of the fundamental group of M on the homotopy group [pi][subscript]n(M) tells how a one-dimensional defect acts on a n-dimensional defect. This information is given by the Abe homotopy group of the order parameter manifold. We look to describe the Abe homotopy groups associated to some interesting physical cases and expand the results to a simplification of the computation of the Abe homotopy groups of all connected homogeneous spaces.