Global Structure in the Presence of a Topological Defect

We investigate the global structure of topological defects which wrap a submanifold $F\subset M$ in a quantum field theory defined on a closed manifold $M$. The Pontryagin-Thom construction oversees the interplay between the global structure of $F$ and the global structure of $M$. We will employ this construction to two distinct mathematical frameworks with physical applications. The first framework is the concept of a characteristic structure, consisting of the data of pairs of manifolds $(M,F)$ where $F$ is Poincar\'e dual to some characteristic class. This concept is discussed in the mathematics literature, and shown here to have meaningful physical interpretations related to defects. In our examples we will mainly focus on the case where $M$ is 4-dimensional and $F$ has codimension 2. The second framework uses obstruction theory and the fact that spontaneously broken finite symmetries leave behind domain walls, to determine the conditions on which dimensions a higher-form finite symmetry can spontaneously break. We explicitly study the cases of higher-form $\mathbb Z/2$ symmetry, but the method can be generalized to other groups.