Bordism and the Pontryagin-Thom Theorem

Given the classification of low dimensional manifolds up to equivalence relations such as diffeomorphism or homeomorphism, one would hope to be able to continue to classify higher dimensional manifolds. Unfortunately, this turns out to be difficult or impossible, and so one solution would be turn to some weaker equivalence relation. One such equivalence relation would be to consider manifolds up to bordism, which we will define below. Though it is a weaker notion of equivalence, bordism still captures some important invariants of manifolds, which we will not discuss here. Furthermore, it turns out that bordism has a very rich structure and deep connections to algebraic topology, which one can exploit and makes it possible to classify manifolds up to bordism.