Open Neighborhood Seminar: The Pontryagin-Thom construction

The problem of classifying smooth manifolds up to diffeomorphism is notoriously hard. However, classifying them up to a coarser but still interesting equivalence relation, like cobordism… is still very hard. In fact, it is just as hard as classifying homotopy classes of maps between spheres. In this talk we will discuss a construction due to Pontryagin and Thom that connects smooth manifolds and maps of spheres, with the goal being to prove a certain isomorphism of groups, and if time permits we will talk about possible categorifications of this phenomenon.