230915 Michael Jung

It is a classical fact that for closed manifolds X the homotopy classes of maps X^n→S^n are classified by their degree. The Pontryagin-Thom construction provides a similar construction when X and the sphere have also different dimensions, and thus generalizes the notion of degree. In particular, the homotopy classes of maps X^n+1→S^n are in one-to-one correspondence with framed circles up to framed cobordism in X, and the corresponding set comes equipped with a group structure.

In this talk, we introduce the Pontryagin-Thom construction and the concept of framed cobordism classes, and we compute the group of homotopy classes of maps X^n+1→S^n in terms of geometric and topological information of X. If time permits, we delve into some ideas of the proof, and discuss applications to vector bundles.