Pontryagin-Thom construction: a peek into the role of cobordism in homotopy theory.

Tuesday, September 27, 2016 - 14:00 to 14:50
Thackeray 427

Abstract or Additional Information

One of the ways to approach the problem of classification of manifolds is through cobordism: two manifolds are cobordant if they make up a boundary of a higher-dimensional manifold. The equivalence classes of cobordant manifolds can be endowed with group structure, and thus we can talk about cobordism groups.

Lev Pontryagin proved in 1950. that the cobordism groups of framed manifolds - i.e. manifolds with a fixed choice of trivialization of the tangent bundle  - are isomorphic to the stable homotopy groups of spheres. By requiring the manifold to have other types of structure - orientation, almost complex structure, spin structure etc we can define different types of cobordism (oriented, complex, Spin...).  We can also avoid putting any additional restrictions on manifold structure, and consider what is called the unoriented cobordism.

Motivated by Pontryagin's work, Rene Thom in his seminal 1954 thesis computed the unoriented cobordism groups completely, thus laying  down the foundation for the cobordism theory which later earned him a Fields medal. In this talk I will present a part of Thom's work that illustrates the relationship between cobordism and homotopy theory, namely that the unoriented cobordism groups are isomorphic to homotopy groups of so-called Thom spaces, constructed from the normal bundles of manifolds. 

This approach, called the Pontryagin-Thom construction, can actually be generalized to cobordims with additional structure to construct spaces whose homotopy theory is equivalent to cobordism rings. I will finish the talk with a brief overview of the importance of various cobordism theories in modern homotopy theory. I will try to avoid mentioning frightening words like "spectrum" or "category" or "cohomology" until possibly the last 10 minutes of the talk.