The multiplicative structure of higher cobordism categories

Abstract: The Pontryagin-Thom theorem identifies the $n$-th stable homotopy group of spheres with framed $n$-manifolds up to cobordism, together with their graded ring structure, where disjoint union and Cartesian product correspond to addition and multiplication, respectively. The Baez-Dolan cobordism hypothesis provides a higher-categorical lift, stating that the $(∞, n)$-category ${\rm Bord}_n$ of framed cobordisms is the free $(∞, n)$-category generated by a fully dualizable object. While the additive structure lifts naturally on both the geometric and categorical sides, the multiplicative structure is more subtle. In joint work in progress with Leor Neuhauser, we construct a compatible multiplicative structure on both sides and show that they coincide under the cobordism hypothesis.

mathematical physicsalgebraic topologycategory theory

Audience: researchers in the topic

Topology and Geometry Seminar (Texas, Kansas)