Algebra/Topology seminar

Speaker: Shai Keidar

Title: Multiplicative structure of higher cobordism categories

Abstract: The Pontryagin-Thom theorem identifies the stable homotopy groups of spheres with the cobordism groups of framed manifolds. This identification is compatible with the graded ring structure on both sides, where disjoint union and Cartesian product of manifolds correspond to addition and multiplication, respectively. A higher categorical refinement of the cobordism groups is given by the (∞,n)-categories of cobordisms, which the Baez-Dolan cobordism hypothesis identifies with the free symmetric monoidal (∞,n)-category generated by a fully dualizable object.
While the additive structure lifts naturally to this setting on both the geometric and categorical sides, the multiplicative structure is more subtle. In joint work in progress with Leor Neuhauser and Lior Yanovski, we construct a compatible multiplicative structure on both sides and show that they agree under the cobordism hypothesis.