Defect TQFTs, the Generalized Cobordism Hypothesis, and Algebraic $K$-Theory

Abstract

By constructing a fully dualizable category $\mathrm{Bord}^{\mathcal{C}}_n$ as the left adjoint of the forgetful functor, we state a generalized cobordism hypothesis. As a corollary, we establish a generalized Pontryagin-Thom isomorphism that gives a geometric interpretation of higher algebraic $K$-theory using cobordism classes of framed manifolds and stratified spaces decorated by categorical string diagrams, which we call lasagnas. For finite-dimensional vector spaces over a field, these decorations are branched local systems. We also examine low-dimensional representatives, including $K_1$, Steinberg-group presentations of $K_2$, and fillings related to the Steinberg relation, providing a more explicit construction of Ghys' manifold.