Topology Seminar: Jacob Lurie

Let C be a triangulated category. The Grothendieck group K_0(C) is defined as the abelian group generated by symbols [X], where X is an object of C, moduli the relation [X] = [X'] + [X''] for every exact triangle X' -> X -> X'' in C.

A simple consequence of this relation is that the double suspension functor X -> X[2] induces the identity map from K_0(C) to itself. In this talk, I will explain how this observation can be seen as the shadow of a certain rotation-invariance phenomenon in algebraic K-theory, and describe the connection of this phenomenon with the theory of "topological Fukaya categories" introduced by Dyckerhoff and Kapranov.