Hermitian K-theory for stable ∞-categories II: Cobordism categories and additivity

1^st appeared online on the 16-09-2020

This is the second article in a series about hermitian K-theory in the context of stable ∞-categories endowed with quadratic functors. The first one containing the basic formalism is here: Foundations.

In this second article, we associate to a Poincaré ∞-category a Grothendieck-Witt spectrum. It can be obtained by a type of hermitian Q-construction that we view through the angle of cobordism categories. The functor Grothendieck-Witt spectrum is also characterized by a universal property in terms of additivity.

We show a list of fundamental properties of the Grothendieck-Witt spectrum, and in particular:

• it sits in a fiber sequence in between coinvariants of the duality action on K-theory and L-theory;
• it is a fiber product in a cartesian square containing L-theory, invariants and Tate fixed poins of K-theory by the duality action;
• it sits in a fiber sequence (that we call Bott-Genauer sequence) that permits Karoubi induction.

We express and prove Karoubi periodicity in a general framework of genuine equivariant spectra and derive its avatars in concrete categories of modules.