The connecting homomorphism for Hermitian $K$-theory

We provide a geometric interpretation for the connecting homomorphism in the localization sequence of Hermitian $K$-theory. As an application, we compute the Hermitian $K$-theory of projective bundles and Grassmannians in the regular case. We provide an explicit basis for Hermitian $K$-theory of Grassmannians, which is indexed by even Young diagrams together with another special class of Young diagrams, that we call $\mathit{\text{buffalo-check}}$ Young diagrams. To achieve this, we develop pushforwards and pullbacks in Hermitian $K$-theory using Grothendieck’s residue complexes, and we establish fundamental theorems for those pushforwards and pullbacks, including base change, projection, and excess intersection formulas.