Derived categories of hyper-K\"ahler manifolds: extended Mukai vector and integral structure

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-K\"ahler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-K\"ahler manifolds as well as an integral lattice associated to the derived category of hyper-K\"ahler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.