  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 an application, we obtain 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.
