  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.
