  Inserting a magnetic flux into a two-dimensional one-particle Hamiltonian
leads to a spectral flow through a given gap which is equal to the Chern number
of the associated Fermi projection. This paper establishes a generalization to
higher even dimension by inserting non-abelian monopoles of the Wu-Yang type.
The associated spectral flow is then equal to a higher Chern number. For the
study of odd spacial dimensions, a new so-called `chirality flow' is introduced
which, for the insertion of a monopole, is then linked to higher winding
numbers. This latter fact follows from a new index theorem for the spectral
flow between two unitaries which are conjugates of each other by a self-adjoint
unitary.
