  Mickelsson's invariant is an invariant of certain odd twisted K-classes of
compact oriented three dimensional manifolds. We reformulate the invariant as a
natural homomorphism taking values in a quotient of the third cohomology, and
provide a generalization taking values in a quotient of the fifth cohomology.
These homomorphisms are related to the Atiyah-Hirzebruch spectral sequence. We
also construct some characteristic classes for odd twisted K-theory in a
similar vein.
