Mickelsson's twisted K-theory invariant and its generalizations

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.