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

Mickelsson's twisted K-theory invariant and its generalizations