  There is an equivalence relation on the set of smooth maps of a manifold into
the stable unitary group, defined using a Chern-Simons type form, whose
equivalence classes form an abelian group under ordinary block sum of matrices.
This construction is functorial, and defines a differential extension of odd
K-theory, fitting into natural commutative diagrams and exact sequences
involving K-theory and differential forms. To prove this we obtain along the
way several results concerning even and odd Chern and Chern-Simons forms.
