  This is a study of twisted K-theory on a product space $T \times M$. The
twisting comes from a decomposable cup product class which applies the
1-cohomology of $T$ and the 2-cohomology of $M$. In the case of a topological
product, we give a concrete realization for the gerbe associated to a cup
product characteristic class and use this to realize twisted $K^1$-theory
elements in terms of supercharge sections in a Fredholm bundle. The
nontriviality of this construction is proved. Equivariant twisted K-theory and
gerbes are studied in the product case as well. This part applies Lie groupoid
theory. Superconnection formalism is used to provide a construction for
characteristic polynomials which are used to extract information from the
twisted K-theory classes.
