  In this paper, we construct for higher twists that arise from cohomotopy
classes, the Chern character in higher twisted K-theory, that maps into higher
twisted cohomology. We show that it gives rise to an isomorphism between higher
twisted K-theory and higher twisted cohomology over the reals. Finally we
compute spherical T-duality in higher twisted K-theory and higher twisted
cohomology in very general cases.
