  We give a complete description of the potential failure of the surjectivity
of the Thom morphism from complex cobordism to integral cohomology for compact
Lie groups via a detailed study of the Atiyah-Hirzebruch spectral sequence and
the action of the Steenrod algebra. We show how the failure of the surjectivity
of the topological Thom morphism can be used to find examples of non-trivial
elements in the kernel of the induced differential Thom morphism from
differential cobordism of Hopkins and Singer to differential cohomology. These
arguments are based on the particular algebraic structure and interplay of the
torsion and non-torsion parts of the cohomology and cobordism rings of a given
compact Lie group. We then use the geometry of special orthogonal groups to
construct concrete cobordism classes in the non-trivial part of the kernel of
the differential Thom morphism.
