  Let $M$ be a closed connected smooth manifold and $G=\textmd{Diff}_0(M)$
denote the connected component of the diffeomorphism group of $M$ containing
the identity. The natural action of $G$ on $M$ induces the trace homomorphism
on homology. We show that the image of trace homomorphism is annihilated by the
subalgebra of the cohomology ring of $M$, generated by the characteristic
classes of $M$. Analogously, if $J$ is an almost complex structure on $M$ and
$G$ denotes the identity component of the group of diffeomorphisms of $M$
preserving $J$ then the image of the corresponding trace homomorphism is
annihilated by subalgebra generated by the Chern classes of $(M,J)$.
