  In this paper, we discuss the cancellation and splitting of the symplectic
modules. The symplectic cancellation result presented here can be thought of as
an analog of the Projective module cancellation result of Fasel. The symplectic
splitting is similar to Murthy's splitting theorem. To prove the cancellation
and splitting, we carefully analyze the Postnikov towers in the
$\mathbb{A}^1$-homotopy category. Then we prove the vanishing of top cohomology
with coefficients in some homotopy sheaf. As another application of the
vanishing results, we answer partially a question of Mrinal Das about the
isomorphism of $(d-1)$-th Euler class group and $(d-1)$-th Chow group, where
$d$ is the dimension of the underlying smooth affine variety.
