  We consider an embedding of a $2$-dimensional CW complex into the $3$-sphere,
and construct it's dual graph. Then we obtain a homogeneous system of linear
equations from the $2$-dimensional CW complex in the first homology group of
the complement of the dual graph. By checking that the homogeneous system of
linear equations does not have an integral solution, we show that some
$2$-dimensional CW complexes cannot be embedded into the 3-sphere.
