  In this article, I prove the following statement: Every compact complex
surface with even first Betti number is deformation equivalent to one which
admits an extremal K\"ahler metric. In fact, this extremal K\"ahler metric can
even be taken to have constant scalar curvature in all but two cases: the
deformation equivalence classes of the blow-up of $\p_2$ at one or two points.
The explicit construction of compact complex surfaces with constant scalar
curvature K\"ahler metrics in different deformation equivalence classes is
given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo
and F. Pacard which states that the blow-up/resolution of a compact
manifold/orbifold of discrete type, which admits cscK metrics, still admits
cscK metrics.
