  It is well-known that there is a close relationship between the dynamics of
diffeomorphisms satisfying the axiom $A$ and the topology of the ambient
manifold. In the given article, this statement is considered for the class
$\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable surfaces such that
their non-wandering set consists of $k_f\geq 2$ connected components of
one-dimensional basic sets (attractors and repellers). We prove that the
ambient surface of every diffeomorphism $f\in \mathbb G(M^2)$ is homeomorphic
to the connected sum of $k_f$ closed orientable surfaces and $l_f$
two-dimensional tori such that the genus of each surface is determined by the
dynamical properties of appropriating connected component of a basic set and
$l_f$ is determined by the number and position of bunches, belonging to all
connected components of basic sets. We also prove that every diffeomorphism
from the class $\mathbb G(M^2)$ is $\Omega$-stable but is not structurally
stable.
