The authors study topological restrictions of symplectically filling 4-manifolds of links around simple singularities. Let \(\Gamma \) be a finite subgroup of \(SU(2)\). They prove that for any symplectically filling 4-manifold of a rational homology 3-sphere with the standard contact structure \((S^3/ \Gamma, \xi_0)\), the intersection form is negative definite. When \(\Gamma \) is of \(E_8\) type then the intersection form of minimal symplectically filling 4-manifolds of the Poincaré integral homology 3-sphere with the standard contact structure \((S^3/ \Gamma, \xi_0)\) is equivalent to the negative definite Cartan matrix of type \(E_8\). Let \((X, \omega) \) be a symplectic manifold with boundary \(\bigcup_{i=1}^N S^3/\Gamma_i\). If \((X, \omega) \) is a symplectic filling of its boundary with the standard contact structure on each boundary component \(S^3/ \Gamma_i\) then the boundary is connected. The authors also consider similar problems for simply elliptic singularities. They obtain similar restrictions for the intersection form of any strongly symplectically filling 4-manifold for this case. The Seiberg-Witten monopole equations are used to study these problems. In the proof, a vanishing theorem of the Seiberg-Witten invariants is discussed.