On the topology of a resolution of isolated singularities

Let $Y$ be a complex projective variety of dimension $n$ with isolated singularities, $pi:Xto Y$ a resolution of singularities, $G:=pi^{-1}{rm{Sing}}(Y)$ the exceptional locus. From Decomposition Theorem one knows that the map $H^{k-1}(G)to H^k(Y,Ybackslash {rm{Sing}}(Y))$ vanishes for $k>n$. Assuming this vanishing, we give a short proof of Decomposition Theorem for $pi$. A consequence is a short proof of the Decomposition Theorem for $pi$ in all cases where one can prove the vanishing directly. This happens when either $Y$ is a normal surface, or when $pi$ is the blowing-up of $Y$ along ${rm{Sing}}(Y)$ with smooth and connected fibres, or when $pi$ admits a natural Gysin morphism. We prove that this last condition is equivalent to say that the map $H^{k-1}(G)to H^k(Y,Ybackslash {rm{Sing}}(Y))$ vanishes for any $k$, and that the pull-back $pi^*_k:H^k(Y)to H^k(X)$ is injective. This provides a relationship between Decomposition Theorem and Bivariant Theory.