  Let $Y$ be a complex projective variety of dimension $n$ with isolated
singularities, $\pi:X\to Y$ a resolution of singularities,
$G:=\pi^{-1}{\rm{Sing}}(Y)$ the exceptional locus. From Decomposition Theorem
one knows that the map $H^{k-1}(G)\to H^k(Y,Y\backslash {\rm{Sing}}(Y))$
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 $H^{k-1}(G)\to H^k(Y,Y\backslash
{\rm{Sing}}(Y))$ 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.
