A new technique is presented for construction of Poisson
manifolds. This technique is inspired by surgery
ideas used to define Poisson structures on 3-manifolds
and Gompf's surgery construction for symplectic manifolds.
As an application of these ideas it is proved that
for all n ≥ d ≥ 4, d even, any finitely presentable group
is the fundamental group of a n-dimensional orientable
closed Poisson manifold of constant rank d. The unimodularity
of some of the Poisson structures thus constructed
is studied.