We consider the group of all polynomial automorphisms 
of the coordinate space and its subgroups, and compute 
the first homologies of their groups. Our main result 
is that the first homology group of the group of all 
polynomial automorphisms of the coordinate plane is 
isomorphic to ${\bf K}^*$, where ${\bf K}={\bf R}$ or 
${\bf C}$ and $ {\bf K}^*={\bf K}- \{ 0 \}$. This fact 
relates mostly to the topology of transversely algebraic 
foliations.