  For S a compact connected oriented surface, we consider homology cylinders
over S: these are homology cobordisms with an extra homological triviality
condition. When considered up to Y_2-equivalence, which is a surgery
equivalence relation arising from the Goussarov-Habiro theory, homology
cylinders form an Abelian group.
  In this paper, when S has one or zero boundary component, we define a surgery
map from a certain space of graphs to this group. This map is shown to be an
isomorphism, with inverse given by some extensions of the first Johnson
homomorphism and Birman-Craggs homomorphisms.
