Comment. Math. Helvetici 58 (1983) 111-114 0010-2571/83/001111-04501.50 + 0.20/0 (~) 1983 Birkh~iuser Verlag, Basel Poincar~ duality groups of dimension two, H BENO ECKMANN and PETER LINNELL 1. Introduction A Poincar6 duality group of dimension n, in short a PD"-group, is a group G acting on Z such that one has natural isomorphisms Hk(G; A) ~ H._k(G; Z@A) for all integers k and all ZG-modules A (where Z(~A is the tensor product over Z with diagonal G-action). G is called orientable or not according to whether or not Z is trivial as a ZG-module. All "surface groups", i.e., fundamental groups of closed surfaces of genus />1 are well-known to be pDa-groups. In Eckmann- Miiller [4] it was proved that a PD2-group with positive first Betti number /31 is isomorphic to a surface group. The purpose of the present paper is to show that the condition on /31 is automatically fulfilled: THEOREM 1. The first Betti number/31 of a PD2-group is positive. As a consequence we thus have a complete classification of pD2-groups. THEOREM 2. A group G is a PD2-group if and only if it is isomorphic to a surface group. For notations and properties concerning PD"-groups, not explicitly mentioned here, we refer to [4] where also several (algebraic and topological) consequences are discussed. 2. Finitely generated projective ZG-modules For the proof of Theorem 1 we need the following fact, which may be of interest in connection with the conjectures of Bass (4.4 and 4.5 of [2]). lll