Let Y be a complex algebraic surface. We say that it is Z-acyclic (respectively Q-acyclic) if its reduced homology with coefficients in Z (resp. in Q) vanishes. Topologically one can represent Y as a compact 4-manifold with boundary (denote the boundary by 5), attached by a collar S x [0,1). Call 5 the boundary of Y. If Y is an affine surface in C m then S is the intersection of Y with a sufficiently large sphere. We say that Y is Λ-acyclic at infinity If 5 is an ^4-homology 3-sphere. (A = Z, Q). If Y is A-acyclic then it is A-acyclic at infinity. If Y is Q-acyclic and Z-acyclic at infinity, then it is Z-acyclic. In the paper [18] Ramanujam proved that the only Z-acyclic surface bounded by a homotopy 3-sphere is C 2 , and he also constructed there the first example of a nontrivial Z-acyclic (and even contractible) surface. Later on Gurjar and Shastri [7] proved that all Z-acyclic surfaces are rationnal. Tom Dieck and Petri [1] classifind all acyclic surfaces which rise out of line configurations on P. Fujita [5] (resp. Miyanishi, Tsunoda [11] and Gurjar, Miyanishi [6]) classified acyclic surfaces with Έ = 0 (resp. —oo and 1), where « denotes the log-Kodaira dimension. Zaidenberg [21] pointed out the connection of Z-acyclic surfaces with exotic algebraic and analytic structures on C, n > 3. Flenner and Zaidenberg [4] studied deformations of acyclic surfaces. A Seίfert fibration (see [19], [17]) on a smooth compact 3-manifold M is a mapping onto a 2-manifold π : M —• B, which is a locally trivial fibration with fiber S over B — {pi,... ,pr} and which looks nearpj like D xS —> D, {zι,z2) »-• Vl¥> where D = {\z 2. The π~(pj) are called multiple fibers; M is called Seifert manifold if it admits a Seifert fibration. Seifert A-homology sphere {A stands for Z or Q) is a Seifert manifold M with H%{M\A) = if*(5 ;A). In this case the base B is a 2-sphere. The question, when a Seifert homology sphere bounds an acyclic 4-manifold, was studied, for instance, in [3], [15]. Our main result is: