A method is shown to construct ANR-topologies on topological groups of suitable homotopy type. It is well known that every simple countable CW-complex with vanishing Postnikov invariants has the homotopy type of a topological Abelian group G, that can be taken as realization of a countable simplicial Abelian group and hence is a countable CW-complex [5, Ch.V]. In particular this means that the homotopy groups 7rn(G) can be chosen as an arbitrarily prescribed sequence of countable Abelian groups. However, there are situations where it would be preferable to have G as an ANR-space instead of a CW-complex. One would guess that a suitable topology exists on G, because by subdivision it can be turned into a polyhedron and then can get the metric topology, but the subdivision process necessarily leaves the range of simplicial Abelian groups and need not produce a group topology. Instead, we follow a method developed by Cauty in [1]. Theorem. Let G be a locally contractible, not necessarily Abelian group, every open subset of which is a-compact, such that G is the union of a sequence of finite dimensional compact metric spaces and has the homotopy type of a CW-complex. Then G carries a metrizable group topology, coarser than the original one but of the same homotopy type, which turns G into an ANR-space. All these requirements are satisfied, for instance, if G itself is a countable CWcomplex. Proof. We construct a decreasing sequence of open neighborhoods Vn of the identity 1 G G such that 1. nn-c=l i n = { 1}, 2. the inclusion map Vn+1 c* Vin is null homotopic, 3. if Hm: Vm+i x I -* Vm is the null homotopy from 2, then for any g G Vm+i and each n > m there exist N > n and E > 0 such that g' E gVN, It'-tj Hm (g/, t') (E Hm (g, t) Vn, 4.Vn-1 =?Vn andVn+,Vn+l CVn, 5. for each n and every g E G there exists N > n with g-1VNg C Vn. Received by the editors April 1, 1996. 1991 Mathematics Subject Classification. Primary 54H11, 54C55, 22A05.