Topological abelian groups and equivariant homology Marcelo A. Aguilar & Carlos Prieto ∗ Instituto de Matem´ aticas, UNAM, 04510 M´ exico, D.F., Mexico Email: marcelo@math.unam.mx, cprieto@math.unam.mx Abstract We prove an equivariant version of the Dold-Thom theorem by giving an explicit isomorphism between Bredon-Illman homology  H G * (X ; L) and equivariant homotopical homology π * (F G (X, L)), where G is a finite group and L is a G -module. We use the homotopical definition to obtain several properties of this theory and we do some calculations. AMS Classification 55N91; 55P91,14F43 Keywords Equivariant homology, equivariant homotopy, coefficient sys- tems 0 Introduction The presentation of homology using the Dold-Thom construction has been very useful in algebraic geometry. Lawson homology (see [14, 15, 9, 10]) was defined using this approach. In this paper we study the Dold-Thom-McCord theorem (see [19]) in the equivariant case. Let G be a finite group. If L is a G-module, then one can define a coefficient system L on the category of canonical orbits of G by L(G/H )= L H , where L H is the subgroup of fixed points of L under H ⊂ G. One then has an ordinary equivariant homology theory H G ∗ (−; L), called Bredon-Illman homology, whose associated coefficient system is precisely L . Let X be a (pointed) G-space and let F (X,L) be the topological abelian group generated by the points of X , with coefficients in L . Consider the subgroup F G (X,L) of equivariant elements, that is, the elements ∑ l x x in F (X,L) such that l gx = g · l x . Then one can associate to X the homotopy groups π q (F G (X,L)), and one has that, if X is a G- CW-complex, then  H G q (X; L) is isomorphic to π q (F G (X,L)). When G is the trivial group, H G ∗ (−; L) is singular homology and this statement is the classical Dold-Thom theorem [7], which was extended to the equivariant case by Lima- Filho [16] (when L = Z with trivial G-action) and by dos Santos [21] (when ∗ Corresponding author, Phone: ++5255-56224489, Fax ++5255-56160348. This au- thor was partially supported by PAPIIT grant IN110902 and by CONACYT grant 43724. 1