G-surgery on 3-dimensional manifolds for homology equivalences

For a finite group G and a G-map f : X → Y of degree one, where X and F are compact, connected, oriented, 3-dimensional, smooth G-manifolds, we give an obstruction element σ(f) in a K-theoretic group called the Bak group, with the property: σ(f) = 0 guarantees that one can perform G-surgery on X so as to convert f to a homology equivalence f′ : X′ → Y. Using this obstruction theory, we determine the G-homeomorphism type of the singular set of a smooth action of A₅ on a 3-dimensional homology sphere having exactly one fixed point, where A₅ is the alternating group on five letters.