Making equivariant maps fixed point free

One of the most crucial questions in (Nielsen) equivariant fixed point theory is the following. Let G be a compact Lie group, and X a G-space. Given an equivariant map f : X → X such that for each subgroup H ⊂ G, fH :XH → XH can be deformed to be fixed point free by a compactly fixed homotopy, is it true that f can be deformed by a compactly fixed G-homotopy to a fixed point free map? Furthermore, if the Nielsen numbers N(fH) vanish, is it true that f can be equivariantly deformed to be fixed point free (converse of the Lefschetz property for nG)? Fadell and Wong [Pacific J. Math. 132 (1988) 277-281] gave positive answers to these questions, under the hypothesis that dim(XsH) + 2 ≤ dim(XH) (Codimension Hypothesis). In this paper we prove that if f is isovariant, and the group G is 2-split, then the converse of the Lefschetz Property for nG holds for G-manifolds of dimension ≠ 2. We also give a counter-example that illustrates why in general the Codimension Hypothesis cannot be removed from the main result of Fadell and Wong, unless assuming further hypotheses. © 2001 Elsevier Science B.V. All rights reserved.