  If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed
sets M^G and N^G are also homotopy equivalent. The replacement problem asks the
converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G
for a G-manifold equivariantly homotopy equivalent to M? We prove that for
locally linear actions on topological or PL manifolds by compact Lie groups,
the replacement is always possible if the normal bundle of the fixed set is
twice of a complex bundle over a 1-skeleton of the fixed set. Moreover, we also
study some specific examples, where the answer to the replacement problem
ranges from always possible to the rigidity.
