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

Replacement of fixed sets for compact group actions: The 2\rho theorem