Replacement of fixed sets for compact group actions: The 2ρtheorem

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If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets Mᵍ and Nᵍ are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set Mᵍ, is F = Nᵍ 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.