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

We study the problem in transformation groups of realizing a manifold simple homotopy equivalent to a component of the fixed point set of a G-manifold as (a component of) the fixed set of another G-manifold equivariantly simple homotopy equivalent to the original one. We show that such replacability of (a component of) the fixed points is very often possible if the normal representation of the fixed point component is twice a complex representation (and a monodromy vanishes). In addition, we discuss for various compact groups some examples displaying topological phenomenona ranging from replacablility to rigidity.