algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety reductive group on affine space; variety diffeomorphic to affine space; fixed point; groups which act smoothly on homotopy spheres with exactly one fixed point; icosahedral group; real algebraic action; G-cobordant to a real algebraic G-variety; G-diffeomorphic to a real algebraic G- variety; Poincaré homology sphere; equivariant surgery 