Icosahedral group actions on \(\mathbb{R}{}^ 3\)
The main result of this paper is Theorem B: Every smooth action of a compact Lie group \(G\) on \(\mathbb{R}^ 3\) or \(D^ 3\) is differentiably equivalent to a linear action. A similar conclusion holds for smooth actions on \(S^ 3\) with nonempty fixed point sets. The theorem had been well-known for actions of positive-dimensional compact Lie groups and was proved earlier for orientation-preserving actions of finite groups non-isomorphic to \(A_ 5\) in [\textit{W. H. Meeks III} and \textit{S.-T. Yau}, Chapt. IX in ``The Smith conjecture'', Pure Appl. Math. 112, 167-180 (1984; Zbl 0599.57007)]. Hence, the authors have to prove the main result for the group \(A_ 5\) and to extend the techniques to the non-orientation preserving case. Using a stabilization with copies of the product cobordism \(T^ 2\times [0,1]\) on an induced action on \(S^ 2\times [0,1]\), the authors show that a smooth \(A_ 5\)-action on \(S^ 3\) with a nonempty fixed point set has an equivariant Heegaard splitting of genus \(60=| A_ 5|\). If one fixed point is removed from \(S^ 3\) the induced action is then shown to be equivariantly diffeomorphic to an invariant open neighborhood of the origin in \(\mathbb{R}^ 3\) with an orthogonal action. Standard cancellation techniques then show the topological equivalence to an orthogonal action. Essentially by proving the uniqueness of equivariant smoothings of 3-manifolds, smooth equivalence to an orthogonal action can be concluded, too.
scientific article; zbMATH DE number 149679
scientific article