Euclidean symmetry of closed surfaces immersed in 3-space (Q260547)

If \(G\) is a group of orientation-preserving isometries of Euclidean 3-space \(E^3\) and \(S\) a closed surface, a map \(f: S\to E^3\) is a \(G\)-general position immersion if every point in \(S\) has a disk neighborhood which is embedded with images in general position, the singular points are disjoint from axes of rotation of \(G\) and \(f(S)\) is \(G\)-invariant. The action of \(G\) on \(f(S)\) induces a pseudo-free action of \(G\) on \(f(S)\) with regular branched covering \(f: S\to T\).NEWLINENEWLINE This paper classifies those Riemann-Hurwitz equations coming from pseudo free actions of a group \(G\) of this type which are realizable by \(G\)-general immersions. The possibilities for \(G\) are the cyclic and dihedral groups of order \(n\) for \(n\)-fold rotation and symmetry of the \(n\)-prism, respectively, and \(A_4\), \(S_4\) and \(A_5\) for symmetry of the tetrahedron, cube and icosahedron, respectively. If the Riemann-Hurwitz equation has the properties that all branch points have full order, constant parity of number of branch points of each type, \(G\) has a subgroup of index two containing all branching and \(\chi(T)\) is even if \(S\) is orientable and \(T\) is not, then it is realizable except when \(\chi(T)\geq -2\) and there is insufficient branching.