On the Class of Real-Analytic, Sense-Preserving Mappings Between Compact Surfaces.
Susan Lacey Tucker · Deep Blue (University of Michigan) · 1980
Let M and N be compact, connected, oriented, real-analytic manifolds of the same dimension. A differentiable mapping f:M(--->)N is called sense-preserving if the derivative mapping Df(p):T(,p)(M)(--->)T(,f(p))(N) is either singular or is non-singular and preserves the orientations on the tangent spaces. Let SP('(omega))(M,N) denote the class of real-analytic, sense preserving mappings from M into N. Such a mapping may fail to be locally finite-to-one. In the case where dim M=dim N=2, every mapping in SP('(omega))(M,N) which is finite-to-one is a branched covering map. For an arbitrary mapping f in SP('(omega))(M('2),N('2)), necessary and sufficient conditions are described for the C('0)-approximation of f by continuous (sense-preserving) branched covering maps. Furthermore, it is shown that every branched covering map of positive degree from M('2) onto N('2) can be approximated by mappings in SP('(omega))(M('2),N('2)) which have only finitely many singular points, each of rank zero. In fact, the approximating mappings may be chosen so that they behave locally like the complex mapping "z(--->)z('2)" at each singular point. As a corollary to the above results, necessary and sufficient conditions are obtained for the C('0)-approximation of a mapping in SP('(omega))(M('2),N('2)) by real-analytic branched covering maps. For mappings in the class SP('(omega))(M('2),N('2)), a relation is derived which is similar to the classical Riemann-Hurwitz relation for branched covering maps. This relation imposes certain necessary conditions on the triple (M, N, d) for the existence of a mapping f in SP('(omega))(M,N) of degree d. As an application of the above, it is observed that for certain triples (M, N, d), in particular for M=N=the torus and d (GREATERTHEQ) 1, or for (chi)(M) (LESSTHEQ) (chi)(N) (LESSTHEQ) -2 and d = (chi)(M)/(chi)(N), every mapping in SP('(omega))(M,N) of degree d must be a covering map, and hence is finite-to-one, even though the mapping may be singular on a set of codimension 1 in M. Furthermore, for certain pairs (M, N), every mapping in SP('(omega))(M, N) must have degree 0, and hence is everywhere singular. Several observations are made concerning the structure of the space SP('(omega))(M('2),N('2)), endowed with the C('0)-topology. In particular, the subset of branched covering maps is open in SP('(omega))(M('2),N('2)), and the earlier approximation result allows a complete characterization of the boundary of the set of branched covering maps in SP('(omega))(M,N). As a tool in proving results concerning the class SP('(omega))(M,N), an extension result is developed for the class of continuous branched covering maps. If f: (PAR-DIFF)M(--->)S('1) is a sense-preserving local homeomorphism from the boundary of the compact, oriented 2-manifold onto the unit circle, and if deg f (GREATERTHEQ) 2, then f may be extended to a branched covering map from M onto the disc D('2) such that f('-1)(S('1)) = (PAR-DIFF)M. If deg f = 1, then the extension exists if and only if M is homeomorphic to the disc.