Multiple-sheeted Spaces and Manifolds of States of Motion* By

Every closed orientable manifold can be represented topologically by a generalized Riemann surface consisting of a number of spaces, each with a single point at infinity, which are considered superimposed and which are joined with each other along certain branch cuts of dimensionality one less than that of the manifold. This has been proved by Professor J. W. Alexander, f who also pointed out that the branch cuts' boundaries may be taken as non-singular non-intersecting manifolds. But the straightforward reduction of a manifold otherwise defined to such a form is likely to be very tedious and to yield a space having an unnecessarily large number of sheets. Consider for example the threedimensional manifolds of states of motion used by Poincaré and Birkhoff in connection with dynamical problems having two degrees of freedom and studied topologically by the present writer in a paperj referred to hereinafter as A. In following the proof by Alexander of the existence theorem mentioned, the most obvious way is to represent the manifold by a polyhedron as in A, and to divide each face into triangles and the manifold into tetrahedra having a common vertex and these triangles as bases. The vertices of each tetrahedron must aU be distinct points. The manifold of states of motion on a surface of genus p is in this way divided into some 24^ — 4 tetrahedra. After fitting together all these cells to form a multiple-sheeted space there will remain the not inconsiderable task of so altering the branch system as to get rid of singularities. An attempt to carry through these operations is enough to show the need of shorter methods in applications. With the aid of the theorems of §§ 1 to 6, multiple-sheeted spaces with non-singular branch systems can easily be set up for many three-dimensional manifolds, including manifolds of states of motion, orientable product manifolds, and all manifolds of genus unity. The application to manifolds