MANIFOLDS. Two geometrical figures are said to be homeo morphic if it is possible to set them into point for point continu ous correspondence with one another. To illustrate : a closed curve (of the simple sort which passes through no point more than once) is homeomorphic with the circumference of a circle; the surface of a sphere is homeomorphic with the surface of an ellip soid or of a cube, but not with a ring shaped surface such as a torus (fig. 1). The topological properties of a figure F are those which are shared by all figures homeomorphic with F. For in stance, one of the topological properties of a line segment is its separability into two pieces by the removal of just one of its points ; a non-topological property of the segment is its length. Analysis Situs (q.v.), or Topology, is the theory of the topological properties of figures. Theorems of analysis situs are very general in char acter and often synthesize results originally obtained in widely separated fields of mathematics.
theory of functions of sev eral variables leads directly to the study of a special class of figures called n-dimensional manifolds. These figures are hard to define with precision in non technical terms. We may say that a one dimensional manifold is a simple, closed curve, a two-dimensional manifold a closed surface without singularities and, broadly speaking, an n-dimensional manifold the generalization to n dimen sions of a closed curve or surface without singularities. One of the important outstanding problems of analysis situs is to classify higher dimensional manifolds into types such that two manifolds are of the same type if, and only if, they are homeomorphic. Up to the present, a complete classification has been carried out only for manifolds of dimensions one and two. The case n=i is trivial : all one-dimensional manifolds are of the same type since every simple, closed curve is homeomorphic with the circumference of a circle. We shall outline below the results of the classification for the case n= 2.
First, it will be necessary to call attention to a rather curiously shaped surface known as a Mobius strip. This surface may be obtained by tak ing a plane rectangular region ABCD, where A and C denote diagonally opposite points, and de forming the region in three di mensional space so as to bring the point A into contact with the point C and the edge AB into contact with the edge CD. The resulting strip will be a belt
shaped surface with a twist in it (fig. 2). It obviously differs in type from an ordinary belt-shaped surface without the twist, since its boundary consists of a single curve, whereas the bound ary of the untwisted surface consists of a pair of curves.
A two dimensional manifold is said to be orientable if no por tion of it is homeomorphic with a Mobius strip;
it is said to be non orientable. The manifold is said to have the connec tivity k if the maximum number of simple, closed curves that may be traced upon it without separating it into two or more pieces is The type of a two dimensional manifold is completely fixed when we know the connectivity of the manifold and whether or not the manifold is orientable.
A model of the most general orientable, two dimensional mani fold M may be obtained by making a target out of a spherical block and shooting a suitable number of bullet holes completely through the block. The sui face of the pierced block will then be homeomorphic with the manifold M. The connectivity of the manifold M is where p is the number of bullet holes
in the target ; consequently. the connectivity of M is always odd in this case. The number p is called the genus of the manifold M. The sphere is of genus zero, the torus of genus one.
notion of the genus of a manifold plays an important role in the theory of algebraic equations of the form (I) F (x, y) = o The solutions (x,y) of an irreducible algebraic equation (I) may always be represented by the points of an orientable two dimen sional manifold of suitable genus p, where p is a function of the expression F(x,y). Many of the properties of equation (I) depend upon the value of the number p. In Riemann's classical theory of the integrals of rational functions, (2)
y)dx, where x and y are connected by a relation of the form (I) the genus of the manifold determined by (I) again comes into evi dence. When p is zero the integral (2) is elementary, in the sense that it may be expressed in terms of rational functions and logarithms; when p is unity, the evaluation of (2) leads to the theory of elliptic functions, and so on.