Manifolds

A non orientable two dimensional manifold M cannot be im mersed in a Euclidean space of less than four dimensions. It may, however, be represented schematically by a plane region R bounded by a suitable finite number q of non intersecting circles, where pairs of opposite points on the various bounding circles are each to be thought of as representing a single point of the mani fold. In four dimensional space it would be possible to reconstruct a model of the manifold M by deforming the region R in such a manner as to bring into coincidence opposite points on the bound ing circles. The connectivity of tne manifold M is k=q+I, so that, in this case, k may be any integer greater than unity. The plane of real, projective geometry (with a line of points at infin ity) is a non orientable manifold of the simplest type, with k equal to two.

Combinatorial

very effective way of studying manifolds is the so-called combinatorial one. Consider a convex polyhedral surface in ordinary space. Regarded as a set of points, the surface is an ordinary manifold of genus zero of the same type as the sphere. But the surface may also be regarded as a collec tion of vertices, edges and faces. Let the number of these vertices, edges and faces be a,, and a, respectively. We then find that a relation of the form (3) = 2 is always satisfied, irrespective of the polyhedron chosen. Now, in reality, relation (3) has an underlying topological significance. Let us use the terms and to denote figures homeo morphic with the vertices, edges and faces respectively of a con vex polyhedron. Then if we take any two dimensional manifold i M and subdivide it in a perfectly arbitrary manner into a finite number of cells we always obtain the relation (4) = where, this time, a, and a2 denote the number of o-, 1- and 2-cells of the subdivision respectively. Relation (3) is a special case of relation (4), where we have k=i, and where the cells are all straight. The number ai+a, is called the Euler number of the manifold M. In view of relation (4), a knowledge of the Euler number of a manifold is equivalent to a knowledge of the connectivity number k.

Manifolds of more than two dimensions were considered by Riemann and Betti, but Poincare was the real founder of the higher dimensional theory. Poincare extended to n dimensions the combinatorial method of cellular subdivision described above and discovered a number of new topological invariants. Of these last, the most important ones are his "Betti numbers" P1, . . P„_, which are generalizations of the connectivity number k. The Betti numbers satisfy a duality relation which reduces to (4) in the two dimensional case. Alexander has shown that the invariants of Poincare are insufficient to fix the type of a higher dimensional manifold, and has found further in variants which appear, also, to be insufficient. Heegaard has studied the theory of three dimensional manifolds by a somewhat different method which would, no doubt, bear further exploitation.

An important class of problems has to do with the possible types of continuous transformations of a manifold into itself, and with the existence of points left invariant by these transformations. Brouwer has studied the transformations of a sphere of n dimen sions and has shown, in particular, that a one-to-one sense pre serving transformation of a two dimensional sphere always leaves at least one point invariant. Lefschetz has obtained a very general theorem about the fixed points of an arbitrary transformation of an arbitrary n-dimensional manifold. Morse, Nielsen and others have made noteworthy contributions to the theory. Numerous ap plications are to be found for this branch of analysis situs. For example, to take a very simple case, the so-called fundamental theorem of algebra to the effect that every algebraic equation has a root obviously reduces to the theorem that the transforma tion has a fixed point. Birkhoff and Kellogg have proved that extremely general systems of differential and integral equations admit solu tions by studying the transformations of a sphere of infinitely many dimensions. (See also KNOTS, ANALYSIS SITUS.)