Classification of finite 2-complexes
E. F. Whittlesey · Proceedings of the American Mathematical Society · 1958
We shall set up, for finite 2-complexes, a complete system of topological invariants, partly numerical and partly of order. The essential part of the classification problem is the characterization of the neighborhoods of the points. The singular points constitute a linear graph whose complement is a collection of bounded 2-manifolds. The knowledge of local structure then enables us to describe the manner in which the boundaries of the manifolds are woven into the singular graph to reassemble the complex. At first, we shall suppose that we are given a connected finite simplicial 2-complex, K, wherein every edge is a face of some 2simplex. (Later, the last restriction will be removed.) Let P be an arbitrary point of K, and let st(P) be the star of the open simplex containing P. There are four possibilities for this neighborhood of P. (1) St(P) is homeomorphic to the euclidean plane. Then P is called regular. The points of K which are not regular are called singular. (2) St(P) is topologically equivalent to the space obtained by identifying the x-axes of a certain n (#2) copies of the closed euclidean half-plane y>0. Then P is called line-singular and a neighborhood, or space, of this sort is a book. (3) St(P) is topologically equivalent to the space obtained by identifying the origins of a certain m (>1) copies of the euclidean plane. Then P is called a conical point, and a neighborhood of this sort is a cone. In both cases, (2) and (3), the regular part of st(P) falls into components called leaves of the book or cone, respectively. (4) A singular point which is neither conical nor line-singular is a node. In this case there are certain singular edges with P as a vertex. The regular part of st(P) falls into components each of which is a cone leaf or is, topologically, an open triangle with P as vertex and with two singular edges, with P as vertex, which may be distinct or they may coincide. If the edges are distinct, the component is called a fan, if they coincide, the component is called a cornet. Thus, the neighborhood st(P) of a node P consists of a certain number ( > 0) of leaves of a cone and a certain number of fans and cornets. To specify st(P) one need only specify the number of cone leaves, the number of singular edges with P as vertex, and the number of fans and/or cornets and specify which singular edges belong to which fans and/or cornets. There are obviously restrictions: there must be some fans or