Labelings Of Some Two-dimensional Complexes And The Application Of Sperner Type Lemma

The labeling problem of complex is a classical problem in combinatorial algebraic topology,it is widely used in digital image processing,genetic engineering,geographic information system and other fields.In this paper,we regard the vertices of the complex K as the labeling set of the complex M and define a kind of vertex labeling of two-dimensional complex-K labeling,and then equate the vertex labeling of complex with the simplicial mapping between the two complexes.We give vertex labeling of some two-dimensional complexes and obtain some results of K labeling extension of two-dimensional complex by means of homotopy extension.The classical Sperner lemma is concerned for its equivalence with the famous Brouwer fixed point theorem.Sperner lemma discusses the Sperner labeling problem of the trian-gulation of simplex and the KyFan lemma considers the problem of the vertex labeling of the centrally symmetric triangulation of the sphere.In this paper,we generalize Sperner labeling,define vertex double labeling of triangulation of planar convex polygon,and obtain some conclusions of Sperner type lemma.This paper is divided into three parts.The first part of this paper introduces some basic knowledge of topology and algebra,and gives the definition of K labeling.In the second part,we study the special K labeling triangulation(?)?2 labeling triangulation of some closed surfaces and its extension.The third part introduces the related content of Sperner lemma,defines the double labeling of triangulation of the convex polygon and studies the application of the labeling of two-dimensional complex in Sperner type lemma.