Abstract：Consider an undirected simple connected graph, in which each vertex is replaced by a manifold (a pipe) and each edge is replaced by the Cartesian product of this manifold (a circlar section). The topological space obtained in this way is called a graphlike manifold. The undirected simple graph is called the contraction of graphlike manifolds correspondingly. If the circlar sections of graphlike manifolds have different maps, we will have different graphlike manifolds. There are infinite graphlike manifolds, and the number of homeomorphic classes is also hard to count, nearly infinity. The problem of counting the numbers of homeomorphics classes of graphlike manifolds, and give each homeomorphics classes a representive graphlike manifold, is just the case for the topological classification of graphlike manifolds. This paper discusses the topological classification of graphlike manifolds with the contraction of Wn and the number of homeomorphic classes of graphlike manifolds of Wn. A representative system is formed by all non-isomorphic colorings, and the necessary cases are counted. According to the graph colouring theory and twist operation, the homeomorphic classes of grapglike manifolds W8 and W9 are found to be 18 and 30, respectively.
2010, Vol. 28 
