Topology of 2-dimensional Complexes

We present results concerning geometric topology of 2-dimensional polyhedra, with emphasis on those related to the problem of embeddability into 3-manifolds (and thickenings), the problem of resolving arbitrary 2-polyhedra by fake surfaces (including an application to reduce the classical Whitehead asphericity conjecture to special polyhedra) and existence of nonhomeomorphic 3-manifolds with equivalent spines. We also consider different regular neighborhoods of codimension 2 embeddings of polyhedra into manifolds of any dimension. A special section is devoted to algebraic topology of 2-polyhedra, cohomology of groups and universal covers. 1. ON EMBEDDABILITY OF 2-POLYHEDRA INTO 3-MANIFOLDS In this section, we present the basic results on embeddability of 2-polyhedra into 3-manifolds. We omit 2-coefficients from the notation of (co)homology groups. In our notation and terminology, we follow [43]. Throughout this paper we shall work in the PL category. By [4] the same results hold in the topological category. A vertex of a graph is hanging if its degree is one. An edge of a graph is hanging if one of its endpoints is hanging. A link of a point of X is its link in some sufficiently small triangulation of X . 1.1. Fake surfaces and special 2-polyhedra. A finite 2-polyhedron Q is called a fake surface if each of its points has a neighborhood homeomorphic to one of the following: D2, the book with three pages (T I), or the cone over the complete graph with four vertices (or over the 1-skeleton of the 3-simplex). See Figure 1.1 for an illustration of these three types of neighborhoods. We will refer to points in fake surfaces as points of type 1, 2 and 3, respectively depending on which of the above three neighborhoods they have. Soap films in 3 exhibit singularities precisely of types 2 and 3. The notion of soap films from differential geometry has proved to be an important tool and object of investigation in algebraic and geometric topology. 2000 Mathematics Subject Classification. Primary: 57M20, 57N10; Secondary: 57N35, 57N40.