ZweierkomplementDarstellung

Zweierkomplexe, in German mathematical usage often referred to as 2-complexes, are a basic object in combinatorial and algebraic topology. They are 2-dimensional cellular spaces built by starting from a graph (the 1-skeleton) and attaching 2-dimensional disks along closed edge-paths.

A 2-complex X consists of a set V of vertices, a set E of oriented edges with incidence data, and a set F of 2-cells (faces). Each 2-cell is attached along its boundary circle to the 1-skeleton via a combinatorial attaching map, typically specified by a closed word in the edges. In this way X is obtained by gluing polygons along their edges according to the chosen identifications.

Relation to other concepts: every 2-complex is a CW complex of dimension at most two. If all attaching maps are given by simplicial identifications, the resulting space may be viewed as a simplicial complex. 2-Complexes provide a convenient framework for modeling surfaces and other spaces in two dimensions and for performing algebraic computations.

Invariants and computations: the Euler characteristic χ of a 2-complex is χ = |V| − |E| + |F|. The fundamental group π1(X) can be described by a presentation with generators corresponding to the edges and relations given by the attaching maps of the 2-cells. Cellular homology groups Hn(X) are computed from the cellular chain complex, yielding information about the shape's holes in each dimension.

Examples and applications: the torus and the projective plane can be described as 2-complexes obtained by gluing edges of polygons. More generally, 2-complexes are used to study group presentations, surface topology, and various constructions in low-dimensional topology. See also CW complex, simplicial complex, fundamental group, and homology.