subcomplex

A subcomplex is a related object that inherits structure from a larger complex in algebraic topology. In the context of a simplicial complex K, a subcomplex K′ is another simplicial complex whose simplices are a subset of those of K and which is closed under taking faces. Equivalently, whenever a simplex is in K′, all of its faces are also in K′. The geometric realization of K′ is a subspace of the realization of K. Subcomplexes may be formed by selecting a subset of vertices and taking all simplices whose vertices lie in that subset, or more generally by selecting a collection of simplices that is closed under faces.

A related notion occurs for chain complexes. A subcomplex C′ of a chain complex C consists of subgroups C′n ≤ Cn for every degree n, such that the boundary maps send C′n into C′n−1. With this, C′ is itself a chain complex, and there is a short exact sequence 0 → C′ → C → C/C′ → 0. The quotient C/C′ is the relative chain complex, and its homology Hn(C/C′) is the relative homology Hn(C, C′). Inclusion maps induce homomorphisms on homology, and subcomplexes are central to long exact sequences relating absolute and relative homology.

Subcomplexes also appear in other settings, such as CW complexes, where a subcomplex is a sub-CW complex formed by a union of cells closed under the attaching maps. In all contexts, subcomplexes provide a means to study a larger object by focusing on a coherent, smaller piece that retains the essential combinatorial or algebraic structure.