Singulärkomplexen

Singulärkomplexen, a term originating from German, translates to singular complexes in English. In mathematics, particularly in algebraic topology, a singular complex is a fundamental construction used to define and study topological spaces. It is derived from a topological space by considering all possible continuous maps from standard simplices into that space.

A standard simplex is a basic geometric building block, essentially a triangle in 2D, a tetrahedron in 3D, and their higher-dimensional generalizations. A singular simplex in a topological space X is defined as a continuous map from a standard simplex (often denoted as Δn for an n-simplex) to X. The set of all singular n-simplices in X, denoted as S_n(X), forms a free abelian group generated by these maps.

The collection of these groups, {S_n(X)}_{n≥0}, together with boundary operators that map singular (n)-simplices to linear combinations of singular (n-1)-simplices, forms a chain complex. This chain complex is called the singular chain complex of X. The homology groups derived from this singular chain complex are known as singular homology groups. These groups are powerful invariants of topological spaces, meaning that if two spaces are homeomorphic, they will have the same singular homology groups. This property makes them incredibly useful for distinguishing between different topological spaces. The construction of singular complexes is a cornerstone of homology theory.