Homologisch

Homologisch is a German adjective describing concepts related to homology, a central construction in algebraic topology and homological algebra. Homology assigns algebraic invariants to topological spaces, capturing information about their shape that is invariant under continuous deformations.

The basic objects are chain complexes Cn with boundary maps ∂n: Cn → Cn−1. The n-th homology group Hn(X) is defined as ker ∂n / im ∂n+1. Different flavors exist—singular, simplicial, and cellular homology—depending on how chains are formed, but they share the same formal pattern.

Examples: for the circle S1, H0 ≅ Z and H1 ≅ Z, with other groups zero; for the 2-sphere S2, H0 ≅ Z and H2 ≅ Z. Relative and reduced variants generalize these constructions to pairs (X, A) and quotient spaces. Variants include cellular and simplicial homology, which facilitate computations in specific settings.

Homology theories are functorial, yield long exact sequences for pairs (X, A) and admit Mayer–Vietoris sequences for decompositions. They are related to cohomology via dual constructions and play a key role in algebraic topology, algebraic geometry, and computational topology (for example, persistent homology in data analysis).

Historically, homology was introduced by Henri Poincaré and later axiomatized in the framework of Eilenberg and Steenrod. The term derives from Greek homologia, meaning agreement or correspondence, reflecting the idea of consistent counting across dimensions. Today, “homologisch” denotes the German usage of these ideas and results.