homologische

Homologische, in mathematical contexts, refers to concepts, methods, and invariants associated with homology theory. Homology assigns algebraic objects, typically abelian groups or modules, to spaces or more general algebraic structures in order to detect holes of various dimensions and to classify their shape in a way that is invariant under suitable deformations.

The fundamental construction uses chain complexes. A chain complex consists of a sequence of objects C_n connected by boundary maps d_n: C_n → C_{n-1} with the property that d_n ∘ d_{n+1} = 0. The n-th homology group is H_n = Ker(d_n) / Im(d_{n+1}); elements of H_n represent n-dimensional features that are not boundaries of higher-dimensional ones. Homology is functorial: a structure-preserving map between spaces induces a homomorphism between their homology groups, and homology is invariant under homotopy, hence a robust topological invariant.

Several concrete homology theories illustrate the field. Singular homology uses all continuous maps from simplices into a space; simplicial and cellular homology arise from decompositions of spaces into simplices or cells; de Rham cohomology provides a dual, differential-form-based perspective. Cohomology theories, while dual to homology, share many formal properties and enable additional structures such as cup products.

Key tools include long exact sequences, such as those of a pair (X,A) or via the Mayer–Vietoris sequence, which relate the homology of a space to that of its subspaces. In algebraic contexts, homological methods are extended to chain complexes in abelian categories, leading to derived functors like Ext and Tor and to powerful computational devices such as spectral sequences.

Applications span topology, algebraic geometry, and data analysis (notably persistent homology). The wider field focusing on these ideas is called homological algebra, while homology itself provides the invariants that these methods study.