HomologieTheorien

HomologieTheorien, or homology theories, are a fundamental concept in algebraic topology. They provide algebraic invariants of topological spaces that are more powerful than simple homology groups. Instead of assigning a single abelian group to each dimension, homology theories assign a sequence of abelian groups, known as the homology groups, to each topological space. These groups capture information about the "holes" of different dimensions within a space. A key feature of homology theories is that they satisfy certain axioms, analogous to those for singular homology. These axioms include the dimension axiom, which states that the homology groups of a point are trivial except in dimension zero, where it is isomorphic to the integers. Another crucial axiom is the exactness axiom, which relates the homology groups of a space to those of its subspaces. Furthermore, homology theories are required to be functorial, meaning they preserve continuous maps between topological spaces. The category of topological spaces and continuous maps is mapped to the category of graded abelian groups and chain maps. A significant result is the Eilenberg-Steenrod theorem, which states that any homology theory satisfying these axioms and the dimension axiom is naturally isomorphic to singular homology. However, there exist generalized homology theories that do not satisfy the dimension axiom, such as K-theory or cobordism theory. These generalized theories are often more flexible and can be applied to a wider range of topological and geometric problems.