Homology class

In mathematics, homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defining and categorizing holes in a manifold. Loosely speaking, a cycle is a closed submanifold, a boundary is a cycle which is also the boundary of an (open or closed) submanifold and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries. A non-trivial equivalence class is thus represented by a cycle which is not the boundary of any submanifold. A hypothetical manifold whose boundary would be that particular cycle is "not there" which is why that cycle is indicative of the presence of a hole.

There are many different homology theories. A particular type of mathematical object, such as a topological space or a group, may have one or more associated homology theories. When the underlying object has a geometric interpretation like topological spaces do, the nth homology group represents behavior unique to dimension n. In general, most homology groups or modules arise as derived functors on appropriate abelian categories. They provide concrete descriptions of the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a derived category.

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