Homology (mathematics)
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely related usages. First, the most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups. Secondly, as chain complexes are obtained from various other types of mathematical objects, this operation allows one to associate various named homologies or homology theories to these objects. Finally, since there are many homology theories for topological spaces that produce the same answer, one also often speaks of the homology of a topological space. There is also a related notion of the cohomology of a cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space.
Homology of chain complexes
We start with a chain complex, which is a sequence of abelian groups and group homomorphisms :such that the composition of any two consecutive maps is zero:
The th group of cycles,, is given by the kernel subgroup
and the th group of boundaries,, is given by the image subgroup
The th homology group of this chain complex is then the quotient group of cycles modulo boundaries.
One can endow chain complexes with additional structure: for example, taking the groups to be modules over a coefficient ring and taking the boundary maps to be -module homomorphisms results in homology groups that are also quotient modules.
Tools from homological algebra can be used to relate homology groups of different chain complexes.
Homology theories
To associate a homology theory to other types of mathematical objects, one first gives a prescription for associating chain complexes to that object, and then takes the homology of such a chain complex. For the homology theory to be valid, all such chain complexes associated to the same mathematical object must have the same homology. The resulting homology theory is often named according to the type of chain complex prescribed. For example, singular homology, Morse homology, Khovanov homology, and Hochschild homology are respectively obtained from singular chain complexes, Morse complexes, Khovanov complexes, and Hochschild complexes. In other cases, such as for group homology, there are multiple common methods to compute the same homology groups.In the language of category theory, a homology theory is a type of functor from the category of the mathematical object being studied to the category of abelian groups and group homomorphisms, or more generally to the category corresponding to the associated chain complexes. One can also formulate homology theories as derived functors on appropriate abelian categories, measuring the failure of an appropriate functor to be exact. One can describe this latter construction explicitly in terms of resolutions, or more abstractly from the perspective of derived categories or model categories.
Regardless of how they are formulated, homology theories help provide information about the structure of the mathematical objects to which they are associated, and can sometimes help distinguish different objects.
Homology of a topological space
Perhaps the most familiar usage of the term homology is for the homology of a topological space. For sufficiently nice topological spaces and compatible choices of coefficient rings, any homology theory satisfying the Eilenberg-Steenrod axioms yields the same homology groups as the singular homology of that topological space, with the consequence that one often simply refers to the "homology" of that space, instead of specifying which homology theory was used to compute the homology groups in question.For 1-dimensional topological spaces, probably the simplest homology theory to use is graph homology, which could be regarded as a 1-dimensional special case of simplicial homology, the latter of which involves a decomposition of the topological space into simplices. Simplicial homology can in turn be generalized to singular homology, which allows more general maps of simplices into the topological space. Replacing simplices with disks of various dimensions results in a related construction called cellular homology.
There are also other ways of computing these homology groups, for example via Morse homology, or by taking the output of the Universal Coefficient Theorem when applied to a cohomology theory such as Čech cohomology or De Rham cohomology.
Inspirations for homology (informal discussion)
One of the ideas that led to the development of homology was the observation that certain low-dimensional shapes can be topologically distinguished by examining their "holes." For instance, a figure-eight shape has more holes than a circle, and a 2-torus has different holes from a 2-sphere .Studying topological features such as these led to the notion of the cycles that represent homology classes. For example, the two embedded circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus and 2-sphere represent 2-cycles. Cycles form a group under the operation of formal addition, which refers to adding cycles symbolically rather than combining them geometrically. Any formal sum of cycles is again called a cycle.
Cycles and boundaries (informal discussion)
Explicit constructions of homology groups are somewhat technical. As mentioned above, an explicit realization of the homology groups of a topological space is defined in terms of the cycles and boundaries of a chain complex associated to, where the type of chain complex depends on the choice of homology theory in use. These cycles and boundaries are elements of abelian groups, and are defined in terms of the boundary homomorphisms of the chain complex, where each is an abelian group, and the are group homomorphisms that satisfy for all.Since such constructions are somewhat technical, informal discussions of homology sometimes focus instead on topological notions that parallel some of the group-theoretic aspects of cycles and boundaries.
For example, in the context of chain complexes, a boundary is any element of the image of the boundary homomorphism, for some. In topology, the boundary of a space is technically obtained by taking the space's closure minus its interior, but it is also a notion familiar from examples, e.g., the boundary of the unit disk is the unit circle, or more topologically, the boundary of is.
Topologically, the boundary of the closed interval is given by the disjoint union, and with respect to suitable orientation conventions, the oriented boundary of is given by the union of a positively oriented with a negatively oriented The simplicial chain complex analog of this statement is that.
In the context of chain complexes, a cycle is any element of the kernel, for some. In other words, is a cycle if and only if. The closest topological analog of this idea would be a shape that has "no boundary," in the sense that its boundary is the empty set. For example, since, and have no boundary, one can associate cycles to each of these spaces. However, the chain complex notion of cycles is more general than the topological notion of a shape with no boundary.
It is this topological notion of no boundary that people generally have in mind when they claim that cycles can intuitively be thought of as detecting holes. The idea is that for no-boundary shapes like,, and, it is possible in each case to glue on a larger shape for which the original shape is the boundary. For instance, starting with a circle, one could glue a 2-dimensional disk to that such that the is the boundary of that. Similarly, given a two-sphere, one can glue a ball to that such that the is the boundary of that. This phenomenon is sometimes described as saying that has a -shaped "hole" or that it could be "filled in" with a.
More generally, any shape with no boundary can be "filled in" with a cone, since if a given space has no boundary, then the boundary of the cone on is given by, and so if one "filled in" by gluing the cone on onto, then would be the boundary of that cone. However, it is sometimes desirable to restrict to nicer spaces such as manifolds, and not every cone is homeomorphic to a manifold. Embedded representatives of 1-cycles, 3-cycles, and oriented 2-cycles all admit manifold-shaped holes, but for example the real projective plane and complex projective plane have nontrivial cobordism classes and therefore cannot be "filled in" with manifolds.
On the other hand, the boundaries discussed in the homology of a topological space are different from the boundaries of "filled in" holes, because the homology of a topological space has to do with the original space, and not with new shapes built from gluing extra pieces onto. For example, any embedded circle in already bounds some embedded disk in, so such gives rise to a boundary class in the homology of. By contrast, no embedding of into one of the 2 lobes of the figure-eight shape gives a boundary, despite the fact that it is possible to glue a disk onto a figure-eight lobe.
Homology groups
Given a sufficiently-nice topological space, a choice of appropriate homology theory, and a chain complex associated to that is compatible with that homology theory, the th homology group is then given by the quotient group of -cycles modulo -dimensional boundaries. In other words, the elements of, called homology classes, are equivalence classes whose representatives are -cycles, and any two cycles are regarded as equal in if and only if they differ by the addition of a boundary. This also implies that the "zero" element of is given by the group of -dimensional boundaries, which also includes formal sums of such boundaries.Informal examples
The homology of a topological space X is a set of topological invariants of X represented by its homology groupswhere the homology group describes, informally, the number of holes in X with a k-dimensional boundary. A 0-dimensional-boundary hole is simply a gap between two components. Consequently, describes the path-connected components of X.
A one-dimensional sphere is a circle. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes. The corresponding homology groups are given as
where is the group of integers and is the trivial group. The group represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.
A two-dimensional sphere has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes. The corresponding homology groups are
In general for an n-dimensional sphere the homology groups are
A two-dimensional ball is a solid disc. It has a single path-connected component, but in contrast to the circle, has no higher-dimensional holes. The corresponding homology groups are all trivial except for. In general, for an n-dimensional ball
The torus is defined as a product of two circles. The torus has a single path-connected component, two independent one-dimensional holes and one two-dimensional hole as the interior of the torus. The corresponding homology groups are
If n products of a topological space X is written as, then in general, for an n-dimensional torus,
.
The two independent 1-dimensional holes form independent generators in a finitely generated abelian group, expressed as the product group
For the projective plane P, a simple computation shows :
corresponds, as in the previous examples, to the fact that there is a single connected component. is a new phenomenon: intuitively, it corresponds to the fact that there is a single non-contractible "loop", but if we do the loop twice, it becomes contractible to zero. This phenomenon is called torsion.