Homology (mathematics)

In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely related usages. First, there is the homology of a chain complex, a sequence of abelian groups, called homology groups, which are regarded as fundamental invariants of the chain complex. Secondly, when one can associate a chain complex to a different mathematical object, one can also associate its homology to that object. Distinct procedures of associating chain complexes to a given object are grouped into homology theories. Finally, homology is important in the study of topological spaces. Under nice conditions in which distinct homology theories for a single topological space produce the same homology groups, one can define a single homology of a topological space. This last notion of homology is closely related to topological ideas frequently discussed in popular mathematics such as the holes of a surface or the cycles of a graph. 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.

We start with a chain complex, which is a sequence \((C_\bullet, d_\bullet)\) of abelian groups \(C_{n}\) (whose elements are called chains) and group homomorphisms \(d_n\) (called boundary maps):

\(\cdots \longrightarrow C_{n+1} \stackrel{d_{n+1}}{\longrightarrow} C_n \stackrel{d_n}{\longrightarrow} C_{n-1} \stackrel{d_{n-1}}{\longrightarrow} \cdots\),

such that the composition of any two consecutive maps is zero:

\(d_n \circ d_{n+1}=0.\)

The \(n\)th group of cycles, \(Z_n\), is given by the kernel subgroup

\(Z_n = \ker d_n =\{c \in C_n \,|\; d_n(c) = 0\}\),

and the \(n\)th group of boundaries, \(B_n\), is given by the image subgroup

\(B_n := \mathrm{im}\, d_{n+1} =\{d_{n+1}(c)\,|\; c\in C_{n+1}\}\).

The \(n\)th homology group \(H_{n}\) of this chain complex is then the quotient group \(H_n = Z_n/B_n\) of cycles modulo boundaries.

One can endow chain complexes with additional structure: for example, taking the groups \(C_n\) to be modules over a coefficient ring \(R\) and taking the boundary maps \(d_n\) to be \(R\)-module homomorphisms results in homology groups \(H_{n}\) that are also quotient modules.

Tools from homological algebra can be used to relate homology groups of different chain complexes.

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.

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 (see below) 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. (Simplices are a generalization of triangles to arbitrary dimension; for example, an edge in a graph is homeomorphic to a one-dimensional simplex, and a triangle-based pyramid is a 3-simplex.) 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 (in the case of real coefficients) De Rham cohomology.

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 \(S^1\), and a 2-torus \(T^2\) (a 2-dimensional surface shaped like an inner tube) has different holes from a 2-sphere \(S^2\) (a 2-dimensional surface shaped like a basketball).

Studying topological features such as these led to the notion of the cycles that represent homology classes (the elements of homology groups). For example, the two embedded circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus \(T^2\) and 2-sphere \(S^2\) 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.

Explicit constructions of homology groups are somewhat technical. As mentioned above, an explicit realization of the homology groups \(H_n(X)\) of a topological space \(X\) is defined in terms of the cycles and boundaries of a chain complex \((C_\bullet, d_\bullet)\) associated to \(X\), 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 \(d_n: C_n \to C_{n-1}\) of the chain complex, where each \(C_n\) is an abelian group, and the \(d_n\) are group homomorphisms that satisfy \(d_{n-1} \circ d_n=0\) for all \(n\).

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 \(B_n := \mathrm{im}\, d_{n+1} :=\{d_{n+1}(c)\,|\; c\in C_{n+1}\}\) of the boundary homomorphism \(d_n: C_n \to C_{n-1}\), for some \(n\). 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 \(D^2\) is \(S^1\).

Topologically, the boundary of the closed interval \([0,1]\) is given by the disjoint union \(\{0\} \, \amalg \, \{1\}\), and with respect to suitable orientation conventions, the oriented boundary of \([0,1]\) is given by the union of a positively oriented \(\{1\}\) with a negatively oriented \(\{0\}.\) The simplicial chain complex analog of this statement is that \(d_1([0,1]) = \{1\} - \{0\}\). (Since \(d_1\) is a homomorphism, this implies \(d_1(k\cdot[0,1]) = k\cdot\{1\} - k\cdot\{0\}\) for any integer \(k\).)

In the context of chain complexes, a cycle is any element of the kernel \(Z_n := \ker d_n :=\{c \in C_n \,|\; d_n(c) = 0\}\), for some \(n\). In other words, \(c \in C_n\) is a cycle if and only if \(d_n(c) = 0\). 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 \(S^1, S^2\), and \(T^2\) have no boundary, one can associate cycles to each of these spaces. However, the chain complex notion of cycles (elements whose boundary is a "zero chain") 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 \(S^1\), \(S^2\), and \(T^2\), 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 \(S^1\), one could glue a 2-dimensional disk \(D^2\) to that \(S^1\) such that the \(S^1\) is the boundary of that \(D^2\). Similarly, given a two-sphere \(S^2\), one can glue a ball \(B^3\) to that \(S^2\) such that the \(S^2\) is the boundary of that \(B^3\). This phenomenon is sometimes described as saying that \(S^2\) has a \(B^3\)-shaped "hole" or that it could be "filled in" with a \(B^3\).

Condensed: the full section is in Wikipedia.

Given a sufficiently-nice topological space \(X\), a choice of appropriate homology theory, and a chain complex \((C_\bullet, d_\bullet)\) associated to \(X\) that is compatible with that homology theory, the \(n\)th homology group \(H_n(X)\) is then given by the quotient group \(H_n(X)=Z_n/B_n\) of \(n\)-cycles (\(n\)-dimensional cycles) modulo \(n\)-dimensional boundaries. In other words, the elements of \(H_n(X)\), called homology classes, are equivalence classes whose representatives are \(n\)-cycles, and any two cycles are regarded as equal in \(H_n(X)\) if and only if they differ by the addition of a boundary. This also implies that the "zero" element of \(H_n(X)\) is given by the group of \(n\)-dimensional boundaries, which also includes formal sums of such boundaries.

The homology of a topological space X is a set of topological invariants of X represented by its homology groups \[H_0(X), H_1(X), H_2(X), \ldots\] where the \(k^{\rm th}\) homology group \(H_k(X)\) 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, \(H_0(X)\) describes the path-connected components of X.

A one-dimensional sphere \(S^1\) 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 \[H_k\left(S^1\right) = \begin{cases} \Z & k = 0, 1 \\ \{0\} & \text{otherwise} \end{cases}\] where \(\Z\) is the group of integers and \(\{0\}\) is the trivial group. The group \(H_1\left(S^1\right) = \Z\) represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.

A two-dimensional sphere \(S^2\) 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 \[H_k\left(S^2\right) = \begin{cases} \Z & k = 0, 2 \\ \{0\} & \text{otherwise} \end{cases}\]

In general for an n-dimensional sphere \(S^n,\) the homology groups are \[H_k\left(S^n\right) = \begin{cases} \Z & k = 0, n \\ \{0\} & \text{otherwise} \end{cases}\]

A two-dimensional ball \(B^2\) 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 \(H_0\left(B^2\right) = \Z\). In general, for an n-dimensional ball \(B^n,\) \[H_k\left(B^n\right) = \begin{cases} \Z & k = 0 \\ \{0\} & \text{otherwise} \end{cases}\]

The torus is defined as a product of two circles \(T^2 = S^1 \times S^1\). The torus has a single path-connected component, two independent one-dimensional holes (indicated by circles in red and blue) and one two-dimensional hole as the interior of the torus. The corresponding homology groups are \[H_k(T^2) = \begin{cases} \Z & k = 0, 2 \\ \Z \times \Z & k = 1 \\ \{0\} & \text{otherwise} \end{cases}\]

If n products of a topological space X is written as \(X^n\), then in general, for an n-dimensional torus \(T^n = (S^1)^n\), \[H_k(T^n) = \begin{cases} \Z^\binom{n}{k} & 0 \le k \le n \\ \{0\} & \text{otherwise} \end{cases}\] (see Torus § n-dimensional torus and Betti number § More examples for more details).

Condensed: the full section is in Wikipedia.

The following text describes a general algorithm for constructing the homology groups. It may be easier for the reader to look at some simple examples first: graph homology and simplicial homology.

The general construction begins with an object such as a topological space X, on which one first defines a chain complex C(X) encoding information about X. A chain complex is a sequence of abelian groups or modules \(C_0, C_1, C_2, \ldots\). connected by homomorphisms \(\partial_n : C_n \to C_{n-1},\) which are called boundary operators. That is,

\(\dotsb \overset{\partial_{n+1}}{\longrightarrow\,} C_n \overset{\partial_n}{\longrightarrow\,} C_{n-1} \overset{\partial_{n-1}}{\longrightarrow\,} \dotsb \overset{\partial_2}{\longrightarrow\,} C_1 \overset{\partial_1}{\longrightarrow\,} C_0 \overset{\partial_0}{\longrightarrow\,} 0\)

where 0 denotes the trivial group and \(C_i\equiv0\) for i < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n,

\(\partial_n \circ \partial_{n+1} = 0_{n+1, n-1},\)

i.e., the constant map sending every element of \(C_{n+1}\) to the group identity in \(C_{n-1}.\)

The statement that the boundary of a boundary is trivial is equivalent to the statement that \(\operatorname{im}(\partial_{n+1})\subseteq\ker(\partial_n)\), where \(\operatorname{im}(\partial_{n+1})\) denotes the image of the boundary operator and \(\ker(\partial_n)\) its kernel. Elements of \(B_n(X) = \mathrm{im}(\partial_{n+1})\) are called boundaries and elements of \(Z_n(X) = \ker(\partial_n)\) are called cycles.

Since each chain group C_n is abelian all its subgroups are normal. Then because \(\ker(\partial_n)\) is a subgroup of C_n, \(\ker(\partial_n)\) is abelian, and since \(\operatorname{im}(\partial_{n+1}) \subseteq\ker(\partial_n)\) therefore \(\operatorname{im}(\partial_{n+1})\) is a normal subgroup of \(\ker(\partial_n)\). Then one can create the quotient group

\(H_n(X) := \ker(\partial_n) / \operatorname{im}(\partial_{n+1}) = Z_n(X)/B_n(X),\)

called the nth homology group of X. The elements of H_n(X) are called homology classes. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous.

\(\dotsb \overset{\partial_{n+1}}{\longrightarrow\,} C_n \overset{\partial_n}{\longrightarrow\,} C_{n-1} \overset{\partial_{n-1}}{\longrightarrow\,} \dotsb \overset{\partial_2}{\longrightarrow\,} C_1 \overset{\partial_1}{\longrightarrow\,} C_0 \overset{\varepsilon}{\longrightarrow\,} \Z {\longrightarrow\,} 0\)

\(\varepsilon \left(\sum_i n_i \sigma_i\right) = \sum_i n_i\)

\(H^n(X) = Z^n(X)/B^n(X),\)

Condensed: the full section is in Wikipedia.

The nth homotopy group \(\pi_n(X)\) of a topological space \(X\) is the group of homotopy classes of basepoint-preserving maps from the \(n\)-sphere \(S^n\) to \(X\), under the group operation of concatenation. The most fundamental homotopy group is the fundamental group \(\pi_1(X)\). For connected \(X\), the Hurewicz theorem describes a homomorphism \(h_*: \pi_n(X) \to H_n(X)\) called the Hurewicz homomorphism. For \(n>1\), this homomorphism can be complicated, but when \(n=1\), the Hurewicz homomorphism coincides with abelianization. That is, \(h_*: \pi_1(X) \to H_1(X)\) is surjective and its kernel is the commutator subgroup of \(\pi_1(X)\), with the consequence that \(H_1(X)\) is isomorphic to the abelianization of \(\pi_1(X)\). Higher homotopy groups are sometimes difficult to compute. For instance, the homotopy groups of spheres are poorly understood and are not known in general, in contrast to the straightforward description given above for the homology groups.

For an \(n=1\) example, suppose \(X\) is the figure eight. As usual, its first homotopy group, or fundamental group, \(\pi_1(X)\) is the group of homotopy classes of directed loops starting and ending at a predetermined point (e.g. its center). It is isomorphic to the free group of rank 2, \(\pi_1(X) \cong \mathbb{Z} * \mathbb{Z}\), which is not commutative: looping around the lefthand cycle and then around the righthand cycle is different from looping around the righthand cycle and then looping around the lefthand cycle. By contrast, the figure eight's first homology group \(H_1(X)\cong \mathbb{Z} \times \mathbb{Z}\) is abelian. To express this explicitly in terms of homology classes of cycles, one could take the homology class \(l\) of the lefthand cycle and the homology class \(r\) of the righthand cycle as basis elements of \(H_1(X)\), allowing us to write \(H_1(X)=\{a_l l + a_r r\,|\; a_l, a_r \in \mathbb{Z}\}\).

The different types of homology theory arise from functors mapping from various categories of mathematical objects to the category of chain complexes. In each case the composition of the functor from objects to chain complexes and the functor from chain complexes to homology groups defines the overall homology functor for the theory.

The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex X. Here the chain group C_n is the free abelian group or free module whose generators are the n-dimensional oriented simplexes of X. The orientation is captured by ordering the complex's vertices and expressing an oriented simplex \(\sigma\) as an n-tuple \((\sigma[0], \sigma[1], \dots, \sigma[n])\) of its vertices listed in increasing order (i.e. \(\sigma[0] < \sigma[1] < \cdots < \sigma[n]\) in the complex's vertex ordering, where \(\sigma[i]\) is the \(i\)th vertex appearing in the tuple). The mapping \(\partial_n\) from C_n to C_n−1 is called the boundary mapping and sends the simplex

\(\sigma = (\sigma[0], \sigma[1], \dots, \sigma[n])\)

to the formal sum

\(\partial_n(\sigma) = \sum_{i=0}^n (-1)^i \left (\sigma[0], \dots, \sigma[i-1], \sigma[i+1], \dots, \sigma[n] \right ),\)\

which is evaluated as 0 if \(n = 0.\) This behavior on the generators induces a homomorphism on all of C_n as follows. Given an element \(c \in C_n\), write it as the sum of generators \(c = \sum_{\sigma_i \in X_n} m_i \sigma_i,\) where \(X_n\) is the set of n-simplexes in X and the m_i are coefficients from the ring C_n is defined over (usually integers, unless otherwise specified). Then define

\(\partial_n(c) = \sum_{\sigma_i \in X_n} m_i \partial_n(\sigma_i).\)

The dimension of the n-th homology of X turns out to be the number of "holes" in X at dimension n. It may be computed by putting matrix representations of these boundary mappings in Smith normal form.

Using simplicial homology example as a model, one can define a singular homology for any topological space X. A chain complex for X is defined by taking C_n to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplices into X. The homomorphisms ∂_n arise from the boundary maps of simplices.

In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module \(F_1\) and a surjective homomorphism \(p_1 : F_1 \to X.\) Then one finds a free module \(F_2\) and a surjective homomorphism \(p_2 : F_2 \to \ker\left(p_1\right).\) Continuing in this fashion, a sequence of free modules \(F_n\) and homomorphisms \(p_n\) can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology \(H_n\) of this complex depends only on F and X and is, by definition, the n-th derived functor of F, applied to X.

A common use of group (co)homology \(H^2(G, M)\) is to classify the possible extension groups E which contain a given G-module M as a normal subgroup and have a given quotient group G, so that \(G = E / M.\)