Homology (mathematics)

In mathematics (especially algebraic topology and abstract algebra), homology (in part from Greek ὁμός homos "identical") 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. However, similar constructions are available in a wide variety of other contexts, such as 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 while the disk does not, and the 2-sphere is not a circle because the 2-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 the boundary of a submanifold with boundary, and a homology class (which represents a hole) is an equivalence class of cycles modulo boundaries.

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.

Origins

Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic. This was followed by Riemann's definition of genus and n-fold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis.

Homology itself was developed as a way to analyse and classify manifolds according to the cycles - closed loops or low-dimensional manifolds - that can be drawn on a given n dimensional manifold but not transformed smoothly into each other. These cycles are also sometimes thought of as cuts which can be glued back together, or as zippers which can be fastened and unfastened.

Cycles on a 2-sphere

A cycle drawn on a surface is a closed loop $S^1$ (1-manifold). On the ordinary sphere $S^2$, the cycle a in the diagram can be shrunk to the pole and even the equatorial great circle b can be shrunk in the same way. The Jordan curve theorem shows that any arbitrary cycle such as c can be similarly shrunk to a point. All cycles on the sphere can therefore be smoothly transformed into each other and the homology of the sphere is trivial.

Cycles on a torus

This is not generally true. The torus $T^2$ has cycles which cannot be smoothly transformed into each other, for example in the diagram none of the cycles a, b or c can be. Cycles a and b cannot be shrunk significantly although the arbitrary cycle c can be.

If we treat the cycles on the torus as cuts, we can open out the surface and flatten it down into a rectangle or, more conveniently, a square. The edges of the square may then be glued back together in different ways. The square can be twisted to allow edges to meet in the opposite direction, as shown by the arrows in the diagram. Consequently, there are four ways to glue them, each leading to a different surface:

The four ways of gluing a square to make a closed surface

Cycles on a Klein bottle

$K^2$ is the Klein bottle, which is a torus with a twist in it (The twist can be seen in the square diagram as reversed double-arrows). because of this, when re-glued the surface must self-intersect. In the diagram, like the torus cycles a and b cannot be shrunk while c can be. But unlike the torus, following b forwards right round and back reverses left and right, because of the twist given to the surface. Because local left and right can be arbitrarily re-oriented in this way, the surface as a whole is said to be non=orientable.

Cycles on a hemispherical projective plane

The projective plane $P^2$ is given a further twist. The uncut form, generally represented as the Boy surface, is visually complex, so a hemispherical embedding is shown in the diagram, in which antipodal points around the rim, for example A and A' are identified as the same point. Again, a and b are non-shrinkable while c is. But this time, both a and b reverse left and right.

Cycles can be joined or added together, as a and b were when the torus was cut open and flattened down. In the Klein bottle diagram, a goes round one way and −a goes round the opposite way. If a is thought of as a cut, then −a can be thought of as a gluing operation. Making a cut and then re-gluing it does not change the surface, so a + (−a) = 0.

Following an orientable cycle such as a around twice gives simply a + a = 2a. But non-orientable cycles behave differently. In the projective plane, following the unshrinkable cycle b round twice remarkably creates a trivial cycle which can be shrunk to a point; that is, b + b = 0. Because b must be followed around twice, the surface is said to have a torsion coefficient of 2.

The first recognisable theory of homology was published by Henri Poincaré in his seminal paper "Analysis situs", J. Ecole polytech. (2) 1. 1–121 (1895). The paper introduced homology classes and relations, according to their Betti numbers and torsion coefficients. Betti numbers are a generalisation of the Euler characteristic to higher dimensions. Orientable cycles are classified by their Betti numbers while non-orientable cycles are also classified by their torsion coefficients.

Topological characteristics of some closed manifolds

Manifold		Euler No. χ	Orientability	Betti numbers			Torsion coefficient (1-dimensional)
Symbol	Name			b0	b1	b2
$S^1$	Circle (1-manifold)	0	Orientable	1	1	N/A	N/A
$S^2$	Sphere	2	Orientable	1	0	1	none
$T^2$	Torus	0	Orientable	1	2	1	none
	2-holed torus	−2	Orientable	1	4	1	none
	g-holed torus (Genus = g)	2 − 2g	Orientable	1	2g	1	none
$P^2$	Projective plane	1	Non-orientable	1	0	0	2
$K^2$	Klein bottle	0	Non-orientable	1	1	0	2
	Sphere with c cross-caps	2 − c	Non-orientable	1	c − 1	0	2
	2-Manifold with g holes and c cross-caps (c > 0)	2 − (2g + c)	Non-orientable	1	(2g + c) − 1	0	2

NOTES:

1. For a non-orientable surface, a hole is equivalent to two cross-caps.
2. Any 2-manifold is the connected sum of g tori and c projective planes. For the sphere, $S^2$, g = c = 0.

This geometric analysis of manifolds is not rigorous. In a search for increased rigour, Poincaré went on to develop the simplicial homology of a triangulated manifold and to create what is now called a chain complex.

Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28. In particular, they formally treated topological classes in combinatorial topology as abelian groups. The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "algebraic topology". Algebraic homology remains the primary method of classifying manifolds.

Informal examples

Informally, 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 the k-dimensional holes in X. A 0-dimensional hole is simply a gap between two components, consequently $H_0(X)$ describes the path-connected components of X.

The circle or 1-sphere $S^1$

A one-dimensional sphere $S^1$ is a circle. It has a single connected component and a one-dimensional hole, but no higher-dimensional holes. The corresponding homology groups are given as

$H_k(S^1) = \begin{cases} \mathbb Z & k=0, 1 \\ \{0\} & \text{otherwise} \end{cases}$

where $\mathbb Z$ is the group of integers and $\{0\}$ is the trivial group. The group $H_1(S^1) = \mathbb Z$ represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.

The 2-sphere $S^2$ is the shell, not the interior, of a ball

A two-dimensional sphere $S^2$ has a single connected component, no one-dimensional holes, a two-dimensional hole, and no higher-dimensional holes. The corresponding homology groups are

$H_k(S^2) = \begin{cases} \mathbb Z & k=0, 2 \\ \{0\} & \text{otherwise} \end{cases}$

In general for an n-dimensional sphere Sⁿ, the homology groups are

$H_k(S^n) = \begin{cases} \mathbb Z & k=0, n \\ \{0\} & \text{otherwise} \end{cases}$

The solid disc or 1-ball $B^1$

A one-dimensional ball B¹ is a solid disc. It has a single path-connected component, but in contrast to the circle, has no one-dimensional or higher-dimensional holes. The corresponding homology groups are all trivial except for $H_0(B^1) = \mathbb Z$. In general, for an n-dimensional ball Bⁿ,

$H_k(B^n) = \begin{cases} \mathbb Z & k=0 \\ \{0\} & \text{otherwise} \end{cases}$

The torus $T = S^1 \times S^1$

The torus is defined as a Cartesian product of two circles $T = S^1 \times S^1$. 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

$H_k(T) = \begin{cases} \mathbb Z & k=0, 2 \\ \mathbb Z\times \mathbb Z & k=1 \\ \{0\} & \text{otherwise} \end{cases}$

The two independent 1D holes form independent generators in a finitely-generated abelian group, expressed as the Cartesian product group $\mathbb Z\times \mathbb Z$.

Construction of homology groups

The 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 C₀, C₁, C₂, ... 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}. That the boundary of a boundary is trivial implies $\mathrm{im}(\partial_{n+1})\subseteq\ker(\partial_n)$, where $\mathrm{im}(\partial_{n+1})$ 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 $\mathrm{im}(\partial_{n+1})$ and $\ker(\partial_n)$ are both subgroups of C_n, $\mathrm{im}(\partial_{n+1})$ is a normal subgroup of $\ker(\partial_n)$. Then one can create the quotient group

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

called the n-th 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.

A chain complex is said to be exact if the image of the (n + 1)-th map is always equal to the kernel of the n-th map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.

The reduced homology groups of a chain complex C(X) are defined as homologies of the augmented chain complex

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

where the boundary operator $\epsilon$ is

$\epsilon \left (\sum_i n_i \sigma_i \right ) = \sum_i n_i$

for a combination Σ n_iσ_i of points σ_i, which are the fixed generators of C₀. The reduced homology groups $\tilde{H}_i(X)$ coincide with $H_i(X)$ for i ≠ 0. The extra $\Z$ in the chain complex represents the unique map $[\emptyset] \longrightarrow X$ from the empty simplex to X.

Computing the cycle $Z_n(X)$ and boundary $B_n(X)$ groups is usually rather difficult since they have a very large number of generators. On the other hand, there are tools which make the task easier.

The simplicial homology groups H_n(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with C(X)_n the free abelian group generated by the n-simplices of X. The singular homology groups H_n(X) are defined for any topological space X, and agree with the simplicial homology groups for a simplicial complex.

Cohomology groups are formally similar to homology groups: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted dⁿ point in the direction of increasing n rather than decreasing n; then the groups $\ker(d^n) = Z^n(X)$ and $\mathrm{im}(d^{n - 1}) = B^n(X)$ follow from the same description. The n-th cohomology group of X is then the quotient group

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

in analogy with the n-th homology group.

Types of homology

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.

Simplicial homology

Main article: Simplicial homology

The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex X. Here A_n is the free abelian group or module whose generators are the n-dimensional oriented simplexes of X. The mappings are called the boundary mappings and send the simplex with vertices

$(a[0], a[1], \dots, a[n])$

to the sum

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

(which is considered 0 if n = 0).

If we take the modules to be over a field, then 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.

Singular homology

Main article: Singular homology

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 A_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.

Group homology

Main article: Group cohomology

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 F₁ and a surjective homomorphism p₁ : F₁ → X. Then one finds a free module F₂ and a surjective homomorphism p₂ : F₂ → ker(p₁). 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.

Other homology theories

• Borel–Moore homology
• Cellular homology
• Cyclic homology
• Hochschild homology
• Floer homology
• Intersection homology
• K-homology
• Khovanov homology
• Morse homology
• Persistent homology
• Steenrod homology

Homology functors

Chain complexes form a category: A morphism from the chain complex (d_n: A_n → A_n-1) to the chain complex (e_n: B_n → B_n-1) is a sequence of homomorphisms f_n: A_n → B_n such that $f_{n-1} \circ d_n = e_{n} \circ f_n$ for all n. The n-th homology H_n can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).

If the chain complex depends on the object X in a covariant manner (meaning that any morphism X → Y induces a morphism from the chain complex of X to the chain complex of Y), then the H_n are covariant functors from the category that X belongs to into the category of abelian groups (or modules).

The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called cohomology groups in this context and denoted by Hⁿ) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.

Properties

If (d_n: A_n → A_n-1) is a chain complex such that all but finitely many A_n are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then we can define the Euler characteristic

$\chi = \sum (-1)^n \, \mathrm{rank}(A_n)$

(using the rank in the case of abelian groups and the Hamel dimension in the case of vector spaces). It turns out that the Euler characteristic can also be computed on the level of homology:

$\chi = \sum (-1)^n \, \mathrm{rank}(H_n)$

and, especially in algebraic topology, this provides two ways to compute the important invariant χ for the object X which gave rise to the chain complex.

Every short exact sequence

$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$

of chain complexes gives rise to a long exact sequence of homology groups

$\cdots \to H_n(A) \to H_n(B) \to H_n(C) \to H_{n-1}(A) \to H_{n-1}(B) \to H_{n-1}(C) \to H_{n-2}(A) \to \cdots$

All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps H_n(C) → H_n-1(A) The latter are called connecting homomorphisms and are provided by the zig-zag lemma. This lemma can be applied to homology in numerous ways that aid in calculating homology groups, such as the theories of relative homology and Mayer-Vietoris sequences.

Applications

Notable theorems proved using homology include the following:

• The Brouwer fixed point theorem: If f is any continuous map from the ball Bⁿ to itself, then there is a fixed point a ∈ Bⁿ with f(a) = a.
• Invariance of domain: If U is an open subset of Rⁿ and f : U → Rⁿ is an injective continuous map, then V = f(U) is open and f is a homeomorphism between U and V.
• The Hairy ball theorem: any vector field on the 2-sphere (or more generally, the 2k-sphere for any k ≥ 1) vanishes at some point.
• The Borsuk–Ulam theorem: any continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.)

Application in science and engineering

In topological data analysis, data sets are regarded as a point cloud sampling of a manifold or algebraic variety embedded in Euclidean space. By linking nearest neighbor points in the cloud into a triangulation, a simplicial approximation of the manifold is created and its simplicial homology may be calculated. Finding techniques to robustly calculate homology using various triangulation strategies over multiple length scales is the topic of persistent homology.

In sensor networks, sensors may communicate information via an ad-hoc network that dynamically changes in time. To understand the global context of this set of local measurements and communication paths, it is useful to compute the homology of the network topology to evaluate, for instance, holes in coverage.

In dynamical systems theory in physics, Poincaré was one of the first to consider the interplay between the invariant manifold of a dynamical system and its topological invariants. Morse theory relates the dynamics of a gradient flow on a manifold to, for example, its homology. Floer homology extended this to infinite-dimensional manifolds. The KAM theorem established that periodic orbits can follow complex trajectories; in particular, they may form braids that can be investigated using Floer homology.

In one class of finite element methods, boundary-value problems for differential equations involving the Hodge-Laplace operator may need to be solved on topologically nontrivial domains, for example, in electromagnetic simulations. In these simulations, solution is aided by fixing the cohomology class of the solution based on the chosen boundary conditions and the homology of the domain. FEM domains can be triangulated, from which the simplicial homology can be calculated.

Software

Various software packages have been developed for the purposes of computing homology groups of finite cell complexes. Linbox is a C++ library for performing fast matrix operations, including Smith normal form; it interfaces with both Gap and Maple. Chomp, CAPD::Redhom and Perseus are also written in C++. All three implement pre-processing algorithms based on Simple-homotopy equivalence and discrete Morse theory to perform homology-preserving reductions of the input cell complexes before resorting to matrix algebra. Kenzo is written in Lisp, and in addition to homology it may also be used to generate presentations of homotopy groups of finite simplicial complexes. Gmsh includes a homology solver for finite element meshes, which can generate Cohomology bases directly usable by finite element software.

See also

• Betti number
• Cycle space
• Eilenberg–Steenrod axioms
• Extraordinary homology theory
• Homological algebra
• Homological conjectures in commutative algebra
• Homological dimension (disambiguation)
• Künneth theorem
• List of cohomology theories - also has a list of homology theories
• Torsion coefficient

References

• Cartan, Henri Paul and Eilenberg, Samuel (1956) Homological Algebra Princeton University Press, Princeton, NJ, OCLC 529171
• Eilenberg, Samuel and Moore, J. C. (1965) Foundations of relative homological algebra (Memoirs of the American Mathematical Society number 55) American Mathematical Society, Providence, R.I., OCLC 1361982
• Hatcher, A., (2002) Algebraic Topology Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
• Homology group at Encyclopaedia of Mathematics
• Homology (Topological space) at PlanetMath.org.
• Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University (2008)
• Spanier, Edwin H. (1966). Algebraic Topology., Springer, p. 155,. ISBN 0-387-90646-0.
• Timothy Gowers, June Barrow-Green, Imre Leader (2010), The Princeton Companion to Mathematics., Princeton University Press, ISBN 9781400830398.
• John Stillwell (1993), Classical Topology and Combinatorial Group Theory, Springer, doi:10.1007/978-1-4612-4372-4_6, ISBN 978-0-387-97970-0.
• Charles A. Weibel (1999), History of Homological Algebra, chapter 28 in the book History of Topology by I.M. James, Elsevier, ISBN 9780080534077.