homology theory

TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, homology is a general way of associating a sequence of algebraic objects, such as
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s or
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
s, with other mathematical objects such as
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s. Homology groups were originally defined in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. Similar constructions are available in a wide variety of other contexts, such as
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
,
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
s,
Lie algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s,
Galois theory In , Galois theory, originally introduced by , provides a connection between and . This connection, the , allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the ...
, and
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

. 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 The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...
. Loosely speaking, a ''cycle'' is a closed submanifold, a ''boundary'' is a cycle which is also the boundary of a submanifold, and a ''homology class'' (which represents a hole) is an equivalence class of cycles modulo boundaries. A homology class is thus represented by a cycle which is not the boundary of any submanifold: the cycle represents a hole, namely a hypothetical manifold whose boundary would be that cycle, but which is "not there". There are many different homology theories. A particular type of mathematical object, such as a topological space or a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
, may have one or more associated homology theories. When the underlying object has a geometric interpretation as topological spaces do, the ''n''th homology group represents behavior in dimension ''n''. Most homology groups or modules may be formulated as
derived functorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s on appropriate
abelian categories In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mot ...
, measuring the failure of a functor to be exact. From this abstract perspective, homology groups are determined by objects of a
derived categoryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

Background

Origins

Homology theory can be said to start with the Euler polyhedron formula, or
Euler characteristic#REDIRECT Euler characteristic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...
. This was followed by
Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of ...
's definition of
genus Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank In biological classification In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure ...
and ''n''-fold connectedness numerical invariants in 1857 and '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
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s according to their ''cycles'' – closed loops (or more generally submanifolds) that can be drawn on a given ''n'' dimensional manifold but not continuously deformed 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 are classified by dimension. For example, a line drawn on a surface represents a 1-cycle, a closed loop or $S^1$ (1-manifold), while a surface cut through a three-dimensional manifold is a 2-cycle.

Surfaces

On the ordinary
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

$S^2$, the cycle ''b'' in the diagram can be shrunk to the pole, and even the equatorial
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

''a'' can be shrunk in the same way. The
Jordan curve theorem 200px, Illustration of the Jordan curve theorem. A Jordan curve (drawn in black) divides the plane into an "inside" region (light blue) and an "outside" region (pink). In topology s, which have only one surface and one edge, are a kind of obje ...

shows that any arbitrary cycle such as ''c'' can be similarly shrunk to a point. All cycles on the sphere can therefore be continuously transformed into each other and belong to the same homology class. They are said to be homologous to zero. Cutting a manifold along a cycle homologous to zero separates the manifold into two or more components. For example, cutting the sphere along ''a'' produces two hemispheres. This is not generally true of cycles on other surfaces. The
torus In , a torus (plural tori, colloquially donut) is a generated by revolving a in about an axis that is with the circle. If the does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of ...

$T^2$ has cycles which cannot be continuously deformed into each other, for example in the diagram none of the cycles ''a'', ''b'' or ''c'' can be deformed into one another. In particular, cycles ''a'' and ''b'' cannot be shrunk to a point whereas cycle ''c'' can, thus making it homologous to zero. If the torus surface is cut along both ''a'' and ''b'', it can be opened out and flattened into a rectangle or, more conveniently, a square. One opposite pair of sides represents the cut along ''a'', and the other opposite pair represents the cut along ''b''. 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. Up to symmetry, there are four distinct ways of gluing the sides, each creating a different surface: $K^2$ is the
Klein bottle in three-dimensional space In topology, a branch of mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; it is a two-dimensional manifold against which a system for determining a normal vec ...

, which is a torus with a twist in it (The twist can be seen in the square diagram as the reversal of the bottom arrow). It is a theorem that the re-glued surface must self-intersect (when immersed in
Euclidean 3-space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...
). 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 ''b'' happens to cross over the twist given to one join. If an equidistant cut on one side of ''b'' is made, it returns on the other side and goes round the surface a second time before returning to its starting point, cutting out a twisted
Möbius strip In mathematics, a Möbius strip, band, or loop ( , ; ), also spelled ''Mobius'' or ''Moebius'', is a Surface (topology), surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary (topology), boundary ...

. Because local left and right can be arbitrarily re-oriented in this way, the surface as a whole is said to be non-orientable. The
projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
$P^2$ has both joins twisted. The uncut form, generally represented as the
Boy surface In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
, is visually complex, so a hemispherical embedding is shown in the diagram, in which antipodal points around the rim such as ''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'' on the torus were when it 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. But now consider two ''a''-cycles. Since the Klein bottle is nonorientable, you can transport one of them all the way round the bottle (along the ''b''-cycle), and it will come back as −''a''. This is because the Klein bottle is made from a cylinder, whose ''a''-cycle ends are glued together with opposite orientations. Hence 2''a'' = ''a'' + ''a'' = ''a'' + (−''a'') = 0. This phenomenon is called torsion. Similarly, 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 to achieve a zero cycle, the surface is said to have a torsion coefficient of 2. However, following a ''b''-cycle around twice in the Klein bottle gives simply ''b'' + ''b'' = 2''b'', since this cycle lives in a torsion-free homology class. This corresponds to the fact that in the fundamental polygon of the Klein bottle, only one pair of sides is glued with a twist, whereas in the projective plane both sides are twisted. A square is a contractible topological space, which implies that it has trivial homology. Consequently, additional cuts disconnect it. The square is not the only shape in the plane that can be glued into a surface. Gluing opposite sides of an octagon, for example, produces a surface with two holes. In fact, all closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2''n''-gons) can be glued to make different manifolds. Conversely, a closed surface with ''n'' non-zero classes can be cut into a 2''n''-gon. Variations are also possible, for example a hexagon may also be glued to form a torus. The first recognisable theory of homology was published by Henri Poincaré in his seminal paper "Analysis Situs (paper), Analysis situs", ''J. Ecole polytech.'' (2) 1. 1–121 (1895). The paper introduced homology classes and relations. The possible configurations of orientable cycles are classified by the Betti numbers of the manifold (Betti numbers are a refinement of the Euler characteristic). Classifying the non-orientable cycles requires additional information about torsion coefficients. The complete classification of 1- and 2-manifolds is given in the table. : Notes :# For a non-orientable surface, a hole is equivalent to two cross-caps. :# Any 2-manifold is the connected sum of ''g'' tori and ''c'' projective planes. For the sphere $S^2$, ''g'' = ''c'' = 0.

Generalization

A manifold with boundary or open manifold is topologically distinct from a closed manifold and can be created by making a cut in any suitable closed manifold. For example, the disk or 1-ball $B^1$ is bounded by a circle $S^1$. It may be created by cutting a trivial cycle in any 2-manifold and keeping the piece removed, by piercing the sphere and stretching the puncture wide, or by cutting the projective plane. It can also be seen as filling-in the circle in the plane. When two cycles can be continuously deformed into each other, then cutting along one produces the same shape as cutting along the other, up to some bending and stretching. In this case the two cycles are said to be or to lie in the same . Additionally, if one cycle can be continuously deformed into a combination of other cycles, then cutting along the initial cycle is the same as cutting along the combination of other cycles. For example, cutting along a figure 8 is equivalent to cutting along its two lobes. In this case, the figure 8 is said to be homologous to the sum of its lobes. Two open manifolds with similar boundaries (up to some bending and stretching) may be glued together to form a new manifold which is their connected sum. 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. These chain complexes (since greatly generalized) form the basis for most modern treatments of homology. In such treatments a cycle need not be continuous: a 0-cycle is a set of points, and cutting along this cycle corresponds to puncturing the manifold. A 1-cycle corresponds to a set of closed loops (an image of the 1-manifold $S^1$). On a surface, cutting along a 1-cycle yields either disconnected pieces or a simpler shape. A 2-cycle corresponds to a collection of embedded surfaces such as a sphere or a torus, and so on. Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28. The new combinatorial topology formally treated topological classes as
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and the non-orientable cycles are described by the torsion part. The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
". Algebraic homology remains the primary method of classifying manifolds.

Informal examples

The homology of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' is a set of Topological property, topological invariants of ''X'' represented by its ''homology groups'' :$H_0\left(X\right), H_1\left(X\right), H_2\left(X\right), \ldots$ where the $k^$ homology group $H_k\left(X\right)$ describes, informally, the number of ''k''-dimensional holes in ''X''. A 0-dimensional hole is simply a gap between two Connected space#Connected components, components. Consequently, $H_0\left(X\right)$ describes the path-connected components of ''X''. A one-dimensional
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

$S^1$ 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 :$H_k\left\left(S^1\right\right) = \begin \Z & k = 0, 1 \\ \ & \text \end$ where $\Z$ is the group of integers and $\$ is the trivial group. The group $H_1\left\left(S^1\right\right) = \Z$ represents a finitely-generated abelian group, with a single Generator (groups), generator representing the one-dimensional hole contained in a circle. A two-dimensional
sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 m ...

$S^2$ has a single connected component, no one-dimensional holes, a two-dimensional hole, and no higher-dimensional holes. The corresponding homology groups are :$H_k\left\left(S^2\right\right) = \begin \Z & k = 0, 2 \\ \ & \text \end$ In general for an ''n''-dimensional sphere $S^n,$the homology groups are :$H_k\left\left(S^n\right\right) = \begin \Z & k = 0, n \\ \ & \text \end$ A two-dimensional Ball (mathematics), ball $B^2$ 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 $H_0\left\left(B^2\right\right) = \Z$. In general, for an ''n''-dimensional ball $B^n,$ :$H_k\left\left(B^n\right\right) = \begin \Z & k = 0 \\ \ & \text \end$ The
torus In , a torus (plural tori, colloquially donut) is a generated by revolving a in about an axis that is with the circle. If the does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of ...

is defined as a product topology, 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 :$H_k\left(T\right) = \begin \Z & k = 0, 2 \\ \Z \times \Z & k = 1 \\ \ & \text \end$ The two independent 1-dimensional holes form independent generators in a finitely-generated abelian group, expressed as the direct product of groups, product group $\Z \times \Z.$ For the
projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
''P'', a simple computation shows (where $\Z_2$ is the cyclic group of order 2): : $H_k\left(P\right) = \begin \Z & k = 0 \\ \Z_2 & k = 1 \\ \ & \text \end$ $H_0\left(P\right) = \Z$ corresponds, as in the previous examples, to the fact that there is a single connected component. $H_1\left(P\right) = \Z_2$ 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.

Construction of homology groups

The construction begins with an object such as a topological space ''X'', on which one first defines a ''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 group homomorphism, homomorphisms $\partial_n : C_n \to C_,$ which are called boundary operators. That is, :$\dotsb \overset C_n \overset C_ \overset \dotsb \overset C_1 \overset C_0 \overset 0$ 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'', :$\partial_n \circ \partial_ = 0_,$ i.e., the constant map sending every element of $C_$ to the group identity in $C_.$The statement that the boundary of a boundary is trivial is equivalent to the statement that $\mathrm\left(\partial_\right)\subseteq\ker\left(\partial_n\right)$, where $\mathrm\left(\partial_\right)$ denotes the image (mathematics), image of the boundary operator and $\ker\left(\partial_n\right)$ its kernel (algebra), kernel. Elements of $B_n\left(X\right) = \mathrm\left(\partial_\right)$ are called boundaries and elements of $Z_n\left(X\right) = \ker\left(\partial_n\right)$ are called cycles. Since each chain group ''Cn'' is abelian all its subgroups are normal. Then because $\ker\left(\partial_n\right)$ is a subgroup of ''Cn'', $\ker\left(\partial_n\right)$ is abelian, and since $\mathrm\left(\partial_\right) \leq \ker\left(\partial_n\right)$ therefore $\mathrm\left(\partial_\right)$ is a normal subgroup of $\ker\left(\partial_n\right)$. Then one can create the quotient group :$H_n\left(X\right) := \ker\left(\partial_n\right) / \mathrm\left(\partial_\right) = Z_n\left(X\right)/B_n\left(X\right),$ called the ''n''th homology group of ''X''. The elements of ''Hn''(''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 sequence, 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, reduced homology groups of a chain complex ''C''(''X'') are defined as homologies of the augmented chain complex :$\dotsb \overset C_n \overset C_ \overset \dotsb \overset C_1 \overset C_0 \overset \Z 0$ where the boundary operator $\epsilon$ is :$\epsilon \left\left(\sum_i n_i \sigma_i\right\right) = \sum_i n_i$ for a combination $\sum n_i \sigma_i,$ of points $\sigma_i,$ which are the fixed generators of ''C''0. The reduced homology groups $\tilde_i\left(X\right)$ coincide with $H_i\left(X\right)$ for $i \neq 0.$ The extra $\Z$ in the chain complex represents the unique map $\left[\emptyset\right] \longrightarrow X$ from the empty simplex to ''X''. Computing the cycle $Z_n\left(X\right)$ and boundary $B_n\left(X\right)$ 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 ''Hn''(''X'') of a ''simplicial complex'' ''X'' are defined using the simplicial chain complex ''C''(''X''), with ''Cn''(''X'') the free abelian group generated by the ''n''-simplices of ''X''. See simplicial homology for details. The ''singular homology'' groups ''Hn''(''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 $d_n,$ point in the direction of increasing ''n'' rather than decreasing ''n''; then the groups $\ker\left\left(d^n\right\right) = Z^n\left(X\right)$ of ''cocycles'' and $\mathrm\left\left(d^\right\right) = B^n\left(X\right)$ of follow from the same description. The ''n''th cohomology group of ''X'' is then the quotient group :$H^n\left(X\right) = Z^n\left(X\right)/B^n\left(X\right),$ in analogy with the ''n''th homology group.

Homology vs. homotopy

Homotopy groups are similar to homology groups in that they can represent "holes" in a topological space. There is a close connection between the first homotopy group $\pi_1\left(X\right)$ and the first homology group $H_1\left(X\right)$: the latter is the abelianization of the former. Hence, it is said that "homology is a commutative alternative to homotopy". The higher homotopy groups are abelian and are related to homology groups by the Hurewicz theorem, but can be vastly more complicated. 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. As an example, let ''X'' be the Rose (topology), figure eight. Its first homotopy group $\pi_1\left(X\right)$ is the group of directed loops starting and ending at a predetermined point (e.g. its center). It is equivalent to the free group of rank 2, which is not commutative: looping around the leftmost cycle and then around the rightmost cycle is different than looping around the rightmost cycle and then looping around the leftmost cycle. In contrast, its first homology group $H_1\left(X\right)$ is the group of cuts made in a surface. This group is commutative, since (informally) cutting the leftmost cycle and then the rightmost cycle leads to the same result as cutting the rightmost cycle and then the leftmost cycle.

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

The motivating example comes from
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
: the simplicial homology of a simplicial complex ''X''. Here the chain group ''Cn'' is the free abelian group or module whose generators are the ''n''-dimensional oriented simplexes of ''X''. The orientation is captured by ordering the complex's vertex (geometry), vertices and expressing an oriented simplex $\sigma$ as an ''n''-tuple $\left(\sigma\left[0\right], \sigma\left[1\right], \dots, \sigma\left[n\right]\right)$ of its vertices listed in increasing order (i.e. $\sigma\left[0\right] < \sigma\left[1\right] < \cdots < \sigma\left[n\right]$ in the complex's vertex ordering, where $\sigma\left[i\right]$ is the $i$th vertex appearing in the tuple). The mapping $\partial_n$ from ''Cn'' to ''Cn−1'' is called the and sends the simplex :$\sigma = \left(\sigma\left[0\right], \sigma\left[1\right], \dots, \sigma\left[n\right]\right)$ to the formal sum :$\partial_n\left(\sigma\right) = \sum_^n \left(-1\right)^i \left \left(\sigma\left[0\right], \dots, \sigma\left[i-1\right], \sigma\left[i+1\right], \dots, \sigma\left[n\right] \right \right),$ which is considered 0 if $n = 0.$ This behavior on the generators induces a homomorphism on all of ''Cn'' as follows. Given an element $c \in C_n$, write it as the sum of generators $c = \sum_ m_i \sigma_i,$ where $X_n$ is the set of ''n''-simplexes in ''X'' and the ''mi'' are coefficients from the ring ''Cn'' is defined over (usually integers, unless otherwise specified). Then define :$\partial_n\left(c\right) = \sum_ m_i \partial_n\left(\sigma_i\right).$ 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 (mathematics), matrix representations of these boundary mappings in Smith normal form.

Singular homology

Using simplicial homology example as a model, one can define a ''singular homology'' for any
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X''. A chain complex for ''X'' is defined by taking ''Cn'' to be the free abelian group (or free module) whose generators are all Continuous function (topology), continuous maps from ''n''-dimensional Simplex, simplices into ''X''. The homomorphisms ∂''n'' arise from the boundary maps of simplexes.

Group homology

In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, one uses homology to define
derived functorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s, 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 $p_2 : F_2 \to \ker\left\left(p_1\right\right).$ 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 $H^2\left(G, M\right)$is to classify the possible Group extension, extension groups ''E'' which contain a given ''G''-module ''M'' as a normal subgroup and have a given quotient group ''G'', so that $G = E / M.$

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 (mathematics), category: A morphism from the chain complex ($d_n : A_n \to A_$) to the chain complex ($e_n : B_n \to B_$) is a sequence of homomorphisms $f_n : A_n \to B_n$ such that $f_ \circ d_n = e_n \circ f_n$ for all ''n''. The ''n''-th homology ''Hn'' 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 $X \to Y$ induces a morphism from the chain complex of ''X'' to the chain complex of ''Y''), then the ''Hn'' 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 ($d_n : A_n \to A_$) is a chain complex such that all but finitely many ''An'' are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then we can define the ''
Euler characteristic#REDIRECT Euler characteristic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...
'' :$\chi = \sum \left(-1\right)^n \, \mathrm\left(A_n\right)$ (using the rank of an abelian group, 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: :$\chi = \sum \left(-1\right)^n \, \mathrm\left(H_n\right)$ and, especially in algebraic topology, this provides two ways to compute the important invariant $\chi$ 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 :$\cdots \to H_n\left(A\right) \to H_n\left(B\right) \to H_n\left(C\right) \to H_\left(A\right) \to H_\left(B\right) \to H_\left(C\right) \to H_\left(A\right) \to \cdots$ All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps $H_n\left(C\right) \to H_\left(A\right)$ The latter are called 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 .

Applications

Application in pure mathematics

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 $a \in B^n$ with $f\left(a\right) = a.$ * Invariance of domain: If ''U'' is an open set, open subset of $\R^n$ and $f : U \to \R^n$ is an injective continuous map, then $V = f\left(U\right)$ 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 2''k''-sphere for any $k \geq 1$) vanishes at some point. * The Borsuk–Ulam theorem: any continuous function from an n-sphere, ''n''-sphere into Euclidean space, 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.) * Invariance of dimension: if non-empty open subsets $U \subseteq \R^m$ and $V \subseteq \R^n$ are homeomorphic, then $m = n.$

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 Braid theory, 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 Computational electromagnetics, 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 bot
Gap
an
MapleChompCAPD::Redhom
an
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 Presentation of a group, 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.

* Betti number * Cycle space * De Rham cohomology * Eilenberg–Steenrod axioms * Cohomology#Axioms and generalized cohomology theories, Extraordinary homology theory * Homological algebra * Homological conjectures in commutative algebra * Homological connectivity * Homological dimension * Homotopy group * Künneth theorem * List of cohomology theories - also has a list of homology theories * Poincaré duality

References

* * *. *. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc. * *. *. *. * *.