TheInfoList

OR:

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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, singular homology refers to the study of a certain set of
algebraic invariant Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
s of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
''X'', the so-called homology groups $H_n\left(X\right).$ Intuitively, singular homology counts, for each dimension ''n'', the ''n''-dimensional holes of a space. Singular homology is a particular example of a
homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...
, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory
simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
). In brief, singular homology is constructed by taking maps of the standard ''n''-simplex to a topological space, and composing them into formal sums, called singular chains. The boundary operation – mapping each ''n''-dimensional simplex to its (''n''−1)-dimensional
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
– induces the singular
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel ...
. The singular homology is then the homology of the chain complex. The resulting homology groups are the same for all
homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology is expressible as a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
from the category of topological spaces to the category of graded
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
s.

Singular simplices

A singular ''n''-simplex in a topological space ''X'' is a continuous function (also called a map) $\sigma$ from the standard ''n''-
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimensio ...
$\Delta^n$ to ''X'', written $\sigma:\Delta^n\to X.$ This map need not be injective, and there can be non-equivalent singular simplices with the same image in ''X''. The boundary of $\sigma,$ denoted as $\partial_n\sigma,$ is defined to be the formal sum of the singular (''n'' − 1)-simplices represented by the restriction of $\sigma$ to the faces of the standard ''n''-simplex, with an alternating sign to take orientation into account. (A formal sum is an element of the
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
on the simplices. The basis for the group is the infinite set of all possible singular simplices. The group operation is "addition" and the sum of simplex ''a'' with simplex ''b'' is usually simply designated ''a'' + ''b'', but ''a'' + ''a'' = 2''a'' and so on. Every simplex ''a'' has a negative −''a''.) Thus, if we designate $\sigma$ by its vertices :

Singular chain complex

The usual construction of singular homology proceeds by defining formal sums of simplices, which may be understood to be elements of a
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
, and then showing that we can define a certain group, the homology group of the topological space, involving the boundary operator. Consider first the set of all possible singular ''n''-simplices $\sigma_n\left(X\right)$ on a topological space ''X''. This set may be used as the basis of a
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
, so that each singular ''n''-simplex is a generator of the group. This set of generators is of course usually infinite, frequently
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
, as there are many ways of mapping a simplex into a typical topological space. The free abelian group generated by this basis is commonly denoted as $C_n\left(X\right)$. Elements of $C_n\left(X\right)$ are called singular ''n''-chains; they are formal sums of singular simplices with integer coefficients. The
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
$\partial$ is readily extended to act on singular ''n''-chains. The extension, called the boundary operator, written as :$\partial_n:C_n\to C_,$ is a homomorphism of groups. The boundary operator, together with the $C_n$, form a
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel ...
of abelian groups, called the singular complex. It is often denoted as $\left(C_\bullet\left(X\right),\partial_\bullet\right)$ or more simply $C_\bullet\left(X\right)$. The kernel of the boundary operator is $Z_n\left(X\right)=\ker \left(\partial_\right)$, and is called the group of singular ''n''-cycles. The image of the boundary operator is $B_n\left(X\right)=\operatorname \left(\partial_\right)$, and is called the group of singular ''n''-boundaries. It can also be shown that $\partial_n\circ \partial_=0$, implying $B_n\left(X\right) \subseteq Z_n\left(X\right)$. The $n$-th homology group of $X$ is then defined as the
factor group Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, s ...
:$H_\left(X\right) = Z_n\left(X\right) / B_n\left(X\right).$ The elements of $H_n\left(X\right)$ are called homology classes.

Homotopy invariance

If ''X'' and ''Y'' are two topological spaces with the same homotopy type (i.e. are
homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...
), then :$H_n\left(X\right) \cong H_n\left(Y\right)\,$ for all ''n'' ≥ 0. This means homology groups are homotopy invariants, and therefore topological invariants. In particular, if ''X'' is a connected contractible space, then all its homology groups are 0, except $H_0\left(X\right) \cong \mathbb$. A proof for the homotopy invariance of singular homology groups can be sketched as follows. A continuous map ''f'': ''X'' → ''Y'' induces a homomorphism :$f_ : C_n\left(X\right) \rightarrow C_n\left(Y\right).$ It can be verified immediately that :$\partial f_ = f_ \partial,$ i.e. ''f''# is a
chain map A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
, which descends to homomorphisms on homology :$f_* : H_n\left(X\right) \rightarrow H_n\left(Y\right).$ We now show that if ''f'' and ''g'' are homotopically equivalent, then ''f''* = ''g''*. From this follows that if ''f'' is a homotopy equivalence, then ''f''* is an isomorphism. Let ''F'' : ''X'' ×
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
→ ''Y'' be a homotopy that takes ''f'' to ''g''. On the level of chains, define a homomorphism :$P : C_n\left(X\right) \rightarrow C_\left(Y\right)$ that, geometrically speaking, takes a basis element σ: Δ''n'' → ''X'' of ''Cn''(''X'') to the "prism" ''P''(σ): Δ''n'' × ''I'' → ''Y''. The boundary of ''P''(σ) can be expressed as :$\partial P\left(\sigma\right) = f_\left(\sigma\right) - g_\left(\sigma\right) - P\left(\partial \sigma\right).$ So if ''α'' in ''Cn''(''X'') is an ''n''-cycle, then ''f''#(''α'' ) and ''g''#(''α'') differ by a boundary: :$f_ \left(\alpha\right) - g_\left(\alpha\right) = \partial P\left(\alpha\right),$ i.e. they are homologous. This proves the claim.

Homology groups of common spaces

The table below shows the k-th homology groups $H_k\left(X\right)$ of n-dimensional real projective spaces RP''n'', complex projective spaces, CP''n'', a point, spheres ''S''n($n\ge 1$), and a 3-torus ''T''3 with integer coefficients.

Functoriality

The construction above can be defined for any topological space, and is preserved by the action of continuous maps. This generality implies that singular homology theory can be recast in the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
. In particular, the homology group can be understood to be a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
from the category of topological spaces Top to the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab i ...
Ab. Consider first that $X\mapsto C_n\left(X\right)$ is a map from topological spaces to free abelian groups. This suggests that $C_n\left(X\right)$ might be taken to be a functor, provided one can understand its action on the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s of Top. Now, the morphisms of Top are continuous functions, so if $f:X\to Y$ is a continuous map of topological spaces, it can be extended to a homomorphism of groups :$f_*:C_n\left(X\right)\to C_n\left(Y\right)\,$ by defining :$f_*\left\left(\sum_i a_i\sigma_i\right\right)=\sum_i a_i \left(f\circ \sigma_i\right)$ where $\sigma_i:\Delta^n\to X$ is a singular simplex, and $\sum_i a_i\sigma_i\,$ is a singular ''n''-chain, that is, an element of $C_n\left(X\right)$. This shows that $C_n$ is a functor :$C_n:\mathbf \to \mathbf$ from the category of topological spaces to the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab i ...
. The boundary operator commutes with continuous maps, so that $\partial_n f_*=f_*\partial_n$. This allows the entire chain complex to be treated as a functor. In particular, this shows that the map $X\mapsto H_n \left(X\right)$ is a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
:$H_n:\mathbf\to\mathbf$ from the category of topological spaces to the category of abelian groups. By the homotopy axiom, one has that $H_n$ is also a functor, called the homology functor, acting on hTop, the quotient
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed be ...
: :$H_n:\mathbf\to\mathbf.$ This distinguishes singular homology from other homology theories, wherein $H_n$ is still a functor, but is not necessarily defined on all of Top. In some sense, singular homology is the "largest" homology theory, in that every homology theory on a
subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively ...
of Top agrees with singular homology on that subcategory. On the other hand, the singular homology does not have the cleanest categorical properties; such a cleanup motivates the development of other homology theories such as cellular homology. More generally, the homology functor is defined axiomatically, as a functor on an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abel ...
, or, alternately, as a functor on
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel ...
es, satisfying axioms that require a boundary morphism that turns short exact sequences into
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the contex ...
s. In the case of singular homology, the homology functor may be factored into two pieces, a topological piece and an algebraic piece. The topological piece is given by :$C_\bullet:\mathbf\to\mathbf$ which maps topological spaces as $X\mapsto \left(C_\bullet\left(X\right),\partial_\bullet\right)$ and continuous functions as $f\mapsto f_*$. Here, then, $C_\bullet$ is understood to be the singular chain functor, which maps topological spaces to the category of chain complexes Comp (or Kom). The category of chain complexes has chain complexes as its objects, and
chain map A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
s as its
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s. The second, algebraic part is the homology functor :$H_n:\mathbf\to\mathbf$ which maps :$C_\bullet\mapsto H_n\left(C_\bullet\right)=Z_n\left(C_\bullet\right)/B_n\left(C_\bullet\right)$ and takes chain maps to maps of abelian groups. It is this homology functor that may be defined axiomatically, so that it stands on its own as a functor on the category of chain complexes. Homotopy maps re-enter the picture by defining homotopically equivalent chain maps. Thus, one may define the
quotient category In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but ...
hComp or K, the homotopy category of chain complexes.

Coefficients in ''R''

Given any unital ring ''R'', the set of singular ''n''-simplices on a topological space can be taken to be the generators of a free ''R''-module. That is, rather than performing the above constructions from the starting point of free abelian groups, one instead uses free ''R''-modules in their place. All of the constructions go through with little or no change. The result of this is :$H_n\left(X; R\right)\$ which is now an ''R''-module. Of course, it is usually ''not'' a free module. The usual homology group is regained by noting that :$H_n\left(X;\mathbb\right)=H_n\left(X\right)$ when one takes the ring to be the ring of integers. The notation ''H''''n''(''X''; ''R'') should not be confused with the nearly identical notation ''H''''n''(''X'', ''A''), which denotes the relative homology (below). The
universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': : completely ...
provides a mechanism to calculate the homology with ''R'' coefficients in terms of homology with usual integer coefficients using the short exact sequence :$0\to H_n\left(X; \mathbb\right) \otimes R \to H_n\left(X; R\right) \to Tor\left(H_\left(X; \mathbb\right), R\right) \to 0.$ where ''Tor'' is the
Tor functor In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to cons ...
. Of note, if ''R'' is torsion-free, then ''Tor(G, R)'' = 0 for any G, so the above short exact sequence reduces to an isomorphism between $H_n\left(X; \mathbb\right) \otimes R$ and $H_n\left(X; R\right).$

Relative homology

For a subspace $A\subset X$, the relative homology ''H''''n''(''X'', ''A'') is understood to be the homology of the quotient of the chain complexes, that is, :$H_n\left(X,A\right)=H_n\left(C_\bullet\left(X\right)/C_\bullet\left(A\right)\right)$ where the quotient of chain complexes is given by the short exact sequence :$0\to C_\bullet\left(A\right) \to C_\bullet\left(X\right) \to C_\bullet\left(X\right)/C_\bullet\left(A\right) \to 0.$

Reduced homology

The
reduced homology In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise sta ...
of a space ''X'', annotated as $\tilde_n\left(X\right)$ is a minor modification to the usual homology which simplifies expressions of some relationships and fulfils the intuiton that all homology groups of a point should be zero. For the usual homology defined on a chain complex: :$\dotsb\oversetC_n \oversetC_ \overset \dotsb \overset C_1 \overset C_0\overset 0$ To define the reduced homology, we augment the chain complex with an additional $\mathbb$ between $C_0$ and zero: $\dotsb\oversetC_n \oversetC_ \overset \dotsb \overset C_1 \overset C_0\overset \mathbb \to 0$ where $\epsilon \left\left( \sum_i n_i \sigma_i \right\right) = \sum_i n_i$. This can be justified by interpreting the empty set as "(-1)-simplex", which means that $C_ \simeq \Z$. The ''reduced'' homology groups are now defined by $\tilde_n\left(X\right) = \ker\left(\partial_n\right) / \mathrm\left(\partial_\right)$ for positive ''n'' and $\tilde_0\left(X\right) = \ker\left(\epsilon\right) / \mathrm\left(\partial_1\right)$. Hatcher, 110 For n > 0, $H_n\left(X\right) = \tilde_n\left(X\right)$, while for n = 0, $H_0\left(X\right) = \tilde_0\left(X\right) \oplus \mathbb.$

Cohomology

By dualizing the homology
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel ...
(i.e. applying the functor Hom(-, ''R''), ''R'' being any ring) we obtain a cochain complex with coboundary map $\delta$. The cohomology groups of ''X'' are defined as the homology groups of this complex; in a quip, "cohomology is the homology of the co he dual complex. The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a differential graded algebra as follows: * the graded set of groups form a graded ''R''- module; * this can be given the structure of a graded ''R''-
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
using the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commuta ...
; * the Bockstein homomorphism ''β'' gives a differential. There are additional cohomology operations, and the cohomology algebra has addition structure mod ''p'' (as before, the mod ''p'' cohomology is the cohomology of the mod ''p'' cochain complex, not the mod ''p'' reduction of the cohomology), notably the
Steenrod algebra In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology. For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, ...
structure.

Betti homology and cohomology

Since the number of homology theories has become large (see :Homology theory), the terms ''Betti homology'' and ''Betti cohomology'' are sometimes applied (particularly by authors writing on
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
) to the singular theory, as giving rise to the
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s of the most familiar spaces such as
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
es and
closed manifold In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example i ...
s.

Extraordinary homology

If one defines a homology theory axiomatically (via the
Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homo ...
), and then relaxes one of the axioms (the ''dimension axiom''), one obtains a generalized theory, called an extraordinary homology theory. These originally arose in the form of
extraordinary cohomology theories In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
, namely
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
and
cobordism theory In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same d ...
. In this context, singular homology is referred to as ordinary homology.

*
Derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proc ...
* Excision theorem *
Hurewicz theorem In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results ...
*
Simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case ...
* Cellular homology

References

* Allen Hatcher
''Algebraic topology.''
Cambridge University Press, and * J.P. May, ''A Concise Course in Algebraic Topology'', Chicago University Press * Joseph J. Rotman, ''An Introduction to Algebraic Topology'', Springer-Verlag, {{isbn, 0-387-96678-1 Homology theory