# Singular homology

Last updated

In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups$H_{n}(X).$ 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, 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.

## Contents

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 (n1)-dimensional boundary induces the singular chain complex. The singular homology is then the homology of the chain complex. The resulting homology groups are the same for all homotopy equivalent spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of category theory, where homology is expressible as a functor from the category of topological spaces to the category of graded abelian groups.

## 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 $\Delta ^{n}$ to X, written $\sigm$ :\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 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 = 2a and so on. Every simplex a has a negative a.) Thus, if we designate $\sigma$ by its vertices

$[p_{0},p_{1},\ldots ,p_{n}]=[\sigma (e_{0}),\sigma (e_{1}),\ldots ,\sigma (e_{n})]$ corresponding to the vertices $e_{k}$ of the standard n-simplex $\Delta ^{n}$ (which of course does not fully specify the singular simplex produced by $\sigma$ ), then

$\partial _{n}\sigma =\sum _{k=0}^{n}(-1)^{k}\sigma \mid _{[p_{0},\ldots ,p_{k-1},p_{k+1},\ldots ,p_{n}]}$ is a formal sum of the faces of the simplex image designated in a specific way. (That is, a particular face has to be the restriction of $\sigma$ to a face of $\Delta ^{n}$ which depends on the order that its vertices are listed.) Thus, for example, the boundary of $\sigma =[p_{0},p_{1}]$ (a curve going from $p_{0}$ to $p_{1}$ ) is the formal sum (or "formal difference") $[p_{1}]-[p_{0}]$ .

## 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, 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 $\sigma _{n}(X)$ on a topological space X. This set may be used as the basis of a free abelian group, so that each singular n-simplex is a generator of the group. This set of generators is of course usually infinite, frequently uncountable, 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 $C_{n}(X)$ . Elements of $C_{n}(X)$ are called singular n-chains; they are formal sums of singular simplices with integer coefficients.

The boundary $\partial$ is readily extended to act on singular n-chains. The extension, called the boundary operator, written as

$\partial _{n}:C_{n}\to C_{n-1},$ is a homomorphism of groups. The boundary operator, together with the $C_{n}$ , form a chain complex of abelian groups, called the singular complex. It is often denoted as $(C_{\bullet }(X),\partial _{\bullet })$ or more simply $C_{\bullet }(X)$ .

The kernel of the boundary operator is $Z_{n}(X)=\ker(\partial _{n})$ , and is called the group of singular n-cycles. The image of the boundary operator is $B_{n}(X)=\operatorname {im} (\partial _{n+1})$ , and is called the group of singular n-boundaries.

It can also be shown that $\partial _{n}\circ \partial _{n+1}=0$ . The $n$ -th homology group of $X$ is then defined as the factor group

$H_{n}(X)=Z_{n}(X)/B_{n}(X).$ The elements of $H_{n}(X)$ are called homology classes.

## Homotopy invariance

If X and Y are two topological spaces with the same homotopy type (i.e. are homotopy equivalent), then

$H_{n}(X)\cong H_{n}(Y)\,$ for all n 0. This means homology groups are topological invariants.

In particular, if X is a connected contractible space, then all its homology groups are 0, except $H_{0}(X)\cong \mathbb {Z}$ .

A proof for the homotopy invariance of singular homology groups can be sketched as follows. A continuous map f: XY induces a homomorphism

$f_{\sharp }:C_{n}(X)\rightarrow C_{n}(Y).$ It can be verified immediately that

$\partial f_{\sharp }=f_{\sharp }\partial ,$ i.e. f# is a chain map, which descends to homomorphisms on homology

$f_{*}:H_{n}(X)\rightarrow H_{n}(Y).$ 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× [0, 1] Y be a homotopy that takes f to g. On the level of chains, define a homomorphism

$P:C_{n}(X)\rightarrow C_{n+1}(Y)$ that, geometrically speaking, takes a basis element σ: ΔnX of Cn(X) to the "prism" P(σ): Δn×IY. The boundary of P(σ) can be expressed as

$\partial P(\sigma )=f_{\sharp }(\sigma )-g_{\sharp }(\sigma )-P(\partial \sigma ).$ So if α in Cn(X) is an n-cycle, then f#(α ) and g#(α) differ by a boundary:

$f_{\sharp }(\alpha )-g_{\sharp }(\alpha )=\partial P(\alpha ),$ i.e. they are homologous. This proves the claim.

## 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. In particular, the homology group can be understood to be a functor from the category of topological spaces Top to the category of abelian groups Ab.

Consider first that $X\mapsto C_{n}(X)$ is a map from topological spaces to free abelian groups. This suggests that $C_{n}(X)$ might be taken to be a functor, provided one can understand its action on the morphisms 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

$f_{*}:C_{n}(X)\to C_{n}(Y)\,$ by defining

$f_{*}\left(\sum _{i}a_{i}\sigma _{i}\right)=\sum _{i}a_{i}(f\circ \sigma _{i})$ 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 $C_{n}(X)$ . This shows that $C_{n}$ is a functor

$C_{n}:\mathbf {Top} \to \mathbf {Ab}$ from the category of topological spaces to the category of abelian groups.

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 $X\mapsto H_{n}(X)$ is a functor

$H_{n}:\mathbf {Top} \to \mathbf {Ab}$ 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:

$H_{n}:\mathbf {hTop} \to \mathbf {Ab} .$ 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 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, or, alternately, as a functor on chain complexes, satisfying axioms that require a boundary morphism that turns short exact sequences into long exact sequences. 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

$C_{\bullet }:\mathbf {Top} \to \mathbf {Comp}$ which maps topological spaces as $X\mapsto (C_{\bullet }(X),\partial _{\bullet })$ 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 maps as its morphisms.

The second, algebraic part is the homology functor

$H_{n}:\mathbf {Comp} \to \mathbf {Ab}$ which maps

$C_{\bullet }\mapsto H_{n}(C_{\bullet })=Z_{n}(C_{\bullet })/B_{n}(C_{\bullet })$ 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 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}(X,R)\$ which is now an R-module. Of course, it is usually not a free module. The usual homology group is regained by noting that

$H_{n}(X,\mathbb {Z} )=H_{n}(X)$ when one takes the ring to be the ring of integers. The notation Hn(X, R) should not be confused with the nearly identical notation Hn(X, A), which denotes the relative homology (below).

## Relative homology

For a subspace $A\subset X$ , the relative homology Hn(X, A) is understood to be the homology of the quotient of the chain complexes, that is,

$H_{n}(X,A)=H_{n}(C_{\bullet }(X)/C_{\bullet }(A))$ where the quotient of chain complexes is given by the short exact sequence

$0\to C_{\bullet }(A)\to C_{\bullet }(X)\to C_{\bullet }(X)/C_{\bullet }(A)\to 0.$ ## Cohomology

By dualizing the homology chain complex (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 [the 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:

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 structure.

## Betti homology and cohomology

Since the number of homology theories has become large (see Category:Homology theory), the terms Betti homology and Betti cohomology are sometimes applied (particularly by authors writing on algebraic geometry) to the singular theory, as giving rise to the Betti numbers of the most familiar spaces such as simplicial complexes and closed manifolds.

## Extraordinary homology

If one defines a homology theory axiomatically (via the Eilenberg–Steenrod axioms), 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, namely K-theory and cobordism theory. In this context, singular homology is referred to as ordinary homology.

## Related Research Articles

In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels. Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

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 topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-moduleM to elucidate the properties of the group. By treating the G-module as a kind of topological space with elements of representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group G and G-module M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called group homology. The techniques of group cohomology can also be extended to the case that instead of a G-module, G acts on a nonabelian G-group; in effect, a generalization of a module to non-Abelian coefficients. In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.

In mathematics, the derived categoryD(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 proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.

In mathematics, a simplicial set is an object made up of "simplices" in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and J. A. Zilber.

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. This means that, given a cohomology theory

,

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.

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 categories, as discussed below.

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 homology, developed by Samuel Eilenberg and Norman Steenrod.

In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960.

In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra for any Abelian group pg 134. Note, this construction can be generalized to commutative rings as well from its underlying Abelian group. These are an important class of spectra because they model ordinary integral cohomology and cohomology with coefficients in an abelian group. In addition, they are a lift of the homological structure in the derived category of abelian groups in the homotopy category of spectra. In addition, these spectra can be used to construct resolutions of spectra, called Adams resolutions, which are used in the construction of the Adams spectral sequence.

In algebraic topology, the pushforward of a continuous function  : between two topological spaces is a homomorphism between the homology groups for .

In mathematics, especially in the area of topology known as algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y.

In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane. They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem to generalize this process.

In mathematics, a Δ-setS, often called a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.

In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets. It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism.

This is a glossary of properties and concepts in algebraic topology in mathematics.

• Allen Hatcher, Algebraic topology. Cambridge University Press, ISBN   0-521-79160-X and ISBN   0-521-79540-0
• J.P. May, A Concise Course in Algebraic Topology, Chicago University Press ISBN   0-226-51183-9
• Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, ISBN   0-387-96678-1