Homology (mathematics)

The homology of a topological space X is a set of topological invariants of X represented by its homology groups

$${\displaystyle H_{0}(X),H_{1}(X),H_{2}(X),\ldots }$$

where the ${\displaystyle k^{\rm {th}}}$ homology group ${\displaystyle H_{k}(X)}$ describes, informally, the number of holes in X with a k-dimensional boundary. A 0-dimensional-boundary hole is simply a gap between two components. Consequently, ${\displaystyle H_{0}(X)}$ describes the path-connected components of X.[14]

The circle or 1-sphere ${\displaystyle S^{1}}$

The 2-sphere ${\displaystyle S^{2}}$ is the outer shell, not the interior, of a ball

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

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

where ${\displaystyle \mathbb {Z} }$ is the group of integers and ${\displaystyle \{0\}}$ is the trivial group. The group ${\displaystyle H_{1}\left(S^{1}\right)=\mathbb {Z} }$ represents a finitely-generated abelian group, with a single generator representing the one-dimensional hole contained in a circle.[15]

A two-dimensional sphere ${\displaystyle S^{2}}$ has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes. The corresponding homology groups are[15][16]

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

In general for an n-dimensional sphere ${\displaystyle S^{n},}$the homology groups are

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

The solid disc or 2-ball ${\displaystyle B^{2}}$

The torus ${\displaystyle T=S^{1}\times S^{1}}$

A two-dimensional ball ${\displaystyle B^{2}}$ is a solid disc. It has a single path-connected component, but in contrast to the circle, has no higher-dimensional holes. The corresponding homology groups are all trivial except for ${\displaystyle H_{0}\left(B^{2}\right)=\mathbb {Z} }$. In general, for an n-dimensional ball ${\displaystyle B^{n},}$[15]

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

The torus is defined as a product of two circles ${\displaystyle T=S^{1}\times S^{1}}$. 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[17]

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

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

For the projective plane P, a simple computation shows (where ${\displaystyle \mathbb {Z} _{2}}$ is the cyclic group of order 2):[18]

$${\displaystyle H_{k}(P)={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} _{2}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}$$

${\displaystyle H_{0}(P)=\mathbb {Z} }$ corresponds, as in the previous examples, to the fact that there is a single connected component. ${\displaystyle H_{1}(P)=\mathbb {Z} _{2}}$ 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.

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

The general construction begins with an object such as a topological space X, on which one first defines a chain complex C(X) encoding information about X. A chain complex is a sequence of abelian groups or modules ${\displaystyle C_{0},C_{1},C_{2},\ldots }$. connected by homomorphisms ${\displaystyle \partial _{n}:C_{n}\to C_{n-1},}$ which are called boundary operators.[17] That is,

${\displaystyle \dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\partial _{0}}{\longrightarrow \,}}0}$

where 0 denotes the trivial group and ${\displaystyle C_{i}\equiv 0}$ for i < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n,

${\displaystyle \partial _{n}\circ \partial _{n+1}=0_{n+1,n-1},}$

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

The statement that the boundary of a boundary is trivial is equivalent to the statement that ${\displaystyle \mathrm {im} (\partial _{n+1})\subseteq \ker(\partial _{n})}$, where ${\displaystyle \mathrm {im} (\partial _{n+1})}$ denotes the image of the boundary operator and ${\displaystyle \ker(\partial _{n})}$ its kernel. Elements of ${\displaystyle B_{n}(X)=\mathrm {im} (\partial _{n+1})}$ are called boundaries and elements of ${\displaystyle Z_{n}(X)=\ker(\partial _{n})}$ are called cycles.

Since each chain group C_n is abelian all its subgroups are normal. Then because ${\displaystyle \ker(\partial _{n})}$ is a subgroup of C_n, ${\displaystyle \ker(\partial _{n})}$ is abelian, and since ${\displaystyle \mathrm {im} (\partial _{n+1})\leq \ker(\partial _{n})}$ therefore ${\displaystyle \mathrm {im} (\partial _{n+1})}$ is a normal subgroup of ${\displaystyle \ker(\partial _{n})}$. Then one can create the quotient group

${\displaystyle H_{n}(X):=\ker(\partial _{n})/\mathrm {im} (\partial _{n+1})=Z_{n}(X)/B_{n}(X),}$

called the nth homology group of X. The elements of H_n(X) are called homology classes. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous.[19]

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 nth map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.[20]

The reduced homology groups of a chain complex C(X) are defined as homologies of the augmented chain complex[21]

${\displaystyle \dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\epsilon }{\longrightarrow \,}}\mathbb {Z} {\longrightarrow \,}0}$

where the boundary operator ${\displaystyle \epsilon }$ is

${\displaystyle \epsilon \left(\sum _{i}n_{i}\sigma _{i}\right)=\sum _{i}n_{i}}$

for a combination ${\displaystyle \sum n_{i}\sigma _{i},}$ of points ${\displaystyle \sigma _{i},}$ which are the fixed generators of C₀. The reduced homology groups ${\displaystyle {\tilde {H}}_{i}(X)}$ coincide with ${\displaystyle H_{i}(X)}$ for ${\displaystyle i\neq 0.}$ The extra ${\displaystyle \mathbb {Z} }$ in the chain complex represents the unique map ${\displaystyle [\emptyset ]\longrightarrow X}$ from the empty simplex to X.

Computing the cycle ${\displaystyle Z_{n}(X)}$ and boundary ${\displaystyle B_{n}(X)}$ 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_n(X) the free abelian group generated by the n-simplices of X. See simplicial homology for details.

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 ${\displaystyle d_{n},}$ point in the direction of increasing n rather than decreasing n; then the groups ${\displaystyle \ker \left(d^{n}\right)=Z^{n}(X)}$ of cocycles and ${\displaystyle \mathrm {im} \left(d^{n-1}\right)=B^{n}(X)}$ of coboundaries follow from the same description. The nth cohomology group of X is then the quotient group

${\displaystyle H^{n}(X)=Z^{n}(X)/B^{n}(X),}$

in analogy with the nth homology group.

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 ${\displaystyle \pi _{1}(X)}$ and the first homology group ${\displaystyle H_{1}(X)}$: the latter is the abelianization of the former. Hence, it is said that "homology is a commutative alternative to homotopy".[22]^: 4:00 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 figure eight. Its first homotopy group ${\displaystyle \pi _{1}(X)}$ 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 ${\displaystyle H_{1}(X)}$ 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.

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.[23]

Chain complexes form a category: A morphism from the chain complex (${\displaystyle d_{n}:A_{n}\to A_{n-1}}$) to the chain complex (${\displaystyle e_{n}:B_{n}\to B_{n-1}}$) is a sequence of homomorphisms ${\displaystyle f_{n}:A_{n}\to B_{n}}$ such that ${\displaystyle f_{n-1}\circ d_{n}=e_{n}\circ f_{n}}$ 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 ${\displaystyle X\to Y}$ 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.

If (${\displaystyle d_{n}:A_{n}\to A_{n-1}}$) 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

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

(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:

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

and, especially in algebraic topology, this provides two ways to compute the important invariant ${\displaystyle \chi }$ for the object X which gave rise to the chain complex.

Every short exact sequence

${\displaystyle 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0}$

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

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

All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps ${\displaystyle H_{n}(C)\to H_{n-1}(A)}$ 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.

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.[28]

• Betti number
• Cycle space
• De Rham cohomology
• Eilenberg–Steenrod axioms
• 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

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• Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century", Mathematics Magazine, Mathematical Association of America, 60 (5): 282–291, doi:10.1080/0025570X.1988.11977391, JSTOR 2689545
• Richeson, D. (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University.
• Spanier, Edwin H. (1966), Algebraic Topology, Springer, p. 155, ISBN 0-387-90646-0.
• Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Springer, doi:10.1007/978-1-4612-4372-4_6, ISBN 978-0-387-97970-0.
• Teicher, M., ed. (1999), The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings, Bar-Ilan University/American Mathematical Society/Oxford University Press, ISBN 978-0-19-851045-1, OCLC 223099225
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• Homology group at Encyclopaedia of Mathematics
• N.J. Windberger intro to algebraic topology, last six lectures with an easy intro to homology
• Algebraic topology Allen Hatcher - Chapter 2 on homology