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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 ...
, simplicial homology is the sequence of homology groups of a
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 ...
. 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 of dimension 0). Simplicial homology arose as a way to study
topological spaces 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 point ...
whose building blocks are ''n''- simplices, the ''n''-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to a
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 ...
(more precisely, the geometric realization of an
abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...
). Such a homeomorphism is referred to as a ''
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
'' of the given space. Many topological spaces of interest can be triangulated, including every smooth
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
(Cairns and Whitehead). Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simplicial homology only depends on the associated topological space. As a result, it gives a computable way to distinguish one space from another.


Definitions


Orientations

A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a ''k''-simplex is given by an ordering of the vertices, written as (), with the rule that two orderings define the same orientation if and only if they differ by an
even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total o ...
. Thus every simplex has exactly two orientations, and switching the order of two vertices changes an orientation to the opposite orientation. For example, choosing an orientation of a 1-simplex amounts to choosing one of the two possible directions, and choosing an orientation of a 2-simplex amounts to choosing what "counterclockwise" should mean.


Chains

Let be a simplicial complex. A simplicial -chain is a finite formal sum :\sum_^N c_i \sigma_i, \, where each is an integer and is an oriented -simplex. In this definition, we declare that each oriented simplex is equal to the negative of the simplex with the opposite orientation. For example, : (v_0,v_1) = -(v_1,v_0). The group of -chains on is written . This is 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 ...
which has a basis in one-to-one correspondence with the set of -simplices in . To define a basis explicitly, one has to choose an orientation of each simplex. One standard way to do this is to choose an ordering of all the vertices and give each simplex the orientation corresponding to the induced ordering of its vertices.


Boundaries and cycles

Let be an oriented -simplex, viewed as a basis element of . The boundary operator :\partial_k: C_k \rightarrow C_ is the
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
defined by: :\partial_k(\sigma)=\sum_^k (-1)^i (v_0 , \dots , \widehat , \dots ,v_k), where the oriented simplex :(v_0 , \dots , \widehat , \dots ,v_k) is the face of , obtained by deleting its vertex. In , elements of the subgroup :Z_k := \ker \partial_k are referred to as cycles, and the subgroup :B_k := \operatorname \partial_ is said to consist of boundaries.


Boundaries of boundaries

Because (-1)^(v_0 , \dots , \widehat , \dots , \widehat\widehat ,\dots , v_k) = - (-1)^(v_0 , \dots , \widehat\widehat , \dots , \widehat ,\dots , v_k), where \widehat\widehat is the ''second'' face removed, \partial^2 = 0. In geometric terms, this says that the boundary of anything has no boundary. Equivalently, the abelian groups :(C_k, \partial_k) 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 t ...
. Another equivalent statement is that is contained in . As an example, consider a tetrahedron with vertices oriented as . By definition, its boundary is given by: . The boundary of the boundary is given by: .


Homology groups

The homology group of is defined to be the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
abelian group :H_k(S) = Z_k/B_k\, . It follows that the homology group is nonzero exactly when there are -cycles on ''S'' which are not boundaries. In a sense, this means that there are -dimensional holes in the complex. For example, consider the complex obtained by gluing two triangles (with no interior) along one edge, shown in the image. The edges of each triangle can be oriented so as to form a cycle. These two cycles are by construction not boundaries (since every 2-chain is zero). One can compute that the homology group is isomorphic to , with a basis given by the two cycles mentioned. This makes precise the informal idea that has two "1-dimensional holes". Holes can be of different dimensions. The
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of the homology group, the number :\beta_k = \operatorname (H_k(S))\, is called 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 simplici ...
of . It gives a measure of the number of -dimensional holes in .


Example


Homology groups of a triangle

Let be a triangle (without its interior), viewed as a simplicial complex. Thus has three vertices, which we call , and three edges, which are 1-dimensional simplices. To compute the homology groups of , we start by describing the chain groups : * is isomorphic to with basis * is isomorphic to with a basis given by the oriented 1-simplices , , and . * is the trivial group, since there is no simplex like (v_0, v_1, v_2) because the triangle has been supposed without its interior. So are the chain groups in other dimensions. The boundary homomorphism : is given by: : \partial(v_0,v_1) = (v_1)-(v_0) : \partial(v_0,v_2) = (v_2)-(v_0) : \partial(v_1,v_2) = (v_2)-(v_1) Since , every 0-chain is a cycle (i.e. ); moreover, the group of the 0-boundaries is generated by the three elements on the right of these equations, creating a two-dimensional subgroup of . So the 0th homology group is isomorphic to , with a basis given (for example) by the image of the 0-cycle (). Indeed, all three vertices become equal in the quotient group; this expresses the fact that is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
. Next, the group of 1-cycles is the kernel of the homomorphism ∂ above, which is isomorphic to , with a basis given (for example) by . (A picture reveals that this 1-cycle goes around the triangle in one of the two possible directions.) Since , the group of 1-boundaries is zero, and so the 1st homology group is isomorphic to . This makes precise the idea that the triangle has one 1-dimensional hole. Next, since by definition there are no 2-cycles, (the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
). Therefore the 2nd homology group is zero. The same is true for for all not equal to 0 or 1. Therefore, the
homological connectivity In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups. Definitions Background ''X'' is ''homologically-connected'' if its 0-th homology group equals Z, i.e. H_0(X)\cong \math ...
of the triangle is 0 (it is the largest for which the reduced homology groups up to are trivial).


Homology groups of higher-dimensional simplices

Let be a
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
(without its interior), viewed as a simplicial complex. Thus has four 0-dimensional vertices, six 1-dimensional edges, and four 2-dimensional faces. The construction of the homology groups of a tetrahedron is described in detail here. It turns out that is isomorphic to , is isomorphic to too, and all other groups are trivial.Therefore, the
homological connectivity In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups. Definitions Background ''X'' is ''homologically-connected'' if its 0-th homology group equals Z, i.e. H_0(X)\cong \math ...
of the tetrahedron is 0. If the tetrahedron contains its interior, then is trivial too. In general, if is a -dimensional simplex, the following holds: * If is considered without its interior, then and and all other homologies are trivial; * If is considered with its interior, then and all other homologies are trivial.


Simplicial maps

Let ''S'' and ''T'' be
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. A simplicial map ''f'' from ''S'' to ''T'' is a function from the vertex set of ''S'' to the vertex set of ''T'' such that the image of each simplex in ''S'' (viewed as a set of vertices) is a simplex in ''T''. A simplicial map ''f'': determines a homomorphism of homology groups for each integer ''k''. This is the homomorphism associated to 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 ...
from the chain complex of ''S'' to the chain complex of ''T''. Explicitly, this chain map is given on ''k''-chains by :f((v_0, \ldots, v_k)) = (f(v_0),\ldots,f(v_k)) if are all distinct, and otherwise . This construction makes simplicial homology 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 simplicial complexes to abelian groups. This is essential to applications of the theory, including the
Brouwer fixed point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simple ...
and the topological invariance of simplicial homology.


Related homologies

Singular homology 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''- ...
is a related theory that is better adapted to theory rather than computation. Singular homology is defined for all topological spaces and depends only on the topology, not any triangulation; and it agrees with simplicial homology for spaces which can be triangulated. Nonetheless, because it is possible to compute the simplicial homology of a simplicial complex automatically and efficiently, simplicial homology has become important for application to real-life situations, such as
image analysis Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as soph ...
,
medical imaging Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to re ...
, and
data analysis Data analysis is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, enc ...
in general. Another related theory is
Cellular homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex ...
.


Applications

A standard scenario in many computer applications is a collection of points (measurements, dark pixels in a bit map, etc.) in which one wishes to find a topological feature. Homology can serve as a qualitative tool to search for such a feature, since it is readily computable from combinatorial data such as a simplicial complex. However, the data points have to first be triangulated, meaning one replaces the data with a simplicial complex approximation. Computation of persistent homology
involves analysis of homology at different resolutions, registering homology classes (holes) that persist as the resolution is changed. Such features can be used to detect structures of molecules, tumors in X-rays, and cluster structures in complex data. More generally, simplicial homology plays a central role in topological data analysis, a technique in the field of data mining.


Implementations

* A
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
toolbox for computing persistent homology, Plex ( Vin de Silva,
Gunnar Carlsson Gunnar E. Carlsson (born August 22, 1952 in Stockholm, Sweden) is an American mathematician, working in algebraic topology. He is known for his work on the Segal conjecture, and for his work on applied algebraic topology, especially topologica ...
), is available a
this site.
* Stand-alone implementations in
C++ C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
are available as part of th
PerseusDionysus
an
PHAT
software projects. * For Python, there are libraries such a
scikit-tdaPersimgiotto-tda
an
GUDHI
the latter aimed at generating topological features for
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
. These can be found at the
PyPI The Python Package Index, abbreviated as PyPI () and also known as the Cheese Shop (a reference to the ''Monty Python's Flying Circus'' sketch " Cheese Shop"), is the official third-party software repository for Python. It is analogous to the C ...
repository.


See also

*
Simplicial homotopy In algebraic topology, a simplicial homotopypg 23 is an analog of a homotopy between topological spaces for simplicial sets. If :f, g: X \to Y are maps between simplicial sets, a simplicial homotopy from ''f'' to ''g'' is a map :h: X \times \Delta^ ...


References

{{reflist


External links


Topological methods in scientific computingComputational homology (also cubical homology)
Computational topology