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A set  $  K = \{ t \} $
of elements  $  t $
that is partially ordered by a reflexive regular transitive relation  $  < $, 
together with an integer-valued function  $   \mathop{\rm dim}  t $, 
called the dimension of the element  $  t $, 
and a number  $  [ t :  t  ^  \prime  ] $, 
called the incidence coefficient of the elements  $  t $
and  $  t  ^  \prime  $, 
satisfying the conditions: 1)  $  t  ^  \prime  < t $
implies  $   \mathop{\rm dim}  t  ^  \prime  <  \mathop{\rm dim}  t $; 
2)  $  [ t :  t  ^  \prime  ] = [ t  ^  \prime  :  t ] $; 
3)  $  [ t :  t  ^  \prime  ] \neq 0 $
implies that either  $  t  ^  \prime  < t $
or  $  t < t  ^  \prime  $, 
and that  $  |  \mathop{\rm dim}  t -  \mathop{\rm dim}  t  ^  \prime  | = 1 $; 
and 4) for any pair of elements  $  t , t  ^ {\prime\prime} $
in  $  K $
the dimensions of which differ by two, there exists in  $  K $
at most a finite number of elements  $  t  ^  \prime  $
such that

$$ 
[ t :  t  ^  \prime  ]
[ t  ^  \prime  :  t  ^ {\prime\prime} ]
 \neq  0,\ \ 
$$

and, moreover,

$$ 
\sum _ {t  ^  \prime  }
[ t :  t  ^  \prime  ]
[ t  ^  \prime  :  t  ^ {\prime\prime} ]  =  0 .
$$

On replacing  $  [ t :  t  ^  \prime  ] $
by  $  \alpha ( t) \alpha ( t  ^  \prime  ) [ t :  t  ^  \prime  ] $, 
where  $  \alpha ( t) $
is a function with values  $  \pm  1 $, 
one obtains a complex that can be identified with  $  K $; 
in other words, the incidences  $  [ t :  t  ^  \prime  ] $
are determined up to factors  $  \alpha ( t) \alpha ( t  ^  \prime  ) $; 
transition from one value to the other is called a change of orientation of the complex  $  K $; 
the element  $  t $
preserves or changes its orientation according to whether  $  \alpha ( t) = + 1 $
or  $  - 1 $, 
respectively.

A complex  $  K $
is called finite dimensional, more precisely,  $  n $-
dimensional, if  $  n $
is the maximum dimension of the elements in  $  K $; 
if there is no element of the maximum dimension  $  k $, 
then  $  K $
is called infinite dimensional. The star of an element  $  t $
in the complex  $  K $
is the set of all elements  $  t  ^  \prime  $
in  $  K $
such that  $  t  ^  \prime  > t $. 
The closure of an element  $  t $
in  $  K $
is the set of all elements  $  t  ^  \prime  $
in  $  K $
such that  $  t  ^  \prime  < t $. 
The boundary of an element  $  t $
in  $  K $
is the set of all elements  $  t  ^  \prime  $
in  $  K $
such that  $  t  ^  \prime  < t $
and  $  t  ^  \prime  \neq t $. 
An element  $  t  ^  \prime  $
is called a face of an element  $  t $
in  $  K $
if  $  t  ^  \prime  < t $; 
a face  $  t  ^  \prime  $
of  $  t $
is called a proper face if  $  t  ^  \prime  \neq t $. 
Two elements  $  t $
and  $  t  ^  \prime  $
in  $  K $
are said to be incident if  $  t  ^  \prime  < t $
or  $  t < t  ^  \prime  $. 
A complex  $  K $
is called finite if the set of its elements is finite. A complex  $  K $
is called star-finite (respectively, closure-finite) if the star (respectively, the closure) of each of its elements consists of a finite number of elements. A complex is said to be locally finite if it is star-finite and closure-finite.

A subcomplex of a complex  $  K $
is any subset of  $  K $
that is a complex under the same dimensions and incidence coefficients as  $  K $. 
A subcomplex is closed closed if it contains the closure of each of its elements, and open if it contains the star of each of its elements. The complement of a closed complex is an open complex, and conversely. The star of each element of any complex is an open subcomplex, while the closure and boundary are closed subcomplexes. The  $  r $-
dimensional skeleton, or  $  r $-
skeleton,  $  K  ^ {r} $
of a complex  $  K $
is the set of all elements  $  t $
in  $  K $
for which  $   \mathop{\rm dim}  t \leq  r $; 
it is a closed subcomplex.

Two complexes  $  K = \{ t \} $
and  $  L $
are said to be isomorphic if there is a bijective mapping  $  f $
of the set  $  K $
to the set  $  L $
such that  $   \mathop{\rm dim}  f ( t) =  \mathop{\rm dim}  t $
and  $  [ t :  t  ^  \prime  ] = [ f ( t) :  f ( t  ^  \prime  ) ] $.

The most important type of complex is a simplicial complex, of which there exist two kinds: an abstract complex and a geometric complex.

An abstract simplicial complex  $  K $
has for its elements abstract simplices (simplexes) of different dimensions. An  $  r $-
dimensional simplex  $  t  ^ {r} $
is a set of  $  r + 1 $
objects  $  a  ^ {0} \dots a  ^ {r} $. 
These objects, that is, the  $  0 $-
dimensional simplices, are called the vertices of the complex  $  K $. 
A simplex is oriented if its vertex set is ordered, where orderings that differ by an [[even permutation]] determine the same orientation. The  $  s $-
dimensional faces of a simplex  $  t  ^ {r} $
are the  $  s $-
dimensional simplices the vertices of which are contained among those of  $  t  ^ {r} $. 
A simplicial complex  $  K $
contains all faces of each of its simplices. The relation  $  t  ^ {s} < t  ^ {r} $
means that  $  t  ^ {s} $
is a face of  $  t  ^ {r} $. 
The faces  $  ( a  ^ {0} \dots a  ^ {s} ) $
and  $  ( a  ^ {s+} 1 \dots a  ^ {r} ) $
are called opposite faces of the simplex  $  t  ^ {r} $. 
If  $  t  ^ {r-} 1 $
is the face of  $  t  ^ {r} $
opposite to the vertex  $  a  ^ {i} $, 
then

$$ 
[ t  ^ {r-} 1 :  t  ^ {r} ]  = \ 
[ t  ^ {r} :  t  ^ {r-} 1 ]  = \ 
\pm  1 ,
$$

according to whether  $  t  ^ {r} $
has the same orientation as  $  a  ^ {i} t  ^ {r-} 1 $
or not. If  $  t  ^ {r-} 1 $
is not a face of  $  t  ^ {r} $, 
then

$$ 
[ t  ^ {r-} 1 :  t  ^ {r} ]  = \ 
[ t  ^ {r} :  t  ^ {r-} 1 ]  =  0 .
$$

By giving an orientation to each simplex of a simplicial complex one obtains an oriented complex  $  K $.

An abstract simplicial complex is defined if the set of its vertices is known as well as the system, called a scheme, of all those finite subsets of this set that are to be taken as the simplices; here it is required that each vertex belongs to at least one element of the system and that each subset of an element belonging to the system also belongs to the system. Dimension, orientation, etc., are defined as before.

A polyhedral (cellular) complex of an  $  n $-
dimensional Euclidean space  $  E  ^ {n} $
is a countable locally finite complex  $  K $
the elements of which are  $  r $-
dimensional cells  $  t  ^ {r} $, 
i.e. bounded convex open subsets of some  $  E  ^ {r} $
in  $  E  ^ {n} $, 
$  0 \leq  r \leq  n $, 
where the cells are pairwise disjoint, the union of the cells belonging to the closure of the element  $  t  ^ {r} $
is the topological closure  $  \overline{t}\; {}  ^ {r} $
of  $  t  ^ {r} $
in  $  E  ^ {r} $, 
and the topological closure of the union of the cells not belonging to the star of  $  t  ^ {r} $
does not intersect  $  t  ^ {r} $. 
Here  $  t  ^ {r} < t  ^ {s} $
means that either  $  t  ^ {r} = t  ^ {s} $
or  $  t  ^ {r} \subset  \overline{t}\; {}  ^ {s} \setminus  t  ^ {s} $, 
and  $  [ t  ^ {r-} 1 :  t  ^ {r} ] $
is defined by the incidence coefficients  $  [ E _ {1}  ^ {r-} 1 :  E  ^ {r} ] = - [ E _ {2}  ^ {r-} 1 :  E  ^ {r} ] $, 
where  $  E _ {1}  ^ {r-} 1 $
and  $  E _ {2}  ^ {r-} 1 $
are the two regions into which the space  $  E  ^ {r-} 1 $
containing  $  t  ^ {r-} 1 $
divides  $  E  ^ {r} $. 
The union of the cells of the polyhedral complex  $  K $
obtained in this manner with the topology induced from  $  E  ^ {n} $
is called a polyhedron and is usually denoted by  $  | K | $. 
A special form of a polyhedral complex is a Euclidean geometric simplicial complex, the elements of which are Euclidean simplices in  $  E  ^ {n} $. 
An  $  r $-
dimensional Euclidean simplex  $  t  ^ {r} $
consists of points  $  x = ( x _ {1} \dots x _ {n} ) \in E  ^ {n} $, 
defined by the relations

$$ 
x _ {k}  = \ 
\sum _ { i= } 0 ^ { r } 
\lambda  ^ {i} a _ {k}  ^ {i} ,\ \ 
k = 1 \dots n ,
$$

where  $  a  ^ {i} = ( a _ {1}  ^ {i} \dots a _ {n}  ^ {i} ) $, 
$  i = 0 \dots r $, 
are independent points of  $  E  ^ {n} $(
i.e. they are not contained in any  $  E  ^ {r-} 1 $
of  $  E  ^ {n} $), 
$  0 < \lambda  ^ {i} < 1 $,

$$ 
\sum _ { i= } 0 ^ { r } 
\lambda  ^ {i}  =  1 \ \ 
( \textrm{ if }  r = 0 ,\ 
x = a  ^ {0} ) ,
$$

$  a  ^ {i} $
are called the vertices of  $  t  ^ {r} $, 
$  \lambda  ^ {i} $
are the barycentric coordinates of the point  $  x $, 
and  $  t  ^ {r} $
is called the geometric simplex formed by the abstract simplex  $  ( a  ^ {0} \dots a  ^ {r} ) $.

Let  $  K $
be a countable locally finite abstract simplicial complex with vertices in  $  E  ^ {n} $, 
where any vertices forming a simplex are independent, any two simplexes of  $  K $
having no vertices in common generate disjoint geometric complexes, and the closure of the union of all those geometric simplices that are generated by simplices of  $  K $
and which do not belong to some generated simplex does not intersect the latter. The notions of dimension, order, incidence, etc., are carried over from  $  K $
to the set of generated geometric simplices; this turns this set into a polyhedral complex, called a Euclidean realization of  $  K $.

A geometric realization, not necessarily Euclidean, is also possible for any abstract simplicial complex. Let  $  \{ a  ^ {i} \} $
be the family of vertices of an arbitrary abstract simplicial complex  $  K $
labelled by indices  $  i $
in a totally well-ordered set  $  I $, 
let  $  | K | $
be the set of all systems  $  \{ \lambda _ {i} \} $, 
$  i \in I $, 
of non-negative real numbers  $  \lambda _ {i} $
such that the vertices corresponding to non-zero coordinates  $  \lambda _ {i _ {0}  } \dots \lambda _ {i _ {r}  } $
of the system  $  \{ \lambda _ {i} \} $
form a simplex  $  ( a ^ {i _ {0} } \dots a ^ {i _ {r} }) $
in  $  K $(
the number of such coordinates is finite), and let  $  \sum _ {i} \lambda _ {i} = 1 $. 
The simplex  $  t  ^ {r} = ( a ^ {i _ {0} } \dots a ^ {i _ {r} } ) $
in  $  K $
is put in correspondence with the set  $  | t  ^ {r} | $
of all systems  $  \{ \lambda _ {i} \} $
such that  $  \lambda _ {i} \neq 0 $
if and only if  $  i $
is one of the  $  i _ {0} \dots i _ {r} $; 
then  $  | K | $
is the union of the sets  $  | t  ^ {r} | $. 
Let  $  | t  ^ {r} | $
be homeomorphically imbedded in  $  E  ^ {r+} 1 $: 
Corresponding to the point  $  \{ \lambda _ {i} \} $
in  $  | t  ^ {r} | $
is the point  $  \{ \lambda _ {i _ {0}  } \dots \lambda _ {i _ {r}  } \} $
in  $  E  ^ {r+} 1 $. 
This introduces a topology in  $  | t  ^ {r} | $
and in  $  | K | $: 
A set in  $  | K | $
is taken to be open if its intersection with each  $  | t  ^ {r} | $
is open in  $  | t  ^ {r} | $. 
The polyhedron  $  | K | $
is called a geometric realization of the complex  $  K $, 
and  $  K $
is called a triangulation of the polyhedron  $  | K | $. 
A simplicial complex  $  K $
is finite (respectively, locally finite) if and only if  $  | K | $
is a compact (respectively, locally compact) space. Local finiteness of a simplicial complex  $  K $
is also a necessary and sufficient condition for the metrizability of  $  | K | $, 
where the metric is defined by the formula

$$ 
\rho ( \{ \lambda _ {i} \} ,\ 
\{ \mu _ {i} \} )  = \ 
\sqrt {\sum _ { i }
( \lambda _ {i} -
\mu _ {i} )  ^ {2} } .
$$

If  $  K $
is a countable locally finite  $  n $-
dimensional complex, then it can be realized in the  $  ( 2 n + 1 ) $-
dimensional Euclidean space  $  E  ^ {2n+} 1 $. 
A complex  $  K $
is realizable in a Hilbert space if  $  | K | $
can be homeomorphically imbedded in this space such that every closed simplex in  $  | K | $
has a Euclidean realization; this is possible if and only if  $  K $
is a countable locally finite simplicial complex.

A finite geometric complex is a set of open geometric simplices that contains all the faces of each of the simplices and is such that the intersection of different simplices is empty. When studying closed simplices the second condition is replaced by the requirement that the intersection of two closed simplices be empty or a closed face of these simplices.

The notion of a complex finds its greatest application in homology theory. The use of simplicial complexes in the calculation of topological invariants of polyhedra is complicated by the fact that under triangulation of a polyhedron one may have to use many simplices. In this respect the [[CW-complex|CW-complex]] is preferable: in the latter the number of cells can be considerably fewer than the number of simplices in an arbitrary simplicial subdivision of the polyhedron. On the other hand, the simplicial complexes and triangulations have their advantages too. For example, in the simplicial approximation of a continuous mapping, in the composition and application of incidence matrices, in the use of complexes for the homological investigation of general topological spaces, etc.

A simplicial mapping from a complex  $  K $
to a complex  $  L $
is a function  $  f :  K \rightarrow L $
that sets up a correspondence between each vertex  $  a $
of  $  K $
and a vertex  $  f ( a) $
of  $  L $, 
such that whenever some vertices  $  a ^ {i _ {0} } \dots a ^ {i _ {r} } $
of  $  K $
form a simplex in  $  K $, 
then the vertices  $  f ( a ^ {i _ {0} } ) \dots f ( a ^ {i _ {r} } ) $, 
some of which may be coincident, must also form simplex in  $  L $. 
The function  $  f $
associates with each simplex  $  t  ^ {r} $
of  $  K $
a simplex  $  t  ^ {s} = f ( t  ^ {r} ) $
of  $  L $. 
A simplicial mapping  $  f :  ( K , L ) \rightarrow ( K ^ { \prime } , L  ^  \prime  ) $
of a pair  $  ( K , L ) $
into a pair  $  ( K ^ { \prime } , L  ^  \prime  ) $, 
where  $  L , L  ^  \prime  $
are closed subcomplexes of  $  K , K ^ { \prime } $, 
respectively, is a simplicial mapping  $  f :  K \rightarrow K ^ { \prime } $
such that  $  f ( L) \subset  L  ^  \prime  $. 
The set of all simplicial complexes and their simplicial mappings forms a category, as does the set of all simplicial pairs and all their simplicial mappings.

The homology of a complex, which, to begin with, was expressed by numerical invariants, subsequently came to be represented by algebraic means such as groups, modules, sheaves, etc. The scheme of their construction is as follows. Let  $  K $
be an arbitrary complex and let  $  G $
be an Abelian group; an  $  r $-
dimensional chain complex (generally infinite)  $  K $
over the group of coefficients  $  G $
is a function  $  c _ {r} $
with domain the set of all  $  r $-
dimensional elements of  $  K $
and with range  $  G $. 
The collection  $  \{ c _ {r} \} $
of all  $  r $-
dimensional chains  $  c _ {r} $
of the complex  $  K $, 
denoted by  $  C _ {r} ( K ;  G ) $, 
forms a group with respect to the operation of addition

$$ 
( c _ {r} + c _ {r}  ^  \prime  )
( t  ^ {r} )  =  c _ {r} ( t  ^ {r} ) + c _ {r}  ^  \prime 
( t  ^ {r} ) ,\  c _ {r} ,\ 
c _ {r}  ^  \prime  \in C _ {r} ( K ;  G ) ,\  t \in K .
$$

It is called the group of  $  r $-
dimensional chains of  $  K $
with coefficients in  $  G $(
or over  $  G $). 
Under the hypothesis that  $  K $
is a star-finite complex, one can introduce on  $  C _ {r} ( K ;  G ) $
a boundary operator  $  \partial  _ {r} $
by means of the formula

$$ 
\partial  c _ {r}  =  \sum _ { j }
\left ( \sum _ { i } c _ {r} ( t _ {i}  ^ {r} ) [ t _ {i}  ^ {r} :  t _ {j}  ^ {r-} 1 ] \right )
t _ {j}  ^ {r-} 1 ,
$$

which defines a homomorphism

$$ 
\partial  _ {r} :  C _ {r} ( K ;  G )  \rightarrow  C _ {r-} 1
( K ;  G ) .
$$

Because the equation  $  \partial  _ {r-} 1 \partial  _ {r} = 0 $
holds, one obtains a chain complex  $  \{ C _ {r} ( K ;  G ) , \partial  _ {r} \} $, 
whose homology group  $  H _ {r} ( K ;  G ) $(
i.e. the quotient group of  $   \mathop{\rm Ker}  \partial  _ {r} $
by the subgroup  $   \mathop{\rm Im}  \partial  _ {r+} 1 $) 
is called the  $  r $-
dimensional homology group of the complex  $  K $
with coefficients in  $  G $. 
(The group  $   \mathop{\rm Ker}  \partial  _ {r} $
is often denoted by  $  Z _ {r} ( K ;  G ) $
and is called the group of  $  r $-
dimensional cycles of the complex  $  K $
with coefficients in  $  G $, 
while the group  $   \mathop{\rm Im}  \partial  _ {r+} 1 $
is denoted by  $  B _ {r} ( K ;  G ) $
and is called the group of  $  r $-
dimensional boundaries of the complex  $  K $
with coefficients in  $  G $.)

As well as homology groups, cohomology groups are also defined for a complex. For their definition, one starts again with a group of chains, called in this case the group of cochains, and denoted by  $  C  ^ {r} ( K ;  G ) $. 
The complex  $  K $
is here assumed to be closed-finite, while the coboundary operator  $  \partial   ^ {r} $
is defined by the formula

$$ 
\delta  ^ {r} c  ^ {r}  = \ 
\sum _ { j } \left ( \sum _ { i }
c  ^ {r} ( t _ {i}  ^ {r} )
[ t _ {i}  ^ {r} :  t _ {j}  ^ {r+} 1 ] \right ) t _ {j}  ^ {r+} 1 ,
$$

defining a homomorphism

$$ 
\delta  ^ {r} :  C  ^ {r}
( K ;  G )  \rightarrow  C  ^ {r+} 1
( K ;  G ) .
$$

For this cochain complex  $  \{ C  ^ {r} ( K ;  G ) , \delta  ^ {r} \} $, 
$  \delta  ^ {r+} 1 \delta  ^ {r} = 0 $, 
the cohomology group  $  H  ^ {r} ( k ;  G ) $, 
i.e. the quotient group of  $   \mathop{\rm Ker}  \delta  ^ {r} $
by the subgroup  $   \mathop{\rm Im}  \delta  ^ {r-} 1 $, 
is called the  $  r $-
dimensional cohomology group of the complex  $  K $
with coefficients in  $  G $. 
(The group  $   \mathop{\rm Ker}  \delta  ^ {r} $
is usually denoted by  $  Z  ^ {r} ( K ;  G ) $
and is called the group of  $  r $-
dimensional cocycles of the complex  $  K $
with coefficients in  $  G $, 
while the group  $   \mathop{\rm Im}  \delta  ^ {r-} 1 $
is denoted by  $  B  ^ {r} ( K ;  G ) $
and is called the group of  $  r $-
dimensional coboundaries of the complex  $  K $
with coefficients in  $  G $.)

Star- (or closed-) finiteness of the complex is required in order that the summation in the definition of the boundary (or coboundary) operator be finite. In the case of a star-finite complex one can define the homology groups of arbitrary (infinite) cycles and the cohomology groups of finite cocycles. In the case of a closed-finite complex one can define the homology groups of infinite cocycles and the homology groups of finite cycles. In the case of a locally finite complex, one can define both finite and infinite homology and cohomology groups. If the complex is arbitrary, then its homology (respectively, cohomology) groups are defined as the direct (respectively, inverse) limit of the spectrum of the homology (respectively, cohomology) groups of all locally finite subcomplexes of the given complex, ordered by increasing size.

In the study of homology and cohomology groups of a complex one can consider the category of simplicial pairs of complexes  $  ( K , L ) $
and simplicial mappings  $  f :  ( K , L ) \rightarrow ( K ^ { \prime } , L  ^  \prime  ) $
between them, and the group  $  C _ {r} ( K , L ;  G ) $
of  $  r $-
dimensional finite chains of  $  K $
modulo  $  L $
over  $  G $, 
this being the quotient group of the group  $  C _ {r} ( K ;  G ) $
of  $  r $-
dimensional chains of  $  K $
with coefficients in  $  G $
by the subgroup  $  C _ {r} ( L ;  G ) $
of  $  r $-
dimensional chains of  $  L $
with coefficients in  $  G $. 
The homology group  $  H _ {r} ( K , L ;  G ) $
of the chain complex  $  \{ C _ {r} ( K ;  L ;  G ) , \partial  _ {r} \} $
is called the  $  r $-
dimensional relative homology group of the complex  $  K $
modulo  $  L $
with coefficient group  $  G $.

A simplicial mapping  $  f $
induces a homomorphism  $  f _ {1} $
of the group  $  C _ {r} ( K ;  G ) $
into the group  $  C _ {r} ( K ^ { \prime } ;  G ) $, 
according to the formula

$$ 
( f _ {1} c _ {r} )
( t _ {K ^ { \prime }  }  ^ {r} )  =  \sum ( \pm  c _ {r} ( t _ {K}  ^ {r} )),
$$

where  $  c _ {r} \in C _ {r} ( K ;  G ) $, 
and the sum extends over all simplices  $  t _ {k}  ^ {r} $
of  $  K $
that are mapped onto the given simplex  $  t _ {K ^ { \prime }  }  ^ {r} $
in  $  K ^ { \prime } $, 
where the sign  $  + $
or  $  - $
is chosen depending on whether or not the orientations of  $  t _ {K ^ { \prime }  }  ^ {r} $
and  $  f ( t _ {K}  ^ {r} ) $
coincide. The homomorphism  $  f _ {1} $, 
extended to the quotient groups, induces a group homomorphism of  $  C _ {r} ( K , L ;  G ) $
into  $  C _ {r} ( K ^ { \prime } , L  ^  \prime  ;  G ) $; 
the latter homomorphism commutes with the boundary operator  $  \partial  _ {r} $, 
so that one obtains a homomorphism of relative homology groups

$$ 
f _ {* r }  :  H _ {r} ( K , L ;  G )  \rightarrow \ 
H _ {r} ( K ^ { \prime } ,\ 
L  ^  \prime  ;  G ) ,
$$

called the homomorphism induced by the simplicial mapping  $  f $. 
The pair  $  ( H _ {r} , f _ {* r }  ) $
is a covariant functor from the category of simplicial pairs and simplicial mappings into the category of Abelian groups.

The inclusion mappings  $  L \subset   ^  \phi  K \subset   ^  \psi  ( K , L ) $, 
where  $  L $
and  $  K $
are the pairs  $  ( L , \emptyset ) $
and  $  ( K , \emptyset ) $, 
induce the exact sequence

$$ 
0  \rightarrow  C _ {r} ( L ;  G )
  \mathop \rightarrow \limits ^  \phi    C _ {r} ( K ;  G )   \mathop \rightarrow \limits ^  \psi   \ 
C _ {r} ( K , L ;  G )  \rightarrow  0 .
$$

Let  $  z _ {r} $
be an arbitrary cycle of the complex  $  K $
modulo  $  L $
from any element  $  h _ {r} $
of the group  $  H _ {r} ( K , L ;  G ) $; 
then there exists a chain  $  c _ {r} $
of  $  K $
such that  $  \psi ( c _ {r} ) = z _ {r} $(
$  \psi $
being an epimorphism), the chain  $  \psi ( \partial  _ {r} c _ {r} ) = \partial  _ {r} \psi ( c _ {r} ) = \partial  _ {r} z _ {r} $
of the complex  $  K $
lies in  $  L $(
that is, it vanishes on the simplices of  $  K \setminus  L $) 
and belongs to  $   \mathop{\rm Ker}  \psi $; 
the chain that is equal to it — the inverse image  $  \phi  ^ {-} 1 ( \partial  _ {r} z _ {r} ) $
under the monomorphism  $  \phi $— 
is a cycle in the complex  $  L $. 
By associating the homology class  $  h _ {r-} 1 \in H _ {r-} 1 ( L ;  G ) $
of the latter cycle with a given element  $  h _ {r} $, 
one obtains a homomorphism

$$ 
\partial  _ {* r }  :  H _ {r} ( K , L ;  G )  \rightarrow \ 
H _ {r-} 1 ( L ;  G ) ,
$$

called the connecting homomorphism. It is compatible with the functor  $  \{ H _ {r} , f _ {* r }  \} $, 
that is, the equation  $  \partial  _ {* r }  f _ {* r }  = ( f \mid  _ {L} ) _ {* r }  \partial  _ {* r }  $
holds, where  $  f \mid  _ {L} $
is the restriction of  $  f $
to  $  L $. 
The inclusion mappings  $  \phi :  L \subset  K $, 
$  \psi :  K \subset  ( K , L ) $
induce the exact sequence of groups

$$ 
\dots \leftarrow ^ { {\phi _ {*(}  r - 1) } } \ 
H _ {r-} 1 ( L ;  G )  \leftarrow ^ { {\partial  _ {*}  r } } \ 
H _ {r} ( K , L ;  G )  \leftarrow ^ { {\psi _ {*}  r } }
$$

$$ 
\leftarrow ^ { {\psi _ {*}  r } }  H _ {r} ( K ;  G )  \leftarrow ^ { {\phi _ {*}  r } } \ 
H _ {r} ( L ;  G )  \leftarrow ^ { {\partial  _ {*(}  r + 1) } } \dots ,
$$

called the homology sequence of pairs  $  ( K , L ) $.

Two simplicial mappings  $  f , g :  ( K , L ) \rightarrow ( K ^ { \prime } , L  ^  \prime  ) $
are said to be contiguous if for each simplex  $  t  ^ {r} $
in  $  K $
the simplices  $  f ( t  ^ {r} ) $
and  $  g ( t  ^ {r} ) $
are faces of the same simplex in  $  K ^ { \prime } $. 
In the category of simplicial pairs and their simplicial mappings, this relation plays the role of that of homotopy: For any contiguous simplicial mappings  $  f , g :  ( K , L ) \rightarrow ( K ^ { \prime } , L  ^  \prime  ) $
and any  $  r $, 
the induced homomorphisms  $  f _ {* r }  , g _ {* r }  $
of the group  $  H _ {r} ( K , L :  G ) $
into the group  $  H _ {r} ( K ^ { \prime } , L  ^  \prime  ;  G ) $
are the same.

An imbedding  $  i :  ( K _ {1} , L _ {1} ) \subset  ( K , L ) $
is called an excision mapping if  $  K _ {1} - L _ {1} $
equals  $  K - L $. 
The excision property is that every excision mapping  $  i $
of simplicial pairs induces, for any  $  r $, 
an isomorphism  $  i _ {* r }  :  H _ {r} ( K _ {1} , L _ {1} ;  G ) \rightarrow H _ {r} ( K , L ;  G ) $. 
The  $  r $-
dimensional homology group, with coefficient group  $  G $, 
of a complex  $  K $
consisting of a single point is the zero group for all  $  r \neq 0 $
and is isomorphic to  $  G $
for  $  r = 0 $.

Thus, the triple  $  ( H _ {r} , f _ {* r }  , \partial  _ {* r }  ) $
forms a homology theory in the sense of Steenrod–Eilenberg (see [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]]).

The cohomology theory is constructed in a similar fashion. The group  $  C  ^ {r} ( K , L ;  G ) $
of  $  r $-
dimensional infinite cochains of a complex  $  K $
modulo the subcomplex  $  L $
with coefficient group  $  G $
is the set of all  $  r $-
dimensional cochains  $  c  ^ {r} $
of  $  K $
that vanish on the simplices  $  t  ^ {r} $
of  $  L $, 
while the  $  r $-
dimensional relative cohomology group  $  H  ^ {r} ( K , L ;  G ) $
of the complex  $  K $
modulo  $  L $
with coefficient group  $  G $
is the cohomology group of the cochain complex  $  \{ C  ^ {r} ( K , L ;  G ) , \delta  ^ {r} \} $.

A simplicial mapping  $  f $
induces a homomorphism  $  f ^ { 1 } $
of the group  $  C  ^ {r} ( K ^ { \prime } ;  G ) $
into the group  $  C  ^ {r} ( K ;  G ) $:

$$ 
( f ^ { 1 } c  ^ {r} )
( t _ {K}  ^ {r} )  = \ 
c  ^ {r} ( f ( t _ {K}  ^ {r} ) ) ,\  t _ {K}  ^ {r} \in K ,\  c _ {r} \in C  ^ {r} ( K ^ { \prime } ;  G ) .
$$

The homomorphism  $  f ^ { 1 } $
also induces a homomorphism of the group  $  C  ^ {r} ( K ^ { \prime } , L  ^  \prime  ;G ) $
into the group  $  C  ^ {r} ( K , L ;  G ) $; 
the latter homomorphism commutes with the coboundary operator  $  \delta  ^ {r} $, 
and one obtains a homomorphism  $  f ^ { * r } $
of the relative cohomology groups,

$$ 
f ^ { * r } :  H  ^ {r} ( K ^ { \prime } , L  ^  \prime  ;  G )
 \rightarrow  H  ^ {r} ( K , L ;  G ) ,
$$

called the homomorphism induced by the simplicial mapping  $  f $. 
The pair  $  ( H  ^ {r} , f ^ { * r } ) $
is a contravariant functor from the category of simplicial pairs and simplicial mappings into the category of Abelian groups.

There is an exact sequence

$$ 
0  \leftarrow  C  ^ {r}
( L ;  G )  \leftarrow ^  \phi   \ 
C  ^ {r} ( K ;  G )  \leftarrow ^  \psi    C  ^ {r} ( K , L ; G)
 \leftarrow  0 ,
$$

induced by the inclusions  $  \phi :  L \subset  K $, 
$  \psi :  K \subset  ( K , L ) $. 
Any cocycle  $  z  ^ {r} \in Z  ^ {r} ( L ;  G ) $
in the cohomology class  $  h  ^ {r} \in H  ^ {r} ( L ;  G ) $
can, in an arbitrary way, be extended to a cochain  $  z _ {1}  ^ {r} \in C  ^ {r} ( K ;  G ) $
when  $  t  ^ {r} $
does not belong to the subcomplex  $  L $
of  $  K $. 
The coboundary  $  \delta  ^ {r} z _ {1}  ^ {r} $
of the cochain thus obtained vanishes on  $  L $
and belongs to the group  $  Z  ^ {r+} 1 ( K , L ;  G ) $. 
The cohomology class  $  \delta  ^ {r} h  ^ {r} \in H  ^ {r+} 1 ( K , L ;  G ) $
of this cocycle is put into correspondence with the selected class  $  h  ^ {r} $. 
This correspondence  $  h  ^ {r} \rightarrow \delta  ^ {r} h  ^ {r} $
defines a homomorphism

$$ 
\delta ^ {* r } :  H  ^ {r}
( L ;  G )  \rightarrow  H  ^ {r+} 1
( K , L ;  G ) ,
$$

called connecting homomorphism. The homomorphism  $  \delta ^ {* r } $
is compatible with the functor  $  \{ H  ^ {r} , f ^ { * r } \} $, 
in other words, the equation  $  \delta  ^ {*} r ( f \mid  _ {L} ) ^ {* r } = f ^ { * r } \delta  ^ {*} r $
holds.

The sequence of groups and homomorphisms

$$ 
\dots \rightarrow ^ { {\phi ^ {*} r } } \ 
H  ^ {r} ( L ;  G )   \mathop \rightarrow \limits ^ { {\delta ^ {* r }}  }  H  ^ {r+} 1
( K , L ;  G )   \mathop \rightarrow \limits ^ { {\psi ^ {* ( r + 1 ) }}  }
$$

$$ 
\rightarrow ^ { {\psi ^ {*} ( r + 1 ) } }  H  ^ {r+} 1 (
K ;  G )  \rightarrow ^ { {\phi ^ {*} ( r + 1 ) } }  H  ^ {r+} 1
( L ;  G )  \rightarrow ^ { {\delta ^ {*} ( r + 1 ) } } \dots ,
$$

where  $  \phi :  L \subset  K $
and  $  \psi :  K \subset  ( K , L ) $
are the inclusion mappings, is an exact sequence and is called a cohomology sequence of the pair  $  ( K , L ) $.

For any contiguous simplicial mappings  $  f , g :  ( K , L ) \rightarrow ( K ^ { \prime } , L  ^  \prime  ) $, 
and any  $  r $, 
the induced group homomorphisms  $  f ^ { * r } $, 
$  g ^ {* r } $
of  $  H  ^ {r} ( K ^ { \prime } , L  ^  \prime  ;  G ) $
into  $  H  ^ {r} ( K , L ;  G ) $
coincide; each excision mapping of simplicial pairs  $  i :  ( K _ {1} , L _ {1} ) \subset  ( K , L ) $
induces an isomorphism  $  i ^ {* r } :  H  ^ {r} ( K , L :  G ) \rightarrow H  ^ {r} ( K _ {1} , L _ {1} ;  G ) $. 
For any complex  $  K $
consisting of a single point,  $  H  ^ {r} ( K ;  G ) = 0 $
for all  $  r \neq 0 $, 
and  $  H  ^ {0} ( K ;  G ) $
is isomorphic to  $  G $. 
Thus, the triple  $  ( H  ^ {r} , f ^ { * r } , \delta ^ {* r } ) $
is a cohomology theory (in the sense of Steenrod–Eilenberg).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Combinatorial topology" , Graylock , Rochester  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.S. Aleksandrov,   "An introduction to homological dimension theory and general combinatorial topology" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lefschetz,   "Algebraic topology" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.J. Hilton,   S. Wylie,   "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press  (1960)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.S. Pontryagin,   "Grundzüge der kombinatorischen Topologie" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.M. Switzer,   "Algebraic topology - homotopy and homology" , Springer  (1975)</TD></TR></table>