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''of topological spaces''
 
''of topological spaces''
  
 
A part of algebraic topology which realizes a connection between topological and algebraic concepts. By associating to each space a certain sequence of groups, and to each continuous mapping of spaces, homomorphisms of the respective groups, homology theory uses the properties of groups and their homomorphisms to clarify the properties of spaces and mappings. Such properties include, for example, the connections between various dimensionalities, the study of which is based on the concept of excision, unlike the other part of algebraic topology — the theory of homotopy, in which deformations are used for the same purpose. Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf. [[Homology of a polyhedron|Homology of a polyhedron]]), but the axiomatic construction (including the precise limits of this concept, which had been indefinite for a long time) was imparted to it only by S. Eilenberg and N. Steenrod [[#References|[3]]] (cf. [[Algebraic topology|Algebraic topology]]; [[Homology group|Homology group]]; [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]]).
 
A part of algebraic topology which realizes a connection between topological and algebraic concepts. By associating to each space a certain sequence of groups, and to each continuous mapping of spaces, homomorphisms of the respective groups, homology theory uses the properties of groups and their homomorphisms to clarify the properties of spaces and mappings. Such properties include, for example, the connections between various dimensionalities, the study of which is based on the concept of excision, unlike the other part of algebraic topology — the theory of homotopy, in which deformations are used for the same purpose. Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf. [[Homology of a polyhedron|Homology of a polyhedron]]), but the axiomatic construction (including the precise limits of this concept, which had been indefinite for a long time) was imparted to it only by S. Eilenberg and N. Steenrod [[#References|[3]]] (cf. [[Algebraic topology|Algebraic topology]]; [[Homology group|Homology group]]; [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]]).
  
According to this construction a homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478601.png" /> consists of three functions: 1) relative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478602.png" />-dimensional homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478603.png" /> of a pair of topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478605.png" />, which assign to each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478606.png" /> and each integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478607.png" /> an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478608.png" />; 2) the homomorphism
+
According to this construction a homology theory $  \{ H, \partial  \} $
 +
consists of three functions: 1) relative $  r $-
 +
dimensional homology groups $  H _ {r} ( X, A) $
 +
of a pair of topological spaces $  ( X, A) $,  
 +
$  A \subset  X $,  
 +
which assign to each pair $  ( X, A) $
 +
and each integer $  r $
 +
an Abelian group $  H _ {r} ( X, A) $;  
 +
2) the homomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h0478609.png" /></td> </tr></table>
+
$$
 +
H _ {r} ( f  )  = \
 +
f _ {*} : H _ {r} ( X, A)  \rightarrow  H _ {r} ( Y, B),
 +
$$
  
which is assigned to a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786010.png" /> and a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786011.png" />, and which is called the homomorphism induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786012.png" />; and 3) the boundary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786013.png" />, which assigns to each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786014.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786015.png" /> a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786017.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786018.png" /> (the so-called absolute group of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786019.png" />, which is the group of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786020.png" />). These functions must satisfy the following axioms.
+
which is assigned to a continuous mapping $  f: ( X, A) \rightarrow ( Y, B) $
 +
and a number $  r $,  
 +
and which is called the homomorphism induced by $  f $;  
 +
and 3) the boundary operator $  \partial  $,  
 +
which assigns to each pair $  ( X, A) $
 +
and each $  r $
 +
a homomorphism $  \partial  $
 +
of $  H _ {r} ( X, A) $
 +
into $  H _ {r-} 1 ( A) $(
 +
the so-called absolute group of the space $  A $,  
 +
which is the group of the pair $  ( A, \emptyset ) $).  
 +
These functions must satisfy the following axioms.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786021.png" /> is the identity mapping, so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786022.png" />.
+
1) If $  f $
 +
is the identity mapping, so is $  f _ {*} $.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786023.png" />.
+
2) $  ( gf  ) _ {*}  = g _ {*} f _ {*} $.
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786024.png" />.
+
3) $  \partial  f _ {*}  = ( f \mid  _ {A} ) _ {*} \partial  $.
  
 
4) The exactness axiom. If
 
4) The exactness axiom. If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786025.png" /></td> </tr></table>
+
$$
 +
i: A  \rightarrow  X \  \textrm{ and } \  j: X  \rightarrow  ( X, A)
 +
$$
  
 
are the natural inclusions, then the sequence
 
are the natural inclusions, then the sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786026.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H _ {r} ( A)  \rightarrow ^ { {i _ *} } \
 +
H _ {r} ( X)  \rightarrow ^ { {j _ *} }  H _ {r} ( X, A)
 +
  \mathop \rightarrow \limits ^  \partial    H _ {r - 1 }  ( A)  \rightarrow \dots ,
 +
$$
  
the so-called homology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786027.png" />, is exact, that is, the image of each incoming homomorphism equals the kernel of the outgoing one.
+
the so-called homology sequence of the pair $  ( X, A) $,
 +
is exact, that is, the image of each incoming homomorphism equals the kernel of the outgoing one.
  
 
5) The homotopy axiom. If the mappings
 
5) The homotopy axiom. If the mappings
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786028.png" /></td> </tr></table>
+
$$
 +
f, g: ( X, A)  \rightarrow  ( Y, B)
 +
$$
  
are homotopic, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786029.png" />.
+
are homotopic, then $  f _ {*} = g _ {*} $.
  
6) The excision axiom. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786030.png" /> is an open subset of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786031.png" /> and its closure is contained in the interior of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786032.png" />, then the inclusion
+
6) The excision axiom. If $  U $
 +
is an open subset of the space $  X $
 +
and its closure is contained in the interior of the subspace $  A $,  
 +
then the inclusion
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786033.png" /></td> </tr></table>
+
$$
 +
e: ( X \setminus  U, A \setminus  U)  \rightarrow  ( X, A)
 +
$$
  
induces an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786034.png" />.
+
induces an isomorphism $  e _ {*} $.
  
7) The dimension axiom. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786035.png" /> is a one-point space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786037.png" />.
+
7) The dimension axiom. If $  X $
 +
is a one-point space, then $  H _ {r} ( X) = 0 $
 +
for all $  r \neq 0 $.
  
Instead of taking the category of all pairs of spaces as the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786038.png" />, it is also possible to take an arbitrary [[Category|category]] of pairs of spaces, e.g. the category of pairs of compact spaces or the category of pairs consisting of polyhedra and their subpolyhedra. However, such a category must contain along with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786039.png" /> also the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786044.png" />, the cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786046.png" />, and some one-point space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786047.png" />, with all their inclusions. Such a category must also include all pairs and mappings which are encountered in the axioms or in the theorems. On the other hand, instead of taking the category of all Abelian groups as the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786048.png" />, it is also possible to use other categories, e.g. the category of topological, in particular that of compact, groups with continuous homomorphisms, or the category of modules over some ring with linear homomorphisms.
+
Instead of taking the category of all pairs of spaces as the domain of definition of $  H _ {r} $,  
 +
it is also possible to take an arbitrary [[Category|category]] of pairs of spaces, e.g. the category of pairs of compact spaces or the category of pairs consisting of polyhedra and their subpolyhedra. However, such a category must contain along with $  ( X, A) $
 +
also the pairs $  ( \emptyset , \emptyset ) $,
 +
$  ( X, \emptyset ) = X $,
 +
$  ( A, \emptyset ) = A $,  
 +
$  ( A, A) $,  
 +
$  ( X, X) $,  
 +
the cylinder $  ( X, A) \times I $,  
 +
where $  I = [ 0, 1] $,
 +
and some one-point space $  P _ {0} $,  
 +
with all their inclusions. Such a category must also include all pairs and mappings which are encountered in the axioms or in the theorems. On the other hand, instead of taking the category of all Abelian groups as the range of $  H _ {r} $,  
 +
it is also possible to use other categories, e.g. the category of topological, in particular that of compact, groups with continuous homomorphisms, or the category of modules over some ring with linear homomorphisms.
  
Axioms 1 and 2 mean that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786049.png" /> is a covariant [[Functor|functor]] from some category of pairs of spaces into the category of groups. Axiom 3 means that the boundary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786050.png" /> is a natural transformation of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786051.png" /> to the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786052.png" />. Axiom 4, which connects the functors of all dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786053.png" />, is sometimes replaced by the weaker requirement: The sequence should only be partially exact, i.e. the image should be included in the kernel (cf. [[Exact sequence|Exact sequence]]); an important example of a partially exact homology theory is the Aleksandrov–Čech homology theory. Axiom 5 may be written in an equivalent form: If
+
Axioms 1 and 2 mean that $  H _ {r} $
 +
is a covariant [[Functor|functor]] from some category of pairs of spaces into the category of groups. Axiom 3 means that the boundary operator $  \partial  $
 +
is a natural transformation of the functor $  H _ {r} ( X , - ) $
 +
to the functor $  H _ {r-} 1 ( - ) $.  
 +
Axiom 4, which connects the functors of all dimensions $  r $,  
 +
is sometimes replaced by the weaker requirement: The sequence should only be partially exact, i.e. the image should be included in the kernel (cf. [[Exact sequence|Exact sequence]]); an important example of a partially exact homology theory is the Aleksandrov–Čech homology theory. Axiom 5 may be written in an equivalent form: If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786054.png" /></td> </tr></table>
+
$$
 +
f _ {0} , f _ {1} : ( X, A)  \rightarrow  ( X, A) \times I
 +
$$
  
are mappings defined by the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786056.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786057.png" />. Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory. Axiom 7, which ensures the geometric meaning of the dimensionality index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786058.png" />, is often omitted in modern studies. One then obtains so-called generalized homology theories, an important example of which is [[Bordism|bordism]] theory.
+
are mappings defined by the formulas $  f _ {0} ( x) = ( x, 0) $,
 +
$  f _ {1} ( x) = ( x, 1) $,  
 +
then $  f _ {0*} = f _ {1*} $.  
 +
Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory. Axiom 7, which ensures the geometric meaning of the dimensionality index $  r $,  
 +
is often omitted in modern studies. One then obtains so-called generalized homology theories, an important example of which is [[Bordism|bordism]] theory.
  
There exists a cohomology theory dual to a homology theory (cf. [[Duality|Duality]] in topology). It is given by relative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786059.png" />-dimensional cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786060.png" />, which are contravariant functors from the category of pairs of topological spaces into the category of Abelian groups, with induced homomorphisms
+
There exists a cohomology theory dual to a homology theory (cf. [[Duality|Duality]] in topology). It is given by relative $  r $-
 +
dimensional cohomology groups $  H  ^ {r} ( X, A) $,
 +
which are contravariant functors from the category of pairs of topological spaces into the category of Abelian groups, with induced homomorphisms
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786061.png" /></td> </tr></table>
+
$$
 +
H  ^ {r} ( f  )  = f ^ { * } : H  ^ {r} ( Y, B)  \rightarrow  H  ^ {r} ( X, A)
 +
$$
  
 
and coboundary operators
 
and coboundary operators
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786062.png" /></td> </tr></table>
+
$$
 +
\delta : H  ^ {r} ( A)  \rightarrow  H ^ {r + 1 } ( X, A).
 +
$$
  
 
The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms. For instance, the exactness axiom requires the existence of an exact cohomology sequence
 
The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms. For instance, the exactness axiom requires the existence of an exact cohomology sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786063.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H ^ {r - 1 } ( A)  \mathop \rightarrow \limits ^  \delta  \
 +
H  ^ {r} ( X, A)  \rightarrow ^ { {j  ^ {*}} }  H  ^ {r} ( X)  \rightarrow ^ { {i  ^ {*}} }
 +
H  ^ {r} ( A)  \rightarrow \dots .
 +
$$
  
And, analogously, there are also generalized cohomology theories, important examples of which are [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786064.png" />-theory]] and [[Cobordism|cobordism]] theory. The facts given below concerning homology theory have cohomology analogues.
+
And, analogously, there are also generalized cohomology theories, important examples of which are [[K-theory| $  K $-
 +
theory]] and [[Cobordism|cobordism]] theory. The facts given below concerning homology theory have cohomology analogues.
  
The coefficient group of a homology or a cohomology theory is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786065.png" /> or, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786066.png" />. The so-called reduced groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786067.png" /> are often conveniently substituted for the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786068.png" />: the reduced zero-dimensional homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786069.png" /> is the kernel of the homomorphism
+
The coefficient group of a homology or a cohomology theory is the group $  H _ {0} ( P _ {0} ) $
 +
or, respectively, $  H  ^ {0} ( P _ {0} ) $.  
 +
The so-called reduced groups $  \widetilde{H}  _ {r} ( X, A) $
 +
are often conveniently substituted for the groups $  H _ {r} ( X, A) $:  
 +
the reduced zero-dimensional homology group $  \widetilde{H}  _ {0} ( X) $
 +
is the kernel of the homomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786070.png" /></td> </tr></table>
+
$$
 +
l _ {*} : H _ {0} ( X)  \rightarrow  H _ {0} ( P _ {0} )
 +
$$
  
induced by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786071.png" />, while the reduced zero-dimensional cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786072.png" /> is the quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786073.png" /> by the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786074.png" />; the reduced groups of other dimensions are identical with the initial ones: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786076.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786078.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786079.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786080.png" />. The replacement of the ordinary by the reduced groups permits one to convert the homology sequence into a reduced homology sequence.
+
induced by the mapping $  l:  X \rightarrow P _ {0} $,  
 +
while the reduced zero-dimensional cohomology group $  \widetilde{H}  {}  ^ {0} ( X) $
 +
is the quotient group of $  H  ^ {0} ( X) $
 +
by the image $  l  ^ {*} ( H  ^ {0} ( P _ {0} )) $;  
 +
the reduced groups of other dimensions are identical with the initial ones: $  \widetilde{H}  _ {r} ( X) = H _ {r} ( X) $,  
 +
$  r \neq 0 $.  
 +
Thus, $  H _ {0} ( X) \sim \widetilde{H}  _ {0} ( X) \oplus G $.  
 +
If $  A \neq \emptyset $,  
 +
then $  \widetilde{H}  _ {r} ( X, A) = H _ {r} ( X, A) $
 +
for all $  r $.  
 +
The replacement of the ordinary by the reduced groups permits one to convert the homology sequence into a reduced homology sequence.
  
The axioms of homology theory are not independent. E.g., axiom 1 is a consequence of the axioms 2, 3 and 4. The system of axioms is compatible, as is seen from the example of the trivial theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786081.png" />; non-trivial examples include the Aleksandrov–Čech cohomology theory, [[Singular homology|singular homology]], etc. Regarding uniqueness, the following holds: A homomorphism of a homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786082.png" /> into a homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786083.png" /> is a system of homomorphisms
+
The axioms of homology theory are not independent. E.g., axiom 1 is a consequence of the axioms 2, 3 and 4. The system of axioms is compatible, as is seen from the example of the trivial theory $  H _ {r} ( X, A) = 0 $;  
 +
non-trivial examples include the Aleksandrov–Čech cohomology theory, [[Singular homology|singular homology]], etc. Regarding uniqueness, the following holds: A homomorphism of a homology theory $  \{ H, \delta \} $
 +
into a homology theory $  \{ H  ^  \prime  , \partial  ^  \prime  \} $
 +
is a system of homomorphisms
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786084.png" /></td> </tr></table>
+
$$
 +
h ( X, A; r): H _ {r} ( X, A)  \rightarrow  H _ {r}  ^  \prime  ( X, A)
 +
$$
  
 
such that
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786085.png" /></td> </tr></table>
+
$$
 +
H  ^  \prime  ( f  ) \circ h ( X, A; r)  = h ( Y, B; r) \circ H ( f  )
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786086.png" /></td> </tr></table>
+
$$
 +
\partial  ^  \prime  \circ h ( X, A; r)  = h ( A; r - 1) \circ \partial  .
 +
$$
  
If all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786087.png" /> are isomorphisms, then the homology theories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786089.png" /> are called isomorphic. A homology theory on finite polyhedra is uniquely determined by its coefficient group. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786090.png" /> is an arbitrary homomorphism of the coefficient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786091.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786092.png" /> into the coefficient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786093.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786094.png" />, then for each polyhedral pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786095.png" /> there exists a unique homomorphism
+
If all h( X, A;  r) $
 +
are isomorphisms, then the homology theories $  \{ H, \partial  \} $
 +
and $  \{ H  ^  \prime  , \partial  ^  \prime  \} $
 +
are called isomorphic. A homology theory on finite polyhedra is uniquely determined by its coefficient group. More precisely, if $  h _ {0} : G \rightarrow G  ^  \prime  $
 +
is an arbitrary homomorphism of the coefficient group $  G $
 +
of $  \{ H, \partial  \} $
 +
into the coefficient group $  G  ^  \prime  $
 +
of $  \{ H  ^  \prime  , \partial  ^  \prime  \} $,  
 +
then for each polyhedral pair $  ( X, A) $
 +
there exists a unique homomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786096.png" /></td> </tr></table>
+
$$
 +
h ( X, A; r): H _ {r} ( X, A)  \rightarrow  H _ {r}  ^  \prime  ( X, A)
 +
$$
  
with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786097.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786098.png" /> is an isomorphism, so are all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h04786099.png" />. Since the homology groups of negative dimension of a triangulable pair are trivial, the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860101.png" />, is valid for any homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860102.png" /> as well. The uniqueness theorem is also valid for wider categories of spaces if the homology theory satisfies appropriate additional axioms.
+
with the property $  h( P _ {0} ;  0) = h _ {0} $.  
 +
If $  h _ {0} $
 +
is an isomorphism, so are all h( X, A;  r) $.  
 +
Since the homology groups of negative dimension of a triangulable pair are trivial, the equality $  H _ {r} ( X, A) = 0 $,  
 +
$  r < 0 $,  
 +
is valid for any homology theory $  \{ H, \partial  \} $
 +
as well. The uniqueness theorem is also valid for wider categories of spaces if the homology theory satisfies appropriate additional axioms.
  
The homology groups are topological and also homotopy invariants: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860103.png" /> is a homotopy equivalence, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860104.png" /> is an isomorphism. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860105.png" /> is a contractible space — a cell, in particular — then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860107.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860108.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860109.png" /> is a homotopy equivalence, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860110.png" /> and, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860112.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860113.png" /> is a retract of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860114.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860115.png" /> is a monomorphism, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860116.png" /> is an epimorphism, the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860117.png" /> is trivial, and
+
The homology groups are topological and also homotopy invariants: If $  f $
 +
is a homotopy equivalence, then $  f _ {*} $
 +
is an isomorphism. If $  X $
 +
is a contractible space — a cell, in particular — then $  H _ {r} ( X) = 0 $,  
 +
$  r \neq 0 $,  
 +
and $  H _ {0} ( X) \sim G $.  
 +
If $  i : A \subset  X $
 +
is a homotopy equivalence, then $  H _ {r} ( X, A) = 0 $
 +
and, for any $  X $,
 +
$  H _ {r} ( X, X) = 0 $.  
 +
If $  A $
 +
is a retract of the space $  X $,  
 +
then $  i _ {*} $
 +
is a monomorphism, $  j _ {*} $
 +
is an epimorphism, the operation $  \partial  $
 +
is trivial, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860118.png" /></td> </tr></table>
+
$$
 +
H _ {r} ( X)  \sim  H _ {r} ( A) \oplus H _ {r} ( X, A).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860119.png" /> is deformable into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860120.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860121.png" /> is an epimorphism, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860122.png" /> is trivial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860123.png" /> is a monomorphism, and
+
If $  X $
 +
is deformable into $  A $,  
 +
then $  i _ {*} $
 +
is an epimorphism, $  j _ {*} $
 +
is trivial, $  \partial  $
 +
is a monomorphism, and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860124.png" /></td> </tr></table>
+
$$
 +
H _ {r} ( A)  \sim  H _ {r} ( X) \oplus H _ {r + 1 }  ( X, A).
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860125.png" /> denote the [[Suspension|suspension]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860126.png" />. The following isomorphism is then valid:
+
Let $  S( X) $
 +
denote the [[Suspension|suspension]] over $  X $.  
 +
The following isomorphism is then valid:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860127.png" /></td> </tr></table>
+
$$
 +
\widetilde{H}  _ {r} ( X)  \sim  \widetilde{H}  _ {r + 1 }  ( S ( X)).
 +
$$
  
This makes it possible to compute the homology groups of the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860128.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860129.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860130.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860131.png" />; consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860132.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860133.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860134.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860135.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860136.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860137.png" />.
+
This makes it possible to compute the homology groups of the spheres $  S  ^ {n} $:  
 +
$  \widetilde{H}  _ {r} ( S  ^ {n} ) = 0 $
 +
if $  r \neq n $,  
 +
and $  \widetilde{H}  _ {n} ( S  ^ {n} ) \sim G $;  
 +
consequently, $  H _ {r} ( S  ^ {n} ) = 0 $
 +
if $  n \neq r \neq 0 $;  
 +
$  H _ {r} ( S  ^ {n} ) \sim G $
 +
if $  n \neq r = 0 $
 +
or $  n = r \neq 0 $;  
 +
and $  H _ {0} ( S  ^ {0} ) \sim G \oplus G $.
  
An important role in homology theory is played by homology sequences of triples and triads. In the case of a [[Triple|triple]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860139.png" />, of spaces, the boundary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860140.png" /> is defined as the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860141.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860142.png" /> is the inclusion. There results the so-called homology sequence of the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860143.png" /> (which, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860144.png" />, reduces to the homology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860145.png" />):
+
An important role in homology theory is played by homology sequences of triples and triads. In the case of a [[Triple|triple]] $  ( X, A, B) $,  
 +
$  X \supset A \supset B $,  
 +
of spaces, the boundary operator $  \partial  ^  \prime  = k _ {*}  ^  \prime  \partial  $
 +
is defined as the composition $  k _ {*}  ^  \prime  \circ \partial  $,  
 +
where $  k  ^  \prime  : A \rightarrow ( A, B) $
 +
is the inclusion. There results the so-called homology sequence of the triple $  ( X, A, B) $(
 +
which, for $  B = \emptyset $,  
 +
reduces to the homology sequence of the pair $  ( X, A) $):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860146.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H _ {r} ( A, B)  \rightarrow ^ { {i _ *}  ^  \prime  } \
 +
H _ {r} ( X, B) \
 +
\rightarrow ^ { {j _ *}  ^  \prime  } \
 +
H _ {r} ( X, A) \
 +
\rightarrow ^ { {\partial  ^  \prime } }
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860147.png" /></td> </tr></table>
+
$$
 +
\rightarrow ^ { {\partial  ^  \prime } }  H _ {r - 1 }  ( A, B)  \rightarrow \dots ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860149.png" /> are the inclusions. This sequence is exact. If the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860150.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860152.png" />, are trivial for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860153.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860154.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860155.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860156.png" />, are isomorphisms, and vice versa. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860157.png" /> is the union of non-intersecting closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860159.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860160.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860161.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860162.png" /> is isomorphic to the direct sum of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860163.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860164.png" />. A triad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860165.png" /> is a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860166.png" /> with an ordered pair of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860167.png" /> (cf. also [[Triads|Triads]]). It is a proper triad if the inclusions
+
where $  i  ^  \prime  : ( A, B) \rightarrow ( X, B) $
 +
and $  j  ^  \prime  : ( X, B) \rightarrow ( X, A) $
 +
are the inclusions. This sequence is exact. If the groups $  H _ {r} ( X, A) $,  
 +
respectively $  H _ {r} ( X, B) $,
 +
$  H _ {r} ( A, B) $,  
 +
are trivial for all $  r $,  
 +
then $  i _ {*}  ^  \prime  $,  
 +
respectively $  \partial  ^  \prime  $,  
 +
$  j _ {*}  ^  \prime  $,  
 +
are isomorphisms, and vice versa. If $  X $
 +
is the union of non-intersecting closed sets $  X _ {i} $,  
 +
$  i = 1 \dots n $,  
 +
and $  A = A _ {1} \cup \dots \cup A _ {n} $,  
 +
where $  A _ {i} \subset  X _ {i} $,  
 +
then $  H _ {r} ( X, A) $
 +
is isomorphic to the direct sum of the groups $  H _ {r} ( X _ {i} , A _ {i} ) $,  
 +
$  i = 1 \dots n $.  
 +
A triad $  ( X;  A, B) $
 +
is a space $  X $
 +
with an ordered pair of subspaces $  A, B $(
 +
cf. also [[Triads|Triads]]). It is a proper triad if the inclusions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860168.png" /></td> </tr></table>
+
$$
 +
k: ( A, A \cap B)  \rightarrow  ( A \cup B, B),\ \
 +
l: ( B, A \cap B)  \rightarrow  ( A \cup B, A)
 +
$$
  
 
induce isomorphisms, or if the decomposition
 
induce isomorphisms, or if the decomposition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860169.png" /></td> </tr></table>
+
$$
 +
H _ {r} ( A \cup B, A \cap B)  \sim \
 +
H _ {r} ( A, A \cap B) \oplus
 +
H _ {r} ( B, A \cap B)
 +
$$
  
 
is valid. Further, the boundary operator
 
is valid. Further, the boundary operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860170.png" /></td> </tr></table>
+
$$
 +
\overline \partial \; : H _ {r} ( X, A \cup B)  \rightarrow \
 +
H _ {r - 1 }  ( A, A \cap B)
 +
$$
  
is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860171.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860172.png" />. This generates the exact homology sequence of the triad:
+
is defined as $  k _ {*}  ^ {-} 1 \circ m _ {*} \circ \partial  $,  
 +
where $  m : A \cup B \subset  ( A \cup B, B) $.  
 +
This generates the exact homology sequence of the triad:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860173.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H _ {r} ( A, A \cap B) \
 +
\rightarrow ^ { {p _ *} } \
 +
H _ {r} ( X, B) \
 +
\rightarrow ^ { {q _ *} } \
 +
H _ {r} ( X, A \cup B) \
 +
\mathop \rightarrow \limits ^ { {\overline \partial \; }}
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860174.png" /></td> </tr></table>
+
$$
 +
\mathop \rightarrow \limits ^ { {\overline \partial \; }}  H _ {r - 1 }  ( A, A \cap B)  \rightarrow \dots ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860176.png" /> are the inclusions (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860177.png" />, this sequence reduces to the homology sequence of the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860178.png" />).
+
where $  p: ( A, A \cap B) \rightarrow ( X, B) $,
 +
$  q: ( X, B) \rightarrow ( X, A \cup B) $
 +
are the inclusions (if $  B \subset  A $,  
 +
this sequence reduces to the homology sequence of the triple $  ( X, A, B) $).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860179.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860180.png" /> and suppose the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860181.png" /> satisfy the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860182.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860183.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860185.png" />. The following addition theorems are then valid.
+
Let $  X = A \cup B $,  
 +
$  A \cap B = C $
 +
and suppose the mappings $  h, h _ {1} , h _ {2} : ( X, C) \rightarrow ( Y, D) $
 +
satisfy the relations $  h _ {1} | _ {A} = h | _ {A} $,  
 +
$  h _ {2} | _ {B} = h | _ {B} $,  
 +
h _ {1} ( B) \subset  D $,  
 +
h _ {2} ( A) \subset  D $.  
 +
The following addition theorems are then valid.
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860186.png" />.
+
1) $  h _ {*} = h _ {1* }  + h _ {2* }  $.
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860187.png" /> is contractible and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860188.png" /> are defined, respectively, by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860189.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860190.png" />, then the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860191.png" /> holds for the induced homomorphisms of the reduced groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860192.png" />.
+
2) If $  D $
 +
is contractible and if $  f, f _ {1} , f _ {2} : X \rightarrow Y $
 +
are defined, respectively, by means of $  h, h _ {1} $
 +
and h _ {2} $,  
 +
then the equality $  f _ {*} = f _ {1* }  + f _ {2* }  $
 +
holds for the induced homomorphisms of the reduced groups $  f _ {*} , f _ {1* }  , f _ {2* }  : \widetilde{H}  _ {r} ( X) \rightarrow \widetilde{H}  _ {r} ( Y) $.
  
 
Define the homomorphism
 
Define the homomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860193.png" /></td> </tr></table>
+
$$
 +
s : H _ {r} ( C)  \rightarrow  H _ {r} ( A) \oplus H _ {r} ( B)
 +
$$
  
by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860194.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860195.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860196.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860197.png" /> are the inclusions. Similarly, define
+
by $  s ( c) = ( s _ {1* }  ( c), - s _ {2* }  ( c)) $,  
 +
$  c \in H _ {r} ( C) $,  
 +
where $  s _ {1} : C \rightarrow A $,  
 +
$  s _ {2} : C \rightarrow B $
 +
are the inclusions. Similarly, define
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860198.png" /></td> </tr></table>
+
$$
 +
t: H _ {r} ( A) \oplus
 +
H _ {r} ( B)  \rightarrow  H _ {r} ( X)
 +
$$
  
by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860199.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860200.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860202.png" /> are the inclusions. Finally, define
+
by $  t ( a, b) = t _ {1* }  ( a) + t _ {2* }  ( b) $,
 +
$  ( a, b) \in H _ {r} ( A) \oplus H _ {r} ( B) $,  
 +
where $  t _ {1} : A \rightarrow X $,  
 +
$  t _ {2} : B \rightarrow X $
 +
are the inclusions. Finally, define
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860203.png" /></td> </tr></table>
+
$$
 +
\Delta : H _ {r} ( X)  \rightarrow  H _ {r} ( C),
 +
$$
  
by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860204.png" />, where
+
by $  \Delta = \partial  u _ {*}  ^ {-} 1 v _ {*} $,  
 +
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860205.png" /></td> </tr></table>
+
$$
 +
v: X  \rightarrow  ( X, B),\ \
 +
u: ( A, C)  \rightarrow  ( X, B)
 +
$$
  
 
are the inclusions. One then obtains the so-called Mayer–Vietoris sequence of the proper triad:
 
are the inclusions. One then obtains the so-called Mayer–Vietoris sequence of the proper triad:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860206.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  H _ {r} ( C) \
 +
\mathop \rightarrow \limits ^ { {s }}  H _ {r} ( A)
 +
\oplus H _ {r} ( B) \
 +
\mathop \rightarrow \limits ^ { {t }}  H _ {r} ( X) \
 +
\mathop \rightarrow \limits ^  \Delta  \
 +
H _ {r - 1 }  ( C)  \rightarrow \dots ,
 +
$$
  
which is exact and which relates the homology groups of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860208.png" /> with the homology groups of their union and intersection. Then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860209.png" />, one may pass to a similar sequence for the reduced groups, which implies that:
+
which is exact and which relates the homology groups of the spaces $  A $
 +
and $  B $
 +
with the homology groups of their union and intersection. Then, if $  C \neq \emptyset $,  
 +
one may pass to a similar sequence for the reduced groups, which implies that:
  
1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860210.png" /> is contractible, then
+
1) if $  A \cap B $
 +
is contractible, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860211.png" /></td> </tr></table>
+
$$
 +
\widetilde{H}  _ {r} ( A \cup B)  \sim \
 +
\widetilde{H}  _ {r} ( A) \oplus \widetilde{H}  _ {r} ( B);
 +
$$
  
2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860212.png" /> is contractible, then
+
2) if $  A \cup B $
 +
is contractible, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860213.png" /></td> </tr></table>
+
$$
 +
\widetilde{H}  _ {r} ( A \cap B)  \sim \
 +
\widetilde{H}  _ {r} ( A) \oplus \widetilde{H}  _ {r} ( B);
 +
$$
  
3) if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860214.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860215.png" /> are contractible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860216.png" /> defines an isomorphism
+
3) if both $  A $
 +
and $  B $
 +
are contractible, then $  \Delta $
 +
defines an isomorphism
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860217.png" /></td> </tr></table>
+
$$
 +
\widetilde{H}  _ {r} ( A \cup B)  \sim  \widetilde{H}  _ {r - 1 }  ( A \cap B).
 +
$$
  
These results make it possible to compute the homology groups of various spaces. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860218.png" /> is a closed orientable surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860219.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860220.png" /> is isomorphic to the coefficient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860221.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860222.png" />; to the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860223.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860224.png" /> copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860225.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860226.png" />; and to 0 in the remaining cases. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860227.png" /> is a closed non-orientable surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860228.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860229.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860230.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860231.png" />; to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860232.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860233.png" /> is the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860234.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860235.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860236.png" />; to the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860237.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860238.png" /> consisting of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860239.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860240.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860241.png" />; and to 0 in the remaining cases. Thus, homology theory gives a topological classification of closed surfaces. For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860242.png" />-dimensional real projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860243.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860244.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860245.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860246.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860247.png" /> and odd; to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860248.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860249.png" /> is odd and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860250.png" />; to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860251.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860252.png" /> is even and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860253.png" />; and to 0 in the remaining cases. The homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860254.png" /> of the complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860255.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860256.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860257.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860258.png" /> is even and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860259.png" />; and to 0 in the remaining cases. The homology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860260.png" /> of the [[Lens space|lens space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860261.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860262.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860263.png" />; to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860264.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860265.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860266.png" />; to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860267.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860268.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860269.png" />; and to 0 in the remaining cases.
+
These results make it possible to compute the homology groups of various spaces. For instance, if $  X $
 +
is a closed orientable surface of genus $  n $,  
 +
then $  H _ {r} ( X) $
 +
is isomorphic to the coefficient group $  G $
 +
if $  r = 0, 2 $;  
 +
to the direct sum $  G  ^ {2n} $
 +
of $  2n $
 +
copies of $  G $
 +
if $  r = 1 $;  
 +
and to 0 in the remaining cases. If $  X $
 +
is a closed non-orientable surface of genus $  n $,  
 +
then $  H _ {r} ( X) $
 +
is isomorphic to $  G $
 +
if $  r = 0 $;  
 +
to $  G  ^ {n-} 1 \oplus G _ {2} $,  
 +
where $  G _ {2} $
 +
is the quotient group $  G/2G $,  
 +
$  2G = \{ {2g } : {g \in G } \} $,  
 +
if $  r = 1 $;  
 +
to the subgroup $  T _ {2} ( G) $
 +
of $  G $
 +
consisting of all elements $  g \in G $
 +
with $  2g = 0 $
 +
if $  r = 2 $;  
 +
and to 0 in the remaining cases. Thus, homology theory gives a topological classification of closed surfaces. For an $  n $-
 +
dimensional real projective space $  P  ^ {n} $
 +
the group $  H _ {r} ( P  ^ {n} ) $
 +
is isomorphic to $  G $
 +
if $  r = 0 $
 +
or $  r = n $
 +
and odd; to $  G _ {2} $
 +
if $  r $
 +
is odd and $  0 < r < n $;  
 +
to $  T _ {2} ( G) $
 +
if $  r $
 +
is even and $  0 < r \leq  n $;  
 +
and to 0 in the remaining cases. The homology group $  H _ {r} ( \mathbf C P  ^ {n} ) $
 +
of the complex projective space $  \mathbf C P  ^ {n} $
 +
of dimension $  2n $
 +
is isomorphic to $  G $
 +
if $  r $
 +
is even and 0 \leq  r \leq  2n $;  
 +
and to 0 in the remaining cases. The homology group $  H _ {r} ( L _ {p,q} ) $
 +
of the [[Lens space|lens space]] $  L _ {p,q} $
 +
is isomorphic to $  G $
 +
if $  r = 0, 3 $;  
 +
to $  G _ {p} = G/pG $,  
 +
where $  pG = \{ {pg } : {g \in G } \} $,  
 +
if $  r = 1 $;  
 +
to $  T _ {p} ( G) $,  
 +
where $  T _ {p} ( G) = \{ {g \in G } : {pg = 0 } \} $,  
 +
if $  r = 2 $;  
 +
and to 0 in the remaining cases.
  
From the various applications of the results described above certain fundamental theorems are stated here. First of all, the invariance of dimension: spheres, as well as Euclidean spaces, of different dimensions are not homeomorphic; in fact, if two polyhedra are homeomorphic, then they have the same dimension. Furthermore, the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860270.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860271.png" /> is an extension of a given mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860272.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860273.png" />, yields various criteria of extendability and retractibility of mappings; for example, a mapping of a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860274.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860275.png" />, of non-zero degree into itself is not extendable to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860276.png" />-dimensional ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860277.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860278.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860279.png" /> is not a retract of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860280.png" /> for any natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860281.png" />. This, in turn, yields Brouwer's fixed-point theorem: Any continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860282.png" /> has a fixed point. Finally, it may be proved that a unit tangent vector field exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860283.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860284.png" /> is odd, while the theory of triads yields several theorems on the degrees of mappings; in particular, it is possibly to give a new proof of the fundamental theorem of algebra.
+
From the various applications of the results described above certain fundamental theorems are stated here. First of all, the invariance of dimension: spheres, as well as Euclidean spaces, of different dimensions are not homeomorphic; in fact, if two polyhedra are homeomorphic, then they have the same dimension. Furthermore, the equality $  f _ {*} i _ {*} = g _ {*} $,  
 +
where $  f: X \rightarrow Y $
 +
is an extension of a given mapping $  g: A \rightarrow Y $,  
 +
$  A \subset  X $,  
 +
yields various criteria of extendability and retractibility of mappings; for example, a mapping of a sphere $  S  ^ {n-} 1 $,  
 +
$  n > 1 $,  
 +
of non-zero degree into itself is not extendable to the $  n $-
 +
dimensional ball $  E  ^ {n} $
 +
with boundary $  S  ^ {n-} 1 $,  
 +
and $  S  ^ {n-} 1 $
 +
is not a retract of $  E  ^ {n} $
 +
for any natural $  n $.  
 +
This, in turn, yields Brouwer's fixed-point theorem: Any continuous mapping $  E  ^ {n} \rightarrow E  ^ {n} $
 +
has a fixed point. Finally, it may be proved that a unit tangent vector field exists on $  S  ^ {n} $
 +
if and only if $  n $
 +
is odd, while the theory of triads yields several theorems on the degrees of mappings; in particular, it is possibly to give a new proof of the fundamental theorem of algebra.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian) {{MR|0076324}} {{ZBL|0441.55002}} {{ZBL|0097.15903}} {{ZBL|0024.08404}} {{ZBL|66.0947.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1942) {{MR|0007093}} {{ZBL|0061.39302}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952) {{MR|0050886}} {{ZBL|0047.41402}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S.-T. Hu, "Homology theory" , Holden-Day (1966) {{MR|0217786}} {{ZBL|0145.19705}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S.-T. Hu, "Cohomology theory" , Markham , Chicago (1968) {{MR|0234448}} {{ZBL|0165.26101}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980) {{MR|0606196}} {{ZBL|0434.55001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian) {{MR|0076324}} {{ZBL|0441.55002}} {{ZBL|0097.15903}} {{ZBL|0024.08404}} {{ZBL|66.0947.03}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1942) {{MR|0007093}} {{ZBL|0061.39302}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952) {{MR|0050886}} {{ZBL|0047.41402}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S.-T. Hu, "Homology theory" , Holden-Day (1966) {{MR|0217786}} {{ZBL|0145.19705}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S.-T. Hu, "Cohomology theory" , Markham , Chicago (1968) {{MR|0234448}} {{ZBL|0165.26101}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A. Dold, "Lectures on algebraic topology" , Springer (1980) {{MR|0606196}} {{ZBL|0434.55001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A sequence of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860285.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860286.png" />, over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860287.png" /> together with homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860288.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860289.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860290.png" /> is often called a chain complex. Thus, in case of a partially exact homology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860291.png" /> the sequence of a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860292.png" /> is a chain complex, or complex, instead of a long exact sequence. Dually one has cochain complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860293.png" />. Many homology and cohomology theories are constructed via chain and cochain complexes: first to a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860294.png" /> there is associated a chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860295.png" /> and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047860/h047860296.png" />, and analogously for cohomology.
+
A sequence of modules $  K _ {n} $,  
 +
$  n \geq  0 $,  
 +
over a ring $  A $
 +
together with homomorphisms $  \partial  _ {n} : K _ {n} \rightarrow K _ {n-} 1 $
 +
such that $  \partial  _ {n} \partial  _ {n+} 1 = 0 $
 +
for all $  n \geq  0 $
 +
is often called a chain complex. Thus, in case of a partially exact homology theory $  ( H _ {n} , \partial  ) $
 +
the sequence of a pair $  ( X , A ) $
 +
is a chain complex, or complex, instead of a long exact sequence. Dually one has cochain complexes $  ( K  ^ {n} , \partial  ^ {n} : K  ^ {n} \rightarrow K  ^ {n+} 1 ) $.  
 +
Many homology and cohomology theories are constructed via chain and cochain complexes: first to a pair $  ( X , A ) $
 +
there is associated a chain complex $  ( C _ {n} ( X , A ) , \partial  _ {n} ) $
 +
and then $  H _ {n} ( X , A ) = = \mathop{\rm Ker} ( \partial  _ {n} ) / \mathop{\rm Im} ( \partial  _ {n+} 1 ) $,  
 +
and analogously for cohomology.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) {{MR|0385836}} {{ZBL|0305.55001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 113ff {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Lefschetz, "Topology" , Chelsea, reprint (1956) {{MR|1674924}} {{MR|0494126}} {{MR|0326703}} {{MR|0247958}} {{MR|0031708}} {{MR|0007094}} {{MR|0007093}} {{MR|1563541}} {{MR|1563085}} {{MR|1545859}} {{MR|1522780}} {{ZBL|0945.55001}} {{ZBL|0328.55001}} {{ZBL|0337.55002}} {{ZBL|0117.16205}} {{ZBL|0045.25902}} {{ZBL|0041.51801}} {{ZBL|0036.12202}} {{ZBL|0061.39303}} {{ZBL|0061.39302}} {{ZBL|0016.41904}} {{ZBL|0011.18002}} {{ZBL|63.0557.02}} {{ZBL|61.1363.01}} {{ZBL|56.0491.08}} {{ZBL|55.0965.01}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C.R.F. Maunder, "Algebraic topology" , v. Nostrand-Reinhold (1970) {{MR|1402473}} {{MR|0694843}} {{MR|1537052}} {{ZBL|0205.27302}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> I. Vaisman, "Cohomology and differential forms" , M. Dekker (1973) {{MR|0341344}} {{ZBL|0267.58001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) {{MR|0385836}} {{ZBL|0305.55001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 113ff {{MR|0210112}} {{MR|1325242}} {{ZBL|0145.43303}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Lefschetz, "Topology" , Chelsea, reprint (1956) {{MR|1674924}} {{MR|0494126}} {{MR|0326703}} {{MR|0247958}} {{MR|0031708}} {{MR|0007094}} {{MR|0007093}} {{MR|1563541}} {{MR|1563085}} {{MR|1545859}} {{MR|1522780}} {{ZBL|0945.55001}} {{ZBL|0328.55001}} {{ZBL|0337.55002}} {{ZBL|0117.16205}} {{ZBL|0045.25902}} {{ZBL|0041.51801}} {{ZBL|0036.12202}} {{ZBL|0061.39303}} {{ZBL|0061.39302}} {{ZBL|0016.41904}} {{ZBL|0011.18002}} {{ZBL|63.0557.02}} {{ZBL|61.1363.01}} {{ZBL|56.0491.08}} {{ZBL|55.0965.01}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C.R.F. Maunder, "Algebraic topology" , v. Nostrand-Reinhold (1970) {{MR|1402473}} {{MR|0694843}} {{MR|1537052}} {{ZBL|0205.27302}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> I. Vaisman, "Cohomology and differential forms" , M. Dekker (1973) {{MR|0341344}} {{ZBL|0267.58001}} </TD></TR></table>

Revision as of 22:11, 5 June 2020


of topological spaces

A part of algebraic topology which realizes a connection between topological and algebraic concepts. By associating to each space a certain sequence of groups, and to each continuous mapping of spaces, homomorphisms of the respective groups, homology theory uses the properties of groups and their homomorphisms to clarify the properties of spaces and mappings. Such properties include, for example, the connections between various dimensionalities, the study of which is based on the concept of excision, unlike the other part of algebraic topology — the theory of homotopy, in which deformations are used for the same purpose. Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf. Homology of a polyhedron), but the axiomatic construction (including the precise limits of this concept, which had been indefinite for a long time) was imparted to it only by S. Eilenberg and N. Steenrod [3] (cf. Algebraic topology; Homology group; Steenrod–Eilenberg axioms).

According to this construction a homology theory $ \{ H, \partial \} $ consists of three functions: 1) relative $ r $- dimensional homology groups $ H _ {r} ( X, A) $ of a pair of topological spaces $ ( X, A) $, $ A \subset X $, which assign to each pair $ ( X, A) $ and each integer $ r $ an Abelian group $ H _ {r} ( X, A) $; 2) the homomorphism

$$ H _ {r} ( f ) = \ f _ {*} : H _ {r} ( X, A) \rightarrow H _ {r} ( Y, B), $$

which is assigned to a continuous mapping $ f: ( X, A) \rightarrow ( Y, B) $ and a number $ r $, and which is called the homomorphism induced by $ f $; and 3) the boundary operator $ \partial $, which assigns to each pair $ ( X, A) $ and each $ r $ a homomorphism $ \partial $ of $ H _ {r} ( X, A) $ into $ H _ {r-} 1 ( A) $( the so-called absolute group of the space $ A $, which is the group of the pair $ ( A, \emptyset ) $). These functions must satisfy the following axioms.

1) If $ f $ is the identity mapping, so is $ f _ {*} $.

2) $ ( gf ) _ {*} = g _ {*} f _ {*} $.

3) $ \partial f _ {*} = ( f \mid _ {A} ) _ {*} \partial $.

4) The exactness axiom. If

$$ i: A \rightarrow X \ \textrm{ and } \ j: X \rightarrow ( X, A) $$

are the natural inclusions, then the sequence

$$ \dots \rightarrow H _ {r} ( A) \rightarrow ^ { {i _ *} } \ H _ {r} ( X) \rightarrow ^ { {j _ *} } H _ {r} ( X, A) \mathop \rightarrow \limits ^ \partial H _ {r - 1 } ( A) \rightarrow \dots , $$

the so-called homology sequence of the pair $ ( X, A) $, is exact, that is, the image of each incoming homomorphism equals the kernel of the outgoing one.

5) The homotopy axiom. If the mappings

$$ f, g: ( X, A) \rightarrow ( Y, B) $$

are homotopic, then $ f _ {*} = g _ {*} $.

6) The excision axiom. If $ U $ is an open subset of the space $ X $ and its closure is contained in the interior of the subspace $ A $, then the inclusion

$$ e: ( X \setminus U, A \setminus U) \rightarrow ( X, A) $$

induces an isomorphism $ e _ {*} $.

7) The dimension axiom. If $ X $ is a one-point space, then $ H _ {r} ( X) = 0 $ for all $ r \neq 0 $.

Instead of taking the category of all pairs of spaces as the domain of definition of $ H _ {r} $, it is also possible to take an arbitrary category of pairs of spaces, e.g. the category of pairs of compact spaces or the category of pairs consisting of polyhedra and their subpolyhedra. However, such a category must contain along with $ ( X, A) $ also the pairs $ ( \emptyset , \emptyset ) $, $ ( X, \emptyset ) = X $, $ ( A, \emptyset ) = A $, $ ( A, A) $, $ ( X, X) $, the cylinder $ ( X, A) \times I $, where $ I = [ 0, 1] $, and some one-point space $ P _ {0} $, with all their inclusions. Such a category must also include all pairs and mappings which are encountered in the axioms or in the theorems. On the other hand, instead of taking the category of all Abelian groups as the range of $ H _ {r} $, it is also possible to use other categories, e.g. the category of topological, in particular that of compact, groups with continuous homomorphisms, or the category of modules over some ring with linear homomorphisms.

Axioms 1 and 2 mean that $ H _ {r} $ is a covariant functor from some category of pairs of spaces into the category of groups. Axiom 3 means that the boundary operator $ \partial $ is a natural transformation of the functor $ H _ {r} ( X , - ) $ to the functor $ H _ {r-} 1 ( - ) $. Axiom 4, which connects the functors of all dimensions $ r $, is sometimes replaced by the weaker requirement: The sequence should only be partially exact, i.e. the image should be included in the kernel (cf. Exact sequence); an important example of a partially exact homology theory is the Aleksandrov–Čech homology theory. Axiom 5 may be written in an equivalent form: If

$$ f _ {0} , f _ {1} : ( X, A) \rightarrow ( X, A) \times I $$

are mappings defined by the formulas $ f _ {0} ( x) = ( x, 0) $, $ f _ {1} ( x) = ( x, 1) $, then $ f _ {0*} = f _ {1*} $. Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory. Axiom 7, which ensures the geometric meaning of the dimensionality index $ r $, is often omitted in modern studies. One then obtains so-called generalized homology theories, an important example of which is bordism theory.

There exists a cohomology theory dual to a homology theory (cf. Duality in topology). It is given by relative $ r $- dimensional cohomology groups $ H ^ {r} ( X, A) $, which are contravariant functors from the category of pairs of topological spaces into the category of Abelian groups, with induced homomorphisms

$$ H ^ {r} ( f ) = f ^ { * } : H ^ {r} ( Y, B) \rightarrow H ^ {r} ( X, A) $$

and coboundary operators

$$ \delta : H ^ {r} ( A) \rightarrow H ^ {r + 1 } ( X, A). $$

The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms. For instance, the exactness axiom requires the existence of an exact cohomology sequence

$$ \dots \rightarrow H ^ {r - 1 } ( A) \mathop \rightarrow \limits ^ \delta \ H ^ {r} ( X, A) \rightarrow ^ { {j ^ {*}} } H ^ {r} ( X) \rightarrow ^ { {i ^ {*}} } H ^ {r} ( A) \rightarrow \dots . $$

And, analogously, there are also generalized cohomology theories, important examples of which are $ K $- theory and cobordism theory. The facts given below concerning homology theory have cohomology analogues.

The coefficient group of a homology or a cohomology theory is the group $ H _ {0} ( P _ {0} ) $ or, respectively, $ H ^ {0} ( P _ {0} ) $. The so-called reduced groups $ \widetilde{H} _ {r} ( X, A) $ are often conveniently substituted for the groups $ H _ {r} ( X, A) $: the reduced zero-dimensional homology group $ \widetilde{H} _ {0} ( X) $ is the kernel of the homomorphism

$$ l _ {*} : H _ {0} ( X) \rightarrow H _ {0} ( P _ {0} ) $$

induced by the mapping $ l: X \rightarrow P _ {0} $, while the reduced zero-dimensional cohomology group $ \widetilde{H} {} ^ {0} ( X) $ is the quotient group of $ H ^ {0} ( X) $ by the image $ l ^ {*} ( H ^ {0} ( P _ {0} )) $; the reduced groups of other dimensions are identical with the initial ones: $ \widetilde{H} _ {r} ( X) = H _ {r} ( X) $, $ r \neq 0 $. Thus, $ H _ {0} ( X) \sim \widetilde{H} _ {0} ( X) \oplus G $. If $ A \neq \emptyset $, then $ \widetilde{H} _ {r} ( X, A) = H _ {r} ( X, A) $ for all $ r $. The replacement of the ordinary by the reduced groups permits one to convert the homology sequence into a reduced homology sequence.

The axioms of homology theory are not independent. E.g., axiom 1 is a consequence of the axioms 2, 3 and 4. The system of axioms is compatible, as is seen from the example of the trivial theory $ H _ {r} ( X, A) = 0 $; non-trivial examples include the Aleksandrov–Čech cohomology theory, singular homology, etc. Regarding uniqueness, the following holds: A homomorphism of a homology theory $ \{ H, \delta \} $ into a homology theory $ \{ H ^ \prime , \partial ^ \prime \} $ is a system of homomorphisms

$$ h ( X, A; r): H _ {r} ( X, A) \rightarrow H _ {r} ^ \prime ( X, A) $$

such that

$$ H ^ \prime ( f ) \circ h ( X, A; r) = h ( Y, B; r) \circ H ( f ) $$

and

$$ \partial ^ \prime \circ h ( X, A; r) = h ( A; r - 1) \circ \partial . $$

If all $ h( X, A; r) $ are isomorphisms, then the homology theories $ \{ H, \partial \} $ and $ \{ H ^ \prime , \partial ^ \prime \} $ are called isomorphic. A homology theory on finite polyhedra is uniquely determined by its coefficient group. More precisely, if $ h _ {0} : G \rightarrow G ^ \prime $ is an arbitrary homomorphism of the coefficient group $ G $ of $ \{ H, \partial \} $ into the coefficient group $ G ^ \prime $ of $ \{ H ^ \prime , \partial ^ \prime \} $, then for each polyhedral pair $ ( X, A) $ there exists a unique homomorphism

$$ h ( X, A; r): H _ {r} ( X, A) \rightarrow H _ {r} ^ \prime ( X, A) $$

with the property $ h( P _ {0} ; 0) = h _ {0} $. If $ h _ {0} $ is an isomorphism, so are all $ h( X, A; r) $. Since the homology groups of negative dimension of a triangulable pair are trivial, the equality $ H _ {r} ( X, A) = 0 $, $ r < 0 $, is valid for any homology theory $ \{ H, \partial \} $ as well. The uniqueness theorem is also valid for wider categories of spaces if the homology theory satisfies appropriate additional axioms.

The homology groups are topological and also homotopy invariants: If $ f $ is a homotopy equivalence, then $ f _ {*} $ is an isomorphism. If $ X $ is a contractible space — a cell, in particular — then $ H _ {r} ( X) = 0 $, $ r \neq 0 $, and $ H _ {0} ( X) \sim G $. If $ i : A \subset X $ is a homotopy equivalence, then $ H _ {r} ( X, A) = 0 $ and, for any $ X $, $ H _ {r} ( X, X) = 0 $. If $ A $ is a retract of the space $ X $, then $ i _ {*} $ is a monomorphism, $ j _ {*} $ is an epimorphism, the operation $ \partial $ is trivial, and

$$ H _ {r} ( X) \sim H _ {r} ( A) \oplus H _ {r} ( X, A). $$

If $ X $ is deformable into $ A $, then $ i _ {*} $ is an epimorphism, $ j _ {*} $ is trivial, $ \partial $ is a monomorphism, and

$$ H _ {r} ( A) \sim H _ {r} ( X) \oplus H _ {r + 1 } ( X, A). $$

Let $ S( X) $ denote the suspension over $ X $. The following isomorphism is then valid:

$$ \widetilde{H} _ {r} ( X) \sim \widetilde{H} _ {r + 1 } ( S ( X)). $$

This makes it possible to compute the homology groups of the spheres $ S ^ {n} $: $ \widetilde{H} _ {r} ( S ^ {n} ) = 0 $ if $ r \neq n $, and $ \widetilde{H} _ {n} ( S ^ {n} ) \sim G $; consequently, $ H _ {r} ( S ^ {n} ) = 0 $ if $ n \neq r \neq 0 $; $ H _ {r} ( S ^ {n} ) \sim G $ if $ n \neq r = 0 $ or $ n = r \neq 0 $; and $ H _ {0} ( S ^ {0} ) \sim G \oplus G $.

An important role in homology theory is played by homology sequences of triples and triads. In the case of a triple $ ( X, A, B) $, $ X \supset A \supset B $, of spaces, the boundary operator $ \partial ^ \prime = k _ {*} ^ \prime \partial $ is defined as the composition $ k _ {*} ^ \prime \circ \partial $, where $ k ^ \prime : A \rightarrow ( A, B) $ is the inclusion. There results the so-called homology sequence of the triple $ ( X, A, B) $( which, for $ B = \emptyset $, reduces to the homology sequence of the pair $ ( X, A) $):

$$ \dots \rightarrow H _ {r} ( A, B) \rightarrow ^ { {i _ *} ^ \prime } \ H _ {r} ( X, B) \ \rightarrow ^ { {j _ *} ^ \prime } \ H _ {r} ( X, A) \ \rightarrow ^ { {\partial ^ \prime } } $$

$$ \rightarrow ^ { {\partial ^ \prime } } H _ {r - 1 } ( A, B) \rightarrow \dots , $$

where $ i ^ \prime : ( A, B) \rightarrow ( X, B) $ and $ j ^ \prime : ( X, B) \rightarrow ( X, A) $ are the inclusions. This sequence is exact. If the groups $ H _ {r} ( X, A) $, respectively $ H _ {r} ( X, B) $, $ H _ {r} ( A, B) $, are trivial for all $ r $, then $ i _ {*} ^ \prime $, respectively $ \partial ^ \prime $, $ j _ {*} ^ \prime $, are isomorphisms, and vice versa. If $ X $ is the union of non-intersecting closed sets $ X _ {i} $, $ i = 1 \dots n $, and $ A = A _ {1} \cup \dots \cup A _ {n} $, where $ A _ {i} \subset X _ {i} $, then $ H _ {r} ( X, A) $ is isomorphic to the direct sum of the groups $ H _ {r} ( X _ {i} , A _ {i} ) $, $ i = 1 \dots n $. A triad $ ( X; A, B) $ is a space $ X $ with an ordered pair of subspaces $ A, B $( cf. also Triads). It is a proper triad if the inclusions

$$ k: ( A, A \cap B) \rightarrow ( A \cup B, B),\ \ l: ( B, A \cap B) \rightarrow ( A \cup B, A) $$

induce isomorphisms, or if the decomposition

$$ H _ {r} ( A \cup B, A \cap B) \sim \ H _ {r} ( A, A \cap B) \oplus H _ {r} ( B, A \cap B) $$

is valid. Further, the boundary operator

$$ \overline \partial \; : H _ {r} ( X, A \cup B) \rightarrow \ H _ {r - 1 } ( A, A \cap B) $$

is defined as $ k _ {*} ^ {-} 1 \circ m _ {*} \circ \partial $, where $ m : A \cup B \subset ( A \cup B, B) $. This generates the exact homology sequence of the triad:

$$ \dots \rightarrow H _ {r} ( A, A \cap B) \ \rightarrow ^ { {p _ *} } \ H _ {r} ( X, B) \ \rightarrow ^ { {q _ *} } \ H _ {r} ( X, A \cup B) \ \mathop \rightarrow \limits ^ { {\overline \partial \; }} $$

$$ \mathop \rightarrow \limits ^ { {\overline \partial \; }} H _ {r - 1 } ( A, A \cap B) \rightarrow \dots , $$

where $ p: ( A, A \cap B) \rightarrow ( X, B) $, $ q: ( X, B) \rightarrow ( X, A \cup B) $ are the inclusions (if $ B \subset A $, this sequence reduces to the homology sequence of the triple $ ( X, A, B) $).

Let $ X = A \cup B $, $ A \cap B = C $ and suppose the mappings $ h, h _ {1} , h _ {2} : ( X, C) \rightarrow ( Y, D) $ satisfy the relations $ h _ {1} | _ {A} = h | _ {A} $, $ h _ {2} | _ {B} = h | _ {B} $, $ h _ {1} ( B) \subset D $, $ h _ {2} ( A) \subset D $. The following addition theorems are then valid.

1) $ h _ {*} = h _ {1* } + h _ {2* } $.

2) If $ D $ is contractible and if $ f, f _ {1} , f _ {2} : X \rightarrow Y $ are defined, respectively, by means of $ h, h _ {1} $ and $ h _ {2} $, then the equality $ f _ {*} = f _ {1* } + f _ {2* } $ holds for the induced homomorphisms of the reduced groups $ f _ {*} , f _ {1* } , f _ {2* } : \widetilde{H} _ {r} ( X) \rightarrow \widetilde{H} _ {r} ( Y) $.

Define the homomorphism

$$ s : H _ {r} ( C) \rightarrow H _ {r} ( A) \oplus H _ {r} ( B) $$

by $ s ( c) = ( s _ {1* } ( c), - s _ {2* } ( c)) $, $ c \in H _ {r} ( C) $, where $ s _ {1} : C \rightarrow A $, $ s _ {2} : C \rightarrow B $ are the inclusions. Similarly, define

$$ t: H _ {r} ( A) \oplus H _ {r} ( B) \rightarrow H _ {r} ( X) $$

by $ t ( a, b) = t _ {1* } ( a) + t _ {2* } ( b) $, $ ( a, b) \in H _ {r} ( A) \oplus H _ {r} ( B) $, where $ t _ {1} : A \rightarrow X $, $ t _ {2} : B \rightarrow X $ are the inclusions. Finally, define

$$ \Delta : H _ {r} ( X) \rightarrow H _ {r} ( C), $$

by $ \Delta = \partial u _ {*} ^ {-} 1 v _ {*} $, where

$$ v: X \rightarrow ( X, B),\ \ u: ( A, C) \rightarrow ( X, B) $$

are the inclusions. One then obtains the so-called Mayer–Vietoris sequence of the proper triad:

$$ \dots \rightarrow H _ {r} ( C) \ \mathop \rightarrow \limits ^ { {s }} H _ {r} ( A) \oplus H _ {r} ( B) \ \mathop \rightarrow \limits ^ { {t }} H _ {r} ( X) \ \mathop \rightarrow \limits ^ \Delta \ H _ {r - 1 } ( C) \rightarrow \dots , $$

which is exact and which relates the homology groups of the spaces $ A $ and $ B $ with the homology groups of their union and intersection. Then, if $ C \neq \emptyset $, one may pass to a similar sequence for the reduced groups, which implies that:

1) if $ A \cap B $ is contractible, then

$$ \widetilde{H} _ {r} ( A \cup B) \sim \ \widetilde{H} _ {r} ( A) \oplus \widetilde{H} _ {r} ( B); $$

2) if $ A \cup B $ is contractible, then

$$ \widetilde{H} _ {r} ( A \cap B) \sim \ \widetilde{H} _ {r} ( A) \oplus \widetilde{H} _ {r} ( B); $$

3) if both $ A $ and $ B $ are contractible, then $ \Delta $ defines an isomorphism

$$ \widetilde{H} _ {r} ( A \cup B) \sim \widetilde{H} _ {r - 1 } ( A \cap B). $$

These results make it possible to compute the homology groups of various spaces. For instance, if $ X $ is a closed orientable surface of genus $ n $, then $ H _ {r} ( X) $ is isomorphic to the coefficient group $ G $ if $ r = 0, 2 $; to the direct sum $ G ^ {2n} $ of $ 2n $ copies of $ G $ if $ r = 1 $; and to 0 in the remaining cases. If $ X $ is a closed non-orientable surface of genus $ n $, then $ H _ {r} ( X) $ is isomorphic to $ G $ if $ r = 0 $; to $ G ^ {n-} 1 \oplus G _ {2} $, where $ G _ {2} $ is the quotient group $ G/2G $, $ 2G = \{ {2g } : {g \in G } \} $, if $ r = 1 $; to the subgroup $ T _ {2} ( G) $ of $ G $ consisting of all elements $ g \in G $ with $ 2g = 0 $ if $ r = 2 $; and to 0 in the remaining cases. Thus, homology theory gives a topological classification of closed surfaces. For an $ n $- dimensional real projective space $ P ^ {n} $ the group $ H _ {r} ( P ^ {n} ) $ is isomorphic to $ G $ if $ r = 0 $ or $ r = n $ and odd; to $ G _ {2} $ if $ r $ is odd and $ 0 < r < n $; to $ T _ {2} ( G) $ if $ r $ is even and $ 0 < r \leq n $; and to 0 in the remaining cases. The homology group $ H _ {r} ( \mathbf C P ^ {n} ) $ of the complex projective space $ \mathbf C P ^ {n} $ of dimension $ 2n $ is isomorphic to $ G $ if $ r $ is even and $ 0 \leq r \leq 2n $; and to 0 in the remaining cases. The homology group $ H _ {r} ( L _ {p,q} ) $ of the lens space $ L _ {p,q} $ is isomorphic to $ G $ if $ r = 0, 3 $; to $ G _ {p} = G/pG $, where $ pG = \{ {pg } : {g \in G } \} $, if $ r = 1 $; to $ T _ {p} ( G) $, where $ T _ {p} ( G) = \{ {g \in G } : {pg = 0 } \} $, if $ r = 2 $; and to 0 in the remaining cases.

From the various applications of the results described above certain fundamental theorems are stated here. First of all, the invariance of dimension: spheres, as well as Euclidean spaces, of different dimensions are not homeomorphic; in fact, if two polyhedra are homeomorphic, then they have the same dimension. Furthermore, the equality $ f _ {*} i _ {*} = g _ {*} $, where $ f: X \rightarrow Y $ is an extension of a given mapping $ g: A \rightarrow Y $, $ A \subset X $, yields various criteria of extendability and retractibility of mappings; for example, a mapping of a sphere $ S ^ {n-} 1 $, $ n > 1 $, of non-zero degree into itself is not extendable to the $ n $- dimensional ball $ E ^ {n} $ with boundary $ S ^ {n-} 1 $, and $ S ^ {n-} 1 $ is not a retract of $ E ^ {n} $ for any natural $ n $. This, in turn, yields Brouwer's fixed-point theorem: Any continuous mapping $ E ^ {n} \rightarrow E ^ {n} $ has a fixed point. Finally, it may be proved that a unit tangent vector field exists on $ S ^ {n} $ if and only if $ n $ is odd, while the theory of triads yields several theorems on the degrees of mappings; in particular, it is possibly to give a new proof of the fundamental theorem of algebra.

References

[1] P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian) MR0076324 Zbl 0441.55002 Zbl 0097.15903 Zbl 0024.08404 Zbl 66.0947.03
[2] S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1942) MR0007093 Zbl 0061.39302
[3] S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952) MR0050886 Zbl 0047.41402
[4] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303
[5] S.-T. Hu, "Homology theory" , Holden-Day (1966) MR0217786 Zbl 0145.19705
[6] S.-T. Hu, "Cohomology theory" , Markham , Chicago (1968) MR0234448 Zbl 0165.26101
[7] A. Dold, "Lectures on algebraic topology" , Springer (1980) MR0606196 Zbl 0434.55001

Comments

A sequence of modules $ K _ {n} $, $ n \geq 0 $, over a ring $ A $ together with homomorphisms $ \partial _ {n} : K _ {n} \rightarrow K _ {n-} 1 $ such that $ \partial _ {n} \partial _ {n+} 1 = 0 $ for all $ n \geq 0 $ is often called a chain complex. Thus, in case of a partially exact homology theory $ ( H _ {n} , \partial ) $ the sequence of a pair $ ( X , A ) $ is a chain complex, or complex, instead of a long exact sequence. Dually one has cochain complexes $ ( K ^ {n} , \partial ^ {n} : K ^ {n} \rightarrow K ^ {n+} 1 ) $. Many homology and cohomology theories are constructed via chain and cochain complexes: first to a pair $ ( X , A ) $ there is associated a chain complex $ ( C _ {n} ( X , A ) , \partial _ {n} ) $ and then $ H _ {n} ( X , A ) = = \mathop{\rm Ker} ( \partial _ {n} ) / \mathop{\rm Im} ( \partial _ {n+} 1 ) $, and analogously for cohomology.

References

[a1] R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) MR0385836 Zbl 0305.55001
[a2] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 113ff MR0210112 MR1325242 Zbl 0145.43303
[a3] S. Lefschetz, "Topology" , Chelsea, reprint (1956) MR1674924 MR0494126 MR0326703 MR0247958 MR0031708 MR0007094 MR0007093 MR1563541 MR1563085 MR1545859 MR1522780 Zbl 0945.55001 Zbl 0328.55001 Zbl 0337.55002 Zbl 0117.16205 Zbl 0045.25902 Zbl 0041.51801 Zbl 0036.12202 Zbl 0061.39303 Zbl 0061.39302 Zbl 0016.41904 Zbl 0011.18002 Zbl 63.0557.02 Zbl 61.1363.01 Zbl 56.0491.08 Zbl 55.0965.01
[a4] C.R.F. Maunder, "Algebraic topology" , v. Nostrand-Reinhold (1970) MR1402473 MR0694843 MR1537052 Zbl 0205.27302
[a5] I. Vaisman, "Cohomology and differential forms" , M. Dekker (1973) MR0341344 Zbl 0267.58001
How to Cite This Entry:
Homology theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_theory&oldid=47263
This article was adapted from an original article by G.S. Chogoshvili (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article