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''Poincaré group''
 
''Poincaré group''
  
The first absolute [[Homotopy group|homotopy group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422101.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422102.png" /> be the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422103.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422104.png" /> be its boundary. The elements of the fundamental group of the pointed topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422105.png" /> are the homotopy classes of closed paths in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422106.png" />, that is, homotopy classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422107.png" /> of continuous mappings of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422108.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f0422109.png" />. The path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221010.png" />:
+
The first absolute [[Homotopy group|homotopy group]] $  \pi _ {1} ( X, x _ {0} ) $.  
 +
Let $  I $
 +
be the interval $  [ 0, 1] $,
 +
and let $  \partial  I = \{ 0, 1 \} $
 +
be its boundary. The elements of the fundamental group of the pointed topological space $  ( X, x _ {0} ) $
 +
are the homotopy classes of closed paths in $  X $,  
 +
that is, homotopy classes $  \mathop{\rm rel} \{ 0, 1 \} $
 +
of continuous mappings of the pair $  ( I, \partial  I) $
 +
into $  ( X, x _ {0} ) $.  
 +
The path $  s _ {1} s _ {2} $:
 +
 
 +
$$
 +
s _ {1} s _ {2} ( t)  = \
 +
\left \{
 +
\begin{array}{ll}
 +
s _ {1} ( 2t),  & t \leq  1/2,  \\
 +
s _ {2} ( 2t - 1),  & t \geq  1/2,  \\
 +
\end{array}
  
<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/f/f042/f042210/f04221011.png" /></td> </tr></table>
+
\right .$$
  
is called the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221013.png" />. The homotopy class of the product depends only on the classes of the factors, and the resulting operation is, generally speaking, non-commutative. The identity is the class of the constant mapping into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221014.png" />, and the inverse of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221015.png" /> containing the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221016.png" /> is the class of the path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221017.png" />. To a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221018.png" /> corresponds the homomorphism
+
is called the product of $  s _ {1} $
 +
and $  s _ {2} $.  
 +
The homotopy class of the product depends only on the classes of the factors, and the resulting operation is, generally speaking, non-commutative. The identity is the class of the constant mapping into $  x _ {0} $,  
 +
and the inverse of the class $  \overline \phi \; $
 +
containing the path $  \phi ( t) $
 +
is the class of the path $  \psi ( t) = \phi ( 1 - t) $.  
 +
To a continuous mapping $  f:  ( X, x _ {0} ) \rightarrow ( Y, y _ {0} ) $
 +
corresponds 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/f/f042/f042210/f04221019.png" /></td> </tr></table>
+
$$
 +
f _ {\#} ( \overline \phi \; = \
 +
\overline{ {f \circ \phi }}\; : \
 +
\pi _ {1} ( X, x _ {0} )  \rightarrow \
 +
\pi _ {1} ( Y, y _ {0} ),
 +
$$
  
that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221020.png" /> is a functor from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221021.png" /> joining the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221023.png" />, one can define an isomorphism
+
that is, $  \pi _ {1} $
 +
is a functor from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path $  \phi $
 +
joining the points $  x _ {1} $
 +
and $  x _ {2} $,  
 +
one can define 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/f/f042/f042210/f04221024.png" /></td> </tr></table>
+
$$
 +
\widehat \phi  : \
 +
\pi _ {1} ( X, x _ {2} )  \rightarrow \
 +
\pi _ {1} ( X, x _ {1} ),
 +
$$
  
<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/f/f042/f042210/f04221025.png" /></td> </tr></table>
+
$$
 +
\widehat \phi  ( u) t  = \left \{
 +
\begin{array}{ll}
 +
\phi
 +
( 3t),  & t \leq  1/3,  \\
 +
\phi ( 3t - 1),  & 1/3 \leq  t \leq  2/3,  \\
 +
\phi ( 3 - 3t),  & 2/3 \leq  t \leq  1,  \\
 +
\end{array}
  
that depends only on the homotopy class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221026.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221027.png" /> acts as a group of automorphisms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221028.png" />, and in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221030.png" /> acts as an inner automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221031.png" />. The Hurewicz homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221032.png" /> is an epimorphism with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221033.png" /> (Poincaré's theorem).
+
\right .$$
  
A path-connected topological space with a trivial fundamental group is called simply connected. The fundamental group of a product of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221034.png" /> is isomorphic to the direct product of the fundamental groups of the factors: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221036.png" /> be a path-connected topological space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221037.png" /> be a covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221038.png" /> by a system of open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221039.png" />, closed under intersection, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221040.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221041.png" /> is the direct limit of the diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221043.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221044.png" /> is induced by the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221045.png" /> (the Seifert–van Kampen theorem). For example, if one is given a covering consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221048.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221049.png" /> is simply connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221050.png" /> is the free product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221052.png" />. In the case of a CW-complex, the assertion of the theorem is also true for closed CW-subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221053.png" />.
+
that depends only on the homotopy class of  $  \phi $.  
 +
The group $  \pi _ {1} ( X, x _ {0} ) $
 +
acts as a group of automorphisms on  $  \pi _ {n} ( X, x _ {0} ) $,  
 +
and in the case  $  n = 1 $,
 +
$  \overline \phi \; $
 +
acts as an inner automorphism  $  \overline{u}\; \rightarrow \overline{ {\phi u \phi }}\; {}  ^ {-} 1 = \widehat \phi  ( \overline{u}\; ) $.  
 +
The Hurewicz homomorphism  $  h: \pi _ {1} ( X, x _ {0} ) \rightarrow H _ {1} ( X) $
 +
is an epimorphism with kernel  $  [ \pi _ {1} , \pi _ {1} ] $(
 +
Poincaré's theorem).
  
For a CW-complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221054.png" /> whose zero-dimensional skeleton consists of a single point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221055.png" />, each one-dimensional cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221056.png" /> gives a generator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221057.png" />, and each two-dimensional cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221058.png" /> gives a relation corresponding to the attaching mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221059.png" />.
+
A path-connected topological space with a trivial fundamental group is called simply connected. The fundamental group of a product of spaces  $  \prod _  \alpha  X _  \alpha  $
 +
is isomorphic to the direct product of the fundamental groups of the factors:  $  \pi _ {1} ( \prod _  \alpha  X _  \alpha  ) = \prod _  \alpha  \pi _ {1} ( X _  \alpha  ) $.
 +
Let  $  ( X, x _ {0} ) $
 +
be a path-connected topological space, and let  $  \{ {U _  \lambda  } : {\lambda \in \Lambda } \} $
 +
be a covering of $  X $
 +
by a system of open sets  $  U _  \lambda  $,
 +
closed under intersection, such that  $  x _ {0} \in \cap _  \lambda  U _  \lambda  $;
 +
then  $  \pi _ {1} ( X, x _ {0} ) $
 +
is the direct limit of the diagram  $  \{ G _  \lambda  , \phi _ {\lambda \mu \# }  \} $,
 +
where  $  G _  \lambda  = \pi _ {1} ( U _  \lambda  , x _ {0} ) $,
 +
and  $  \phi _ {\lambda \mu \# }  $
 +
is induced by the inclusion  $  \phi _ {\lambda \mu }  : U _  \lambda  \rightarrow U _  \mu  $(
 +
the Seifert–van Kampen theorem). For example, if one is given a covering consisting of $  U _ {0} $,
 +
$  U _ {1} $
 +
and  $  U _ {2} $,
 +
and if  $  U _ {0} = U _ {1} \cap U _ {2} $
 +
is simply connected, then  $  \pi _ {1} ( X, x _ {0} ) $
 +
is the free product of  $  \pi _ {1} ( U _ {1} , x _ {0} ) $
 +
and $  \pi _ {1} ( U _ {2} , x _ {0} ) $.  
 +
In the case of a CW-complex, the assertion of the theorem is also true for closed CW-subspaces of $  X $.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221060.png" /> has a covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221061.png" /> such that the inclusion homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221062.png" /> is zero for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221063.png" />. Then there is a [[Covering|covering]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221064.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221065.png" />. In this case the group of homeomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221066.png" /> onto itself that commute with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221067.png" /> (covering transformations) is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221068.png" />, and the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221069.png" /> is equal to the cardinality of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221070.png" />. For a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221071.png" /> of path-connected spaces such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221072.png" /> there is a lifting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221074.png" />. The covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042210/f04221075.png" /> is called universal.
+
For a CW-complex  $  X $
 +
whose zero-dimensional skeleton consists of a single point  $  x _ {0} $,
 +
each one-dimensional cell  $  e _  \lambda  ^ {1} \in X $
 +
gives a generator of  $  \pi _ {1} ( X, x _ {0} ) $,
 +
and each two-dimensional cell  $  e _  \lambda  ^ {2} \in X $
 +
gives a relation corresponding to the attaching mapping of  $  e _  \lambda  ^ {2} $.
 +
 
 +
Suppose that $  X $
 +
has a covering $  \{ {U _  \lambda  } : {\lambda \in \Lambda } \} $
 +
such that the inclusion homomorphism $  \pi _ {1} ( U _  \lambda  , z) \rightarrow \pi _ {1} ( X, z) $
 +
is zero for every point $  z $.  
 +
Then there is a [[Covering|covering]] $  p: \widetilde{X}  \rightarrow X $
 +
with $  \pi _ {1} ( \widetilde{X}  , x) = 0 $.  
 +
In this case the group of homeomorphisms of $  \widetilde{X}  $
 +
onto itself that commute with $  p $(
 +
covering transformations) is isomorphic to $  \pi _ {1} ( X, x _ {0} ) $,  
 +
and the order of $  \pi _ {1} ( X, x _ {0} ) $
 +
is equal to the cardinality of the fibre $  p  ^ {-} 1 x _ {0} $.  
 +
For a mapping $  f:  ( Y, y _ {0} ) \rightarrow ( X, x _ {0} ) $
 +
of path-connected spaces such that $  f _ {\#} ( \pi _ {1} ( Y, y _ {0} )) = 0 $
 +
there is a lifting $  \widetilde{f}  : Y \rightarrow \widetilde{X}  $,
 +
$  p \circ \widetilde{f= f $.  
 +
The covering $  p: \widetilde{X}  \rightarrow X $
 +
is called universal.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.S. Massey,  "Algebraic topology: an introduction" , Springer  (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Rokhlin,  D.B. Fuks,  "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.R. Stallings,  "Group theory and three-dimensional manifolds" , Yale Univ. Press  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.S. Massey,  "Algebraic topology: an introduction" , Springer  (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Rokhlin,  D.B. Fuks,  "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.R. Stallings,  "Group theory and three-dimensional manifolds" , Yale Univ. Press  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Gran,  "Homology theory" , Acad. Press  (1975)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Gran,  "Homology theory" , Acad. Press  (1975)</TD></TR></table>

Revision as of 19:40, 5 June 2020


Poincaré group

The first absolute homotopy group $ \pi _ {1} ( X, x _ {0} ) $. Let $ I $ be the interval $ [ 0, 1] $, and let $ \partial I = \{ 0, 1 \} $ be its boundary. The elements of the fundamental group of the pointed topological space $ ( X, x _ {0} ) $ are the homotopy classes of closed paths in $ X $, that is, homotopy classes $ \mathop{\rm rel} \{ 0, 1 \} $ of continuous mappings of the pair $ ( I, \partial I) $ into $ ( X, x _ {0} ) $. The path $ s _ {1} s _ {2} $:

$$ s _ {1} s _ {2} ( t) = \ \left \{ \begin{array}{ll} s _ {1} ( 2t), & t \leq 1/2, \\ s _ {2} ( 2t - 1), & t \geq 1/2, \\ \end{array} \right .$$

is called the product of $ s _ {1} $ and $ s _ {2} $. The homotopy class of the product depends only on the classes of the factors, and the resulting operation is, generally speaking, non-commutative. The identity is the class of the constant mapping into $ x _ {0} $, and the inverse of the class $ \overline \phi \; $ containing the path $ \phi ( t) $ is the class of the path $ \psi ( t) = \phi ( 1 - t) $. To a continuous mapping $ f: ( X, x _ {0} ) \rightarrow ( Y, y _ {0} ) $ corresponds the homomorphism

$$ f _ {\#} ( \overline \phi \; ) = \ \overline{ {f \circ \phi }}\; : \ \pi _ {1} ( X, x _ {0} ) \rightarrow \ \pi _ {1} ( Y, y _ {0} ), $$

that is, $ \pi _ {1} $ is a functor from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path $ \phi $ joining the points $ x _ {1} $ and $ x _ {2} $, one can define an isomorphism

$$ \widehat \phi : \ \pi _ {1} ( X, x _ {2} ) \rightarrow \ \pi _ {1} ( X, x _ {1} ), $$

$$ \widehat \phi ( u) t = \left \{ \begin{array}{ll} \phi ( 3t), & t \leq 1/3, \\ \phi ( 3t - 1), & 1/3 \leq t \leq 2/3, \\ \phi ( 3 - 3t), & 2/3 \leq t \leq 1, \\ \end{array} \right .$$

that depends only on the homotopy class of $ \phi $. The group $ \pi _ {1} ( X, x _ {0} ) $ acts as a group of automorphisms on $ \pi _ {n} ( X, x _ {0} ) $, and in the case $ n = 1 $, $ \overline \phi \; $ acts as an inner automorphism $ \overline{u}\; \rightarrow \overline{ {\phi u \phi }}\; {} ^ {-} 1 = \widehat \phi ( \overline{u}\; ) $. The Hurewicz homomorphism $ h: \pi _ {1} ( X, x _ {0} ) \rightarrow H _ {1} ( X) $ is an epimorphism with kernel $ [ \pi _ {1} , \pi _ {1} ] $( Poincaré's theorem).

A path-connected topological space with a trivial fundamental group is called simply connected. The fundamental group of a product of spaces $ \prod _ \alpha X _ \alpha $ is isomorphic to the direct product of the fundamental groups of the factors: $ \pi _ {1} ( \prod _ \alpha X _ \alpha ) = \prod _ \alpha \pi _ {1} ( X _ \alpha ) $. Let $ ( X, x _ {0} ) $ be a path-connected topological space, and let $ \{ {U _ \lambda } : {\lambda \in \Lambda } \} $ be a covering of $ X $ by a system of open sets $ U _ \lambda $, closed under intersection, such that $ x _ {0} \in \cap _ \lambda U _ \lambda $; then $ \pi _ {1} ( X, x _ {0} ) $ is the direct limit of the diagram $ \{ G _ \lambda , \phi _ {\lambda \mu \# } \} $, where $ G _ \lambda = \pi _ {1} ( U _ \lambda , x _ {0} ) $, and $ \phi _ {\lambda \mu \# } $ is induced by the inclusion $ \phi _ {\lambda \mu } : U _ \lambda \rightarrow U _ \mu $( the Seifert–van Kampen theorem). For example, if one is given a covering consisting of $ U _ {0} $, $ U _ {1} $ and $ U _ {2} $, and if $ U _ {0} = U _ {1} \cap U _ {2} $ is simply connected, then $ \pi _ {1} ( X, x _ {0} ) $ is the free product of $ \pi _ {1} ( U _ {1} , x _ {0} ) $ and $ \pi _ {1} ( U _ {2} , x _ {0} ) $. In the case of a CW-complex, the assertion of the theorem is also true for closed CW-subspaces of $ X $.

For a CW-complex $ X $ whose zero-dimensional skeleton consists of a single point $ x _ {0} $, each one-dimensional cell $ e _ \lambda ^ {1} \in X $ gives a generator of $ \pi _ {1} ( X, x _ {0} ) $, and each two-dimensional cell $ e _ \lambda ^ {2} \in X $ gives a relation corresponding to the attaching mapping of $ e _ \lambda ^ {2} $.

Suppose that $ X $ has a covering $ \{ {U _ \lambda } : {\lambda \in \Lambda } \} $ such that the inclusion homomorphism $ \pi _ {1} ( U _ \lambda , z) \rightarrow \pi _ {1} ( X, z) $ is zero for every point $ z $. Then there is a covering $ p: \widetilde{X} \rightarrow X $ with $ \pi _ {1} ( \widetilde{X} , x) = 0 $. In this case the group of homeomorphisms of $ \widetilde{X} $ onto itself that commute with $ p $( covering transformations) is isomorphic to $ \pi _ {1} ( X, x _ {0} ) $, and the order of $ \pi _ {1} ( X, x _ {0} ) $ is equal to the cardinality of the fibre $ p ^ {-} 1 x _ {0} $. For a mapping $ f: ( Y, y _ {0} ) \rightarrow ( X, x _ {0} ) $ of path-connected spaces such that $ f _ {\#} ( \pi _ {1} ( Y, y _ {0} )) = 0 $ there is a lifting $ \widetilde{f} : Y \rightarrow \widetilde{X} $, $ p \circ \widetilde{f} = f $. The covering $ p: \widetilde{X} \rightarrow X $ is called universal.

References

[1] W.S. Massey, "Algebraic topology: an introduction" , Springer (1977)
[2] V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)
[3] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[4] J.R. Stallings, "Group theory and three-dimensional manifolds" , Yale Univ. Press (1972)

Comments

References

[a1] B. Gran, "Homology theory" , Acad. Press (1975)
How to Cite This Entry:
Fundamental group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_group&oldid=17041
This article was adapted from an original article by A.V. Khokhlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article