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''Poincaré group''

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}

 \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|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====
<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>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Gran,   "Homology theory" , Acad. Press  (1975)</TD></TR></table>