# Fundamental group

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)