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)
[a1]	B. Gran, "Homology theory" , Acad. Press (1975)