# Fundamental group

Poincaré group

The first absolute homotopy group . Let be the interval , and let be its boundary. The elements of the fundamental group of the pointed topological space are the homotopy classes of closed paths in , that is, homotopy classes of continuous mappings of the pair into . The path :

is called the product of and . 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 , and the inverse of the class containing the path is the class of the path . To a continuous mapping corresponds the homomorphism

that is, is a functor from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path joining the points and , one can define an isomorphism

that depends only on the homotopy class of . The group acts as a group of automorphisms on , and in the case , acts as an inner automorphism . The Hurewicz homomorphism is an epimorphism with kernel (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 is isomorphic to the direct product of the fundamental groups of the factors: . Let be a path-connected topological space, and let be a covering of by a system of open sets , closed under intersection, such that ; then is the direct limit of the diagram , where , and is induced by the inclusion (the Seifert–van Kampen theorem). For example, if one is given a covering consisting of , and , and if is simply connected, then is the free product of and . In the case of a CW-complex, the assertion of the theorem is also true for closed CW-subspaces of .

For a CW-complex whose zero-dimensional skeleton consists of a single point , each one-dimensional cell gives a generator of , and each two-dimensional cell gives a relation corresponding to the attaching mapping of .

Suppose that has a covering such that the inclusion homomorphism is zero for every point . Then there is a covering with . In this case the group of homeomorphisms of onto itself that commute with (covering transformations) is isomorphic to , and the order of is equal to the cardinality of the fibre . For a mapping of path-connected spaces such that there is a lifting , . The covering 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)