Section 52: Problem 1 Solution »

Section 52: The Fundamental Group

Elementary Group Theory

Given two groups and , a function is called a homomorphism if for every , .
  • The image of under the homomorphism is a subgroup of .
  • An injective homomorphism is called monomorphism.
  • A surjective homomorphism is called epimorphism.
  • A bijective homomorphism is called isomorphism.
The preimage of under homomorphism , , is called the kernel of the homomorphism . It is a subgroup of .
The left coset of a subgroup in is the set . Similarly, the right coset of a subgroup in is the set .
Subgroup is called normal if for every , . In other words, is a normal subgroup of if for every and , .
If is a normal subgroup of , the partition of into left (or right) cosets is denoted by . where is a group called the quotient of by . If is a subgroup of that is not normal, means the partition of into right cosets of in .
  • defined by is an epimorphism with the kernel .
  • Vice versa, if is an epimorhism, then is a normal subgroup of , and induces an isomorphism such that for .

The Fundamental Group

Let be a space and .
A loop based at is any path starting and ending at .
The fundamental group of relative to the base point , , is the set of path homotopy classes of loops based at together with the operation .
  • This is also called the first homotopy group of .
  • For a path connected space (or for a path connected component of a space) the choice of the point is not important: if where is path connected, then is isomorphic to .
    • To show this, for a path connecting and , we introduce the map defined by which is a group isomorphism.
  • The reference point is still needed, because the isomorphism between and may depend on the path chosen between and .
    • This is not the case if the fundamental group is abelian (commutative) (Problem 3).
A space is called simply connected if it is path connected and if its fundamental group is trivial.
  • Notation: .
  • In a simply connected space any two paths having the same initial and final points are path homotopic: .
The homomorphism or simply induced by where is continuous, , is a function such that .
  • If is a homeomorphism, then is an isomorphism.
  • A continuous map where is called a retraction of onto if for each . If is a retraction, then is an epimorphism of with where .

Topological Groups

Let ) be a topological group with the identity element .
  • is abelian.
  • where is taken point-wise, defined on paths in is the same as on .