Section 52: The Fundamental Group

Elementary Group Theory

Given two groups $G$ and $G'$ , a function $f:G\rightarrow G'$ is called a homomorphism if for every $x,y\in G$ , $f(x\cdot y)=f(x)\cdot f(y)$ .

The image of $G$ under the homomorphism $f$ is a subgroup of $G'$ .
An injective homomorphism is called monomorphism.
A surjective homomorphism is called epimorphism.
A bijective homomorphism is called isomorphism.

The preimage of $e'\in G'$ under homomorphism $f$ , $f^{-1}(e')$ , is called the kernel $\ker f$ of the homomorphism $f$ . It is a subgroup of $G$ .

The left coset of a subgroup $H$ in $G$ is the set $xH=\{x\cdot h|h\in H\}$ . Similarly, the right coset of a subgroup $H$ in $G$ is the set $Hx=\{h\cdot x|h\in H\}$ .

Subgroup $H$ is called normal if for every $x\in G$ , $xH=Hx$ . In other words, $H$ is a normal subgroup of $G$ if for every $x\in G$ and $h\in H$ , $x\cdot h\cdot x^{-1}\in H$ .

If $H$ is a normal subgroup of $G$ , the partition of $G$ into left (or right) cosets is denoted by $G/H$ . $(G/H,\cdot)$ where $(xH)\cdot(x'H)=(x\cdot x')H$ is a group called the quotient of $G$ by $H$ . If $H$ is a subgroup of $G$ that is not normal, $G/H$ means the partition of $G$ into right cosets of $H$ in $G$ .

$f:G\rightarrow G/H$ defined by $f(x)=xH$ is an epimorphism with the kernel $H$ .
Vice versa, if $f:G\rightarrow G'$ is an epimorhism, then $K=\ker f$ is a normal subgroup of $G$ , and $f$ induces an isomorphism $g:G/K\rightarrow G'$ such that $g(xK)=f(x)$ for $x\in G$ .

The Fundamental Group

Let $X$ be a space and $x_{0}\in X$ .

A loop based at $x_{o}$ is any path starting and ending at $x_{0}$ .

The fundamental group of $X$ relative to the base point $x_{0}$ , $\pi_{1}(X,x_{0})$ , is the set of path homotopy classes of loops based at $x_{0}$ together with the operation $\star$ .

This is also called the first homotopy group of $X$ .
For a path connected space (or for a path connected component of a space) the choice of the point $x_{0}$ is not important: if $x_{0},x_{1}\in X$ where $X$ is path connected, then $\pi_{1}(X,x_{0})$ is isomorphic to $\pi_{1}(X,x_{1})$ .
- To show this, for a path $\alpha$ connecting $x_{0}$ and $x_{1}$ , we introduce the map $\hat{\alpha}:\pi_{1}(X,x_{0})\rightarrow\pi_{1}(X,x_{1})$ defined by $\hat{\alpha}([f])=[\overline{\alpha}]\star[f]\star[\alpha]$ which is a group isomorphism.
The reference point $x_{0}$ is still needed, because the isomorphism between $\pi_{1}(X,x_{0})$ and $\pi_{1}(X,x_{1})$ may depend on the path chosen between $x_{0}$ and $x_{1}$ .
- 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: $\pi_{1}(X,x_{0})=0$ .
In a simply connected space any two paths having the same initial and final points are path homotopic: $[\alpha]=[\alpha\star\overline{\beta}\star\beta]=[\beta]$ .

The homomorphism $(h_{x_{0}})_{*}$ or simply $h_{*}$ induced by $h:X\rightarrow Y$ where $h$ is continuous, $h(x_{0})=y_{0}$ , is a function $h_{*}:\pi_{1}(X,x_{0})\rightarrow\pi_{1}(Y,y_{0})$ such that $h_{*}([f])=[h\circ f]$ .

If $h$ is a homeomorphism, then $h_{*}$ is an isomorphism.
A continuous map $r:X\rightarrow A$ where $A\subset X$ is called a retraction of $X$ onto $A$ if $r(a)=a$ for each $a\in A$ . If $r$ is a retraction, then $r_{*}$ is an epimorphism of $\pi_{1}(X,a)$ with $\pi_{1}(A,a)$ where $a\in A$ .

Topological Groups

Let $(G,\cdot$ ) be a topological group with the identity element $x_{0}$ .

$\pi_{1}(G,x_{0})$ is abelian.
$[f]\otimes[g]=[f\cdot g]$ where $f\cdot g$ is taken point-wise, defined on paths in $\pi_{1}(G,x_{0})$ is the same as $\star$ on $\pi_{1}(G,x_{0})$ .

Subpages

Section 52: The Fundamental Group

Section 52: The Fundamental Group

Elementary Group Theory

The Fundamental Group

Topological Groups