Definition 2.2.1. Group compendium.
A group is a set \(G\) together with a binary operation \(\cdot G\times G\rightarrow G\) such that (i) the operation is associative, (ii) there is an identity element \(e\in G\) satisfying \(e\cdot g=g\cdot e=g\) for all \(g\in G\text{,}\) and (iii) for all \(g\in G\) there is an inverse element \(g^{-1}\in G\) satisfying \(g\cdot g^{-1}=g^{-1}\cdot g=e\text{.}\)
A subgroup of a group \(G\) is a subset \(H\subseteq G\) that contains the identity element \(e\in G\text{,}\) and which is closed under the group operation: i.e., if \(h, h'\in H\text{,}\) then \(h\cdot h'\in H\text{.}\) We write \(H\leq G\) in this case.
A normal subgroup of \(G\) is a subgroup \(N\) that satisfies \(gNg^{-1}=N\) for all \(g\in G\text{:}\) i.e., \(N\) is closed under conjugation. We write \(N\normalin G\) in this case.
Given a subgroup \(H\leq G\text{,}\) we denote by \(G/H\) the set of all left cosets of \(H\text{:}\) i.e., \(G/H=\{gH\colon g\in G\}\text{.}\) Given a normal subgroup \(N\normalin G\) the set of cosets \(G/N\) has a group structure given by the operation
\begin{equation*}
gN\cdot g'N=gg'N\text{.}
\end{equation*}
We call \(G/N\) the quotient of \(G\) by \(N\text{.}\)
A homomorphism between groups \(G, G'\) is a function \(\phi\colon G\rightarrow G'\) satisfying \(\phi(g_1\cdot_G g_2)=\phi(g_1)\cdot_{G'} \phi(g_2)\) for all \(g_1, g_2\in G\text{.}\) (Here we use subscript notation to differentiate the two different group operations at play.)
An isomorphism is an invertible homomorphism \(\phi\colon G\rightarrow G'\text{.}\) We say \(G\) and \(G'\) are isomorphic, denoted \(G\cong G'\text{,}\) if there is an isomorphism between them. (As it turns out, the inverse of an invertible group homomorphism is automatically a group homomorphism.)
Definition 2.2.2. Fundamental group.
Given a topological space \(X\) and element \(x_0\in X\text{,}\) the fundamental group of \(X\) based at \(x_0\text{,}\) denoted \(\pi_1(X,x_0)\text{,}\) is defined as
\begin{equation*}
\pi_1(X,x_0)=P(X; x_0, x_0)/\simeq_p\text{.}
\end{equation*}
In other words, \(\pi_1(X,x_0)\) is the set of all path homotopy equivalence classes \([f]\text{,}\) where \(f\in P(X; x_0, x_0)\) is a closed path (or loop) beginning and ending at \(x_0\text{.}\)
The path product operation
\begin{align*}
\pi_1(X,x_0)\times \pi(X,x_0) \amp \rightarrow \pi_1(X,x_0)\\
([f],[g]) \amp \mapsto [f]*[g]=[f*g]
\end{align*}
endows \(\pi_1(X,x_0)\) with the structure of a group.
Identity element.
The identity element of \(\pi_1(X,x_0)\) is given by \([e_{x_0}]\text{,}\) where \(e_{x_0}\colon I\rightarrow X\) is the constant function \(e_{x_0}(s)=x_0\) for all \(s\in I\text{.}\)
Inverse elements.
Given an element \(g=[f]\in \pi_1(X,x_0)\) its group inverse is \(g^{-1}=[\overline{f}]\in \pi_1(X,x_0)\text{,}\) where \(\overline{f}\) is the reverse of \(f\text{.}\)
Proof.
That the path product operation endows
\(\pi_1(X,x_0)\) with the structure of a group follows directly from
Theorem 2.1.12.
Theorem 2.2.4. Fundamental group and path components.
Assume \(\alpha\in P(X; x_0, x_1)\text{.}\) The map
\begin{align*}
\hat{\alpha}\colon \pi_1(X,x_0) \amp\rightarrow \pi_1(X,x_1) \\
[f] \amp \mapsto \hat{\alpha}([f])=[\overline{\alpha}]*[f]*[\alpha]
\end{align*}
is an isomorphism. As a consequence, if \(C\subseteq X\) is a path component of \(X\text{,}\) then \(\pi_1(X,x)\cong \pi_1(X,x')\) for any \(x,x'\in C\text{.}\)
Definition 2.2.5. Pointed space.
Given a topological space \(X\) and element \(x_0\in X\text{,}\) the pair \((X,x_0)\) is called a pointed space with base point \(x_0\text{.}\) Given pointed spaces \((X,x_0)\) and \((Y,y_0)\text{,}\) a map of pointed spaces (or based map) is a continuous function \(f\colon X\rightarrow Y\) satisfying \(f(x_0)=y_0\text{.}\) We write \(f\colon (X,x_0)\rightarrow (Y,y_0)\) in this case.
Corollary 2.2.7. Fundamental group invariance.
Let \(h\colon X\rightarrow Y\) be a homemorphism, let \(x_0\in X\text{,}\) and let \(y_0=h(x_0)\text{.}\) The map \(h_*\colon \pi_1(X,x_0)\rightarrow (Y,y_0)\) is an isomorphism of groups.
