Defining $\Omega(\mathbb{S}^{1},a) = \left\lbrace \gamma \in C^{0}(\mathbb{S}^{1},X) : \gamma(1) = a\right\rbrace$ with $\mathbb{S}^{1} = \left\lbrace z \in \mathbb{C} : \lvert z\rvert = 1 \right\rbrace \subset \mathbb{C}$ where $1 \in \mathbb{S}^{1}$ is thought as $(1,0) \in \mathbb{R}^{2}$.
We have the following $\textit{facts}$ :
$\textbf{(1)} \hspace{0.1cm} $There is a natural bijection from $\Omega(a,a) = \left\lbrace \gamma \in C^{0}([0,1],X) : \gamma(0) = \gamma(1) = a \right\rbrace$ and $\Omega(\mathbb{S}^{1},a)$ given by $\alpha \to \overset{\sim}{\alpha} \circ \pi$, where $\pi$ is the usual identification of $[0,1]/\left\lbrace 0,1 \right\rbrace \approx \mathbb{S}^{1}$.
Which gives $\alpha \sim \beta \iff \overset{\sim}{\alpha} \sim \overset{\sim}{\beta}$, where $\sim$ denotes the homotopy relation and $\overset{\sim}{\alpha}, \overset{\sim}{\beta}$ denote the maps from $[0,1]/ \sim \hspace{0.1cm} \longmapsto X$.
$\textbf{(2)} \hspace{0.1cm}$ Let $Q = [0,1] \times [0,1]$ and $C \subset Q$ given by $C = \left\lbrace s=1\right\rbrace \cup \left\lbrace t=0\right\rbrace \cup \left\lbrace t=1\right\rbrace$ where $t,s$ are the coordinates of $Q$. We have that $Q/C \approx D^{2}$ (the two dimensinal disk) given homeomorphism which send $[t,0] \to e^{2\pi it} \forall t \in [0,1]$.
The proof I have of this fact is the following : It enough to observe that $\exists f : Q \longmapsto D^{2}$ continuos with $f(C) = \left\lbrace 1 \right\rbrace$, $f(t,0) = e^{2 \pi i t}$ and $f_{|_{Q – \partial Q}} : Q – \partial Q \longmapsto D^{2} – S^{1}$ bijective.
$\textbf{(3)} \hspace{0.1cm}$ Given $\alpha \in \Omega(a,a)$, $[\alpha] = 1 \in \pi_{1}(X,a) \iff \overset{\sim}{\alpha} : S^{1} \longmapsto X$ extends to a map to $D^{2}$, which means that $\exists f : D^{2} \longmapsto X$ with $f_{|_{S^{1}}} = \overset{\sim}{\alpha}$ and $f$ continuos.
Now my $\textit{questions} :$ I didn't find any references between between the relations between these three facts and the fundamental group, neither from a topological nor geometrical point of view. Maybe this concerns more algebraic topology (which I'm unfortunately not familiar with) but still I would be interested in those, since without any further material I cannot link the homotopy theory on the fundamental group and the maps given from $\mathbb{S}^{1}$ to $X$.
As far as concerns the facts $\textbf{(2)}$, $\textbf{(3)}$ I'd like to find a complete proof of $\textbf{(2)}$, (maybe an explicit $f$ ? To visualize it better) and a proof of the third. But I really am more interested in understanding in depth the links or what this facts are telling me in relation with the fundamental group (even with some basic tool of algebraic topology giving the notion if it helps to have a bigger picture) than the demostrations themselves.
Any explanation, thought of reference would be appreciatd.
$\textbf{Edit :}$ I found some references of $\textbf{(2)}$ here : Existence of a simple homeomorphism
Best Answer
I think your facts (1) - (3) are mathematical folklore which means that they are well-known and easy to prove. Sometimes it is difficult to find references in textbooks (although they certainly exist somewhere).
(1) is obvious because the quotient map $p : I \to S^1, p(t) = e^{2\pi it}$, induces a bijection $p^* : \Omega(S^1,a) \to \Omega(a,a)$ given by $p^*(\alpha) = \alpha \circ p$. For a more detailed treatment see for example
Have a look at Chapter 1, Sections 6 and 8.
(2) is covered by Existence of a simple homeomorphism as your edit shows.
(3) is covered by my answer to Can we always view loops as maps from $S^1\to X$? Also see Spanier's book Theorem 7 in Chapter I, Section 6.
See also Hatcher's "Algebraic Topology", Section "Fundamental group" and especially the exercises.