Interpretation of fundamental group

Question

I learned the concept of the fundamental group in Munkres where

For a space $X$ and $x\in X$, the fundamental group is defined by $\pi_1(X,x)=\Omega(X,x)/\sim$ where $\Omega(X,x)=\{\gamma:[0,1]\to X:\gamma(0)=\gamma(1)=x\}$ and $\sim$ is under homotopy.

I interpret it as different kinds of circles up to homotopy. However, on page 339 of Introduction of Topological Manifolds by John M. Lee it's been said that

We can think of nontrivial elements of the fundamental group of a space X as equivalence classes of maps from the circle into X that do not extend to the disk.

My questions are: 1. the path $\gamma: I\to X$ is the same $\gamma':S^1\to X$. Is it because reparametrization preserves homotopy class? 2. What does it mean by not extending to a disk?

Answer 1

1. It is not the same. However, we can identify the quotient space $I/\{0,1\}$ with the circle $S^1$. Hence each closed path $\gamma \in \Omega(X,x)$ induces a map $\gamma' : S^1 \to X$ such that $\gamma'(1) = x$. Concerning the assocation $\gamma \mapsto \gamma'$ have a look at Characterizing simply connected spaces which explains that $\pi_1(X,x)$ can be identified with the set of pointed homotopy classes of pointed maps $(S^1,1) \to (X,x)$.

2. A pointed map $f : (S^1,1) \to (X,x)$ is pointed homotopic to the constant pointed map if and only if $f$ has an extension $F: D^2 \to X$. See again the answer to Characterizing simply connected spaces.