When trying to decide whether or not a given loop on a space $X$ is trivial up to homotopy, it is not sufficient to know the fundamental group of the space. For example, merely knowing that the fundamental group of the circle is $\mathbb Z$ is not enough to conclude that the single turn in $S^1$ is not homotopically trivial (The only argument I know to settle this is the usual covering space one).


A covering space may help decide whether or not a loop is trivial by using lifting and homotopy lifting theorems. But there is this another simple technique which I recently realized.


Fact. Let $X$ and $Y$ be topological spaces and $f, g:X\to Y$ be maps which are homotopic. Let $\gamma$ be a loop based at $x_0$ in $X$. Then $f\circ \gamma$ is homotopically trivial loop based at $f(x_0)$ in $Y$ if and only if $g\circ \gamma$ is homotopically trivial loop based at $g(x_0)$ in $Y$.

A useful corollary to this fact is the following:

Corollary. Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a homotopy equivalence. Then a loop $\gamma$ based at $x_0$ in $X$ is homotopically trivial if and only if $f\circ \gamma$ is a homotopically tirival loop based at $f(x_0)$ in $Y$.

Therefore, a deformation retraction can help us decide whether a given loop is trivial or not, for a deformation retract yields a retraction which is a homotopy equivalence.

This is all that I wanted to say about the matter. Please feel free to comment or add anything in this regard on this thread, especially if it took you also longer than it should have to realize this!

Thank you.