As the title suggests, used in Rotman's proof of the Hurewicz Theorem is the following fact (exercise 4.14, An Introduction to Algebraic Topology): if $f$ is a path (not necessarily closed) in a toplogical space $X$, then $f$ is homologous to $-f^{-1}.$ His hint is a referral back to a previous exercise (which I worked out), that is basically an application of the nullhomotopy theorem, applied to $\Delta^2.$ This may be a softball question, but I do not see the need to use the nullhomotopy theorem, because Rotman has already essentially proved this in his development of the Hurewicz map. That is,
$a).\ \eta:\Delta^1\to I$ is $(1-t)e_0+te_1\mapsto t$ and the Hurewicz map $\varphi$ is $[f]\mapsto f\eta+B_1(X).$ If $e_{x_{0}}$ is the constant loop at $x_0,$ it is immediate that $e_{x_{0}}\eta(\Delta^1)=\left \{ x_0 \right \}.$ But $e_{x_{0}}\eta$ is the boundary of the constant $2-$simplex $t_0e_0+t_1e_1+t_2e_2\mapsto x_0$ so in fact $[e_{x_{0}}]\mapsto 0.$
$b).$ if $f,g: \Delta^1\to X$ and $f\sim g\ $rel $\dot{I}$ then $f$ and $g$ are homologous. This is proved when Rotman shows that $\varphi$ is well-defined.
$c).\ f+g-f\ast g\in B_1(X):\ $ a careful study of Rotman's proof that $\varphi$ is a homomorphism shows that $f$ and $g$ need not be loops for this result to hold.
$d).$ Combining the previous observations, and using $\sim$ to mean "homologous", we have then $f+f^{-1}\sim f\ast f^{-1}\sim e_{x_{0}}\sim 0.$
I never used the nullhomotopy theorem, nor the hint in Rotman's exercise.
edit: I worked the problem out using Rotman's hint, which is a picture:
The proof now consists of defining a $\sigma: \Delta^2\to X$ in such a way that $\sigma$ agrees with a previously defined function on $\dot{\Delta}$ that is equal to $f,f^{-1}$ and $f\ast f^{-1}$ on the corresponding sides of the triangle (the 2-simplex). This is done in the general case in Rotman's Theorem 4.27, and works for any paths $f,g$; i.e. $f,g$ are not necessarily closed. All that is required is that $f$ and $g$ be composable. To finish, one simply invokes the nullhomotopy theorem.
So my question still stands: why not just do the exercise as I did?