Homotopy class of a loop in the $2$-skeleton of a simplicial complex

Question

Suppose I have a loop $\sigma : [0, 1] \rightarrow X$ in a path-connected finite simplicial complex $X$. I know that $\sigma$ can be homotoped so that it lives in $\text{sk}^2(X)$, but is this homotopy class unique in $\text{sk}^2(X)$? Or can I find two loops $\sigma', \sigma'' : [0, 1] \rightarrow \text{sk}^2(X)$ that are homotopic to $\sigma$ in $X$ but not homotopic to one another in $\text{sk}^2(X)$?

The motivation for this question is that given a continuous $f : \text{sk}^2(X) \rightarrow Y$ between path-connected finite complexes that maps a particular loop $\sigma$ in $\text{sk}^2(X)$ to $\tau$ in $Y$, I want to find a homomorphism $h : \pi_1(X) \rightarrow \pi_1(Y)$ that maps $[\sigma]$ to $[\tau]$. It suffices to define $h$ on each homotopy class of loops $[\sigma]$ by first homotoping $\sigma$ to a loop $\sigma'$ in $\text{sk}^2(X)$, then sending it to $[f(\sigma')]$, but I need to show this is well-defined. Can I somehow use the isomorphism between $\pi_1(\text{sk}^2(X))$ and $\pi_1(X)$?

Answer 1

The inclusion of the $2$-skeleton into $X$ induces an isomorphism on $\pi_1$, and in particular an injection. This means that two loops in the skeleton which are homotopic in $X$ are already homotopic in the skeleton!

Answer 2

To expand the context just a bit, all this follows from the cellular approximation theorem, which has the special case:

If $f:X\to Y$ is a continuous map of simplicial complexes with $\dim X=n$, then $f$ is homotopic to a map $g$ with $g(X)\subset \mathrm{sk}^n(Y)$.

So your loop can be homotoped even into $\mathrm{sk}^1$, and any homotopy of a loop, being a map from $I\times I\to Y$, can be homotoped to occur in $\mathrm{sk}^2$.