Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation:
There is a map $f^1:K^1\to Y$ that can be extended to $f^2:K^2\to Y$ and yet no such extension can be further extended to $f^3:K^3\to Y$.
The idea is that there is an obstruction to the existence of $f^3$ already on the one-dimensional level but not by obstructing the existence of $f^2$. It is written in Hilton and Wylie's book that it is possible, yet I was not able to construct an explicit example myself.