Extending cellular maps between aspherical complexes

Question

In a paper I read, the author seemed to use a property similar to:

Let $X, Y$ be two aspherical CW-complexes and $f : X^{(2)} \to Y^{(2)}$ be a cellular map between their 2-skeletons. Then $f$ extends to a cellular map $\tilde{f} : X \to Y$.

Subsidiary question: If $f$ is onto, can $\tilde{f}$ be chosed onto?

I did not really work with homotopy groups, do you have a reference for such a property? (in Hatcher's book?)

Answer 1

If $Y$ is aspherical and path connected, then a map $f : X^{(2)} \to Y$ can be extended to $\tilde{f} : X \to Y$. This is a special case of the extension lemma (lemma 4.7) in Hatcher's Algebraic Topology:

Given a CW pair $(X, A)$ and a map $f : A \to Y$ with $Y$ path-connected, then $f$ can be extended to a map $X \to Y$ if $\pi_{n−1}(Y) = 0$ for all $n$ such that $X - A$ has cells of dimension $n$.

The resulting extension is not onto in general. For example, consider the inclusion $X \hookrightarrow X \times X$, where $X$ is aspherical and $\dim X = 2$.

Edit: As Lano points out in the comments, if you want the extension to be cellular, you can take the cellular approximation of $\tilde{f}$ while keeping $\left.\tilde{f}\right|_A = f$ fixed. This is theorem 4.8 in Hatcher's book.