Existence of a (n-1)-connected map beween CW-spaces

Question

I have two finite CW-spaces $K$ and $L$ (K is n-dimensional and L is (n-1)-dimensional), a topological space $X$ and two maps $\phi:K\to X$ and $\psi: L\to X$, while $\phi$ is n-connected and $\psi$ is (n-1)-connected. I am fairly certain that then there is a (n-1)-connected map $\eta:L \to K$ such that $\phi\circ\eta$ is homotopic to $\psi$, although I have no idea how to prove this. Does anyone have an idea or has anyone already seen such a statement, or can provide a counterexample?

Answer 1

If $L$ is $(n-1)$-dimensional and $\phi:K\to X$ is $n$-connected, the map $\eta:L\to K$ such that $\phi\circ\eta\approx\psi$ always exists. Here doesn't matter, is $\psi$ $(n-1)$-connected or not.

One can prove it by induction. Suppose for some $m<n-1$ we have $\eta_m:sk^m(L)\to K$ such that $\phi\circ\eta_m\approx\psi|_{sk^m(L)}$. Let $D^{m+1}$ be an $(m+1)$-cell of $L$ with a glueing map $f:\partial D^{m+1}=S^m\to sk^m(L)$.

Consider the composition $\psi\circ f:S^m\to X$, as an element of $\pi_m(X)$ this spheroid is trivial, because of we can extend it to the $D^{m+1}$. Since $\phi$ is $n$-connected and $m<n$, the spheroid $\eta_m\circ f$ is trivial too. Therefore we can extend $\eta_m$ to the arbitrary $(m+1)$-cell.

Answer 2

No, such $\eta$ may not exist. For example, let $X=L=\mathbb CP^\infty$, $\psi=\mathrm{Id}_{\mathbb CP^\infty}$, $K=S^2$, $n=2$, and $\phi$ is the inclusion $S^2=\mathbb CP^1\hookrightarrow\mathbb CP^\infty$.

There is no $\eta:\mathbb CP^\infty\to\mathbb CP^1$ such that $\phi\circ\eta\approx\psi=\mathrm{Id}_{\mathbb CP^\infty}$, because $H_4(S^2)=0$ but $H_4(\mathbb CP^\infty)\ne0$.