Let $X$ be a compact Hausdorff space and $S$ a finite dimensional, convex, Hausdorff space. Moreover, let $f:X \to S$ be a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $s\in S$, where $\tilde{H}$ denotes reduced Čech-cohomology with coefficients in a module $G$. Finally, let $\partial X = f^{-1}(\partial S)$.
I am trying to show:
If $f:(X,\partial X) \to (S,\partial S)$ induces a non-zero homomorphism $f^*:\tilde{H}(S,\partial S) \to \tilde{H}(X,\partial X)$ then $f$ is not homotopic to any map $\bar{f}: X \to \partial S$ (that agrees with $f$ on $\partial X$).
I'm not sure how to proceed. My attempt thus far has basically been to observe that if $f$ is injective on $\partial X$, then if such a map $\bar{f}$ existed, $f^{-1} \circ \bar{f}: X \to \partial X$ would be a retract, hence it would follow that:
$$\tilde{H}^n(X) \approx \tilde{H}^n(\partial X) \oplus \tilde{H}^n(X,\partial X)$$
which can yield a contradiction with a step or two of extra work. But clearly this isn't general. I suspect that I need to show that such a map $\bar{f}$ induces the zero homomorphism by showing it's homotopic to a constant map, but I'm not sure how to. Many thanks for any help!
(For those interested, this question is motivated by problem I.4.17.a on pages 54-55 in this book)