Below, "homotopy pushout" is defined via weak homotopy equivalences (as opposed to homotopy equivalences).
Fix a homology theory. In this MO question an excerpt from a paper of Goodwillie is given which states that a homotopy pushout square induces isomorphisms on relative homology.
$$\require{AMScd} \begin{CD} A @>f>> B \\ @VgVV @VV{g^\prime}V\\ C @>>{f^\prime}> X \end{CD}$$
Questions.
- How to prove the above statement?
- In the context of singular homology, is it possible to give a direct proof? That is, can we prove the above assertion using only invariance under weak homotopy equivalence without using the excision theorem (or Mayer-Vietoris)?
Below are the relevant diagrams. The right one is defined as the homology of the left one. The claim is that if the square about is a homotopy pushout then the lower right horizontal arrow is an isomorphism. I just don't see how to get from a homotopy pushout to the map induced by the cokernel (quotient).
$$\require{AMScd} \begin{CD} \mathrm{Ch}(A) @>{\mathrm{Ch}(f)}>> \mathrm{Ch}(B)\\ @V{\mathrm{Ch}(g)}VV @VV{\mathrm{Ch}(g^\prime)}V\\ \mathrm{Ch}(C) @>{\mathrm{Ch}(f^\prime)}>> \mathrm{Ch}(X) \\ @VVV @VVV\\ \operatorname{Coker}\mathrm{Ch}(g) @>>> \operatorname{Coker}\mathrm{Ch}(g^\prime) \end{CD} \qquad \begin{CD} \mathrm{H}(A) @>>> \mathrm{H}(B)\\ @VVV @VVV\\ \mathrm{H}(C) @>>> \mathrm{H}(X) \\ @VVV @VVV\\ \mathrm{H}^\text{rel}(g) @>>> \mathrm{H}^\text{rel}(g^\prime) \end{CD}$$
Here is the cellular approximation idea: First, since you are working with a homotopy push out square, you can assume the maps $C \leftarrow A \rightarrow B$ are cofibrations. Therefore, they are inclusions, and then we can build up a CW approximation to the square above
$$\require{AMScd} \begin{CD} C' @<g'<< A' @>f'>> B' \\ @VVV @VVV @VVV\\ C @<{g}<< A @>f>> B\end{CD}$$ The vertical maps are all weak equivalences.
This is proven in May, Concise: https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf in the theorem on p.78 at the start of section 6. The maps on top, $A' \to B'$ and $A' \to C'$ are subcomplexes. The induced map on pushouts $X' \to X$ is a weak equivalence because each is a homotopy pushout. We can similarly see that the induced vertical maps on quotients:
$$\require{AMScd} \begin{CD} C'/A' @>>> X'/B' \\ @VVV @VVV \\ C/A @>>> X/B \end{CD}$$ are also weak equivalences. The top horizontal map is easily seen to be a bijection ($X' = C'\cup_{A'} B'$), and the CW topology tells you it is a homeomorphism. Thus, the bottom horizontal map is a weak equivalence.