I have been working out a proof of the Barratt Whitehead Lemma. Here it is.
I had it all finished, but then I realised that I had assumed all the 'vertical' homomorphisms connecting the upper and lower exact sequences were isomorphisms, whereas the Lemma's premises only require that every third such map - the one I have labelled $c_n$ - is an isomorphism. So I had to weaken that premise and test if the proof still survived.
On reviewing my proof, I realised that all its parts survived this weakening of the premises except for the proof of exactness at the direct sum node $B_n\oplus A'_n$. For that node, my proof that Im $\alpha_n\subseteq$ Ker $\beta_n$ is still valid, but the proof of the reverse inclusion is not, because it assumes that both of the 'vertical' homomorphisms $a_n$ and $b_n$ are respectively a surjection and an injection, and the premise does not justify that.
I have tried to save the proof but have hit a brick wall. In fact, it doesn't even look to me as though it should be true in general that Ker $\beta_n\subseteq$ Im $\alpha_n$.
I have two third-party partial proofs of the lemma, but both prove exactness at only one of the three types of nodes in the sequence. They both avoid the 'difficult' node - one proving exactness for one of the easy nodes, the other proving it for the other easy one - leaving the difficult node as 'an exercise for the reader'.
Can anybody suggest how to prove Ker $\beta_n\subseteq$ Im $\alpha_n$, or point to a proof that does it. The slightest hint would be helpful (in fact I'd prefer a hint to a 'here it is').
Thank you very much