# Barratt Whitehead Lemma, Proving exactness at the 'direct sum' module

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

• Does the account in merry.io/resources/algtopIII.pdf help? Nov 17 '14 at 11:31
• Yes. That gave me exactly the hint I needed, which was to - quite literally - think outside the square. I really like the way the proof has to first chase around the square to the right and then around the one to the left, to prove the result about the square in the middle. That looks like a nice set of notes by Dr Merry too. Thank you Ronnie. Nov 18 '14 at 8:14
• The link unfortunately does not work anymore. Are there any other references? @RonnieBrown Oct 28 at 9:37