Undestanding why retraction turns the LES of relative homology groups to SES on pg.147 AT.

Question

The following paragraph is written in Hatcher:

But I do not understand this statement: "From this it follows that the boundary maps in the long exact sequence for $(X, A)$ are zero" According to Pg.115 in AT, I do not see where are the boundary maps, could anyone explain this statement in details for me, please?

Answer 1

The "boundary maps" are the maps $\partial:H_n(X,A)\to H_{n-1}(A)$ in the long exact sequence of homology for the pair $(X,A)$. They are so named because they are defined using the boundary map on the chain complexes (as described in the bottom half of page 116). If you know that $i_*:H_{n-1}(A)\to H_{n-1}(X)$ is injective then exactness of the sequence tells you that $\operatorname{im}(\partial)=\ker(i_*)=0$ and so $\partial=0$.