I'd like to prove that if $A$ is a retraction of deformation of $V$, where $A \subset V \subset X$, then $H_n(X,V) \simeq H_n(X,A)$ using the five lemma. I thought I could reach the diagram from the exact sequence of pair of $(X,V)$ and $(X,A)$ in the following way. Consider
$$ \cdots \longrightarrow H_n(V) \longrightarrow H_n(X) \longrightarrow H_n(X,V) \longrightarrow H_{n-1}(V) \longrightarrow H_{n-1}(X) \longrightarrow \cdots$$
$$ \cdots \longrightarrow H_n(A) \longrightarrow H_n(X) \longrightarrow H_n(X,A) \longrightarrow H_{n-1}(A) \longrightarrow H_{n-1}(X) \longrightarrow \cdots$$
As far as concerns the vertical arrow I could use the isomorphism $i_* : H_n(A) \longrightarrow H_n(V)$ induced by the homotopic equivalence $i$. I'd like to conclude that $H_n(X,A) \simeq H_n(X,V)$ completing the diagram with the map induced by the identity of $X$, $id_* : H_n(X) \longrightarrow H_n(X)$ but I don't wheter the diagram commutes to correctly appy the five lemma, to exploit the naturality of the connection homomorphism $\partial : H_n(X,A) \longrightarrow H_{n-1}(A)$, any help would be appreciated.
Best Answer
Consider the map of pairs $\phi : (X,A) \to (X,V), \phi(x) = x$. Its "absolute" part is the identity $id_X : X \to X$ and its "relative" part the inclusion $\phi_A : A \to V$ which is a homotopy equivalence in this case. Hence we get induced maps $(id_X)_* = id : H_n(X) \to H_n(X)$, $(\phi_A)_* : H_n(A) \to H_n(V)$ and $\phi_* : H_n(X,A) \to H_n(X,V)$, where $(\phi_A)_*$ is an isomorphism. These maps connect the two long exact sequences in your question. Each of the resulting squares commutes because the $H_n$ are functors and $\partial$ is natural for maps of pairs.
Now the five lemma applies.