# Homology of triples

A triple of topological spaces $(X, A, B)$ consists of a topological space $X$ and two subspaces $A, B$ with $B \subseteq A \subseteq X$.

A map of triples $f \colon (X, A, B) \rightarrow (Y, G, H)$ is a continuous function $f \colon X \rightarrow Y$ such that $f(A) \subseteq G$ and $f(B) \subseteq H$.

Associated with the triple $(X, A, B)$ are the pairs $(A, B), (X, B), (X, A)$ as well as the natural inclusions $i \colon (A, B) \rightarrow (X, B)$ and $j \colon (X, B) \rightarrow (X,A)$.

Question 1: Show that this induces a long exact sequence in (singular) homology $$\cdots \rightarrow H_{n+1} (A, B) \xrightarrow{i_{\ast}} H_{n+1}(X, B) \xrightarrow{j_{\ast}} H_{n+1}(X,A) \xrightarrow{\partial_{\ast}} H_{n}(A,B) \rightarrow \cdots$$ where $\partial_{\ast}$ is the connecting homomorphism $$\partial_{\ast}^{(X,A)} \colon H_{n+1}(X,A) \rightarrow H_{n}(A)$$ of the pair $(X,A)$ followed by the inclusion $$j_{\ast}^{(A,B)} \colon H_{n}(A) \rightarrow H_{n}(A,B)$$ of the pair $(A,B)$.

Question 2: Show that a map of triples $f \colon (X,A,B) \rightarrow (Y,G,H)$ induces three maps of pairs $f' \colon (A,B) \rightarrow (G,H)$, $f'' \colon (X,B) \rightarrow (Y,H)$, and $f''' \colon (X,A) \rightarrow (Y,G)$ yielding the commutative diagram

$$\begin{array} \rightarrow \rightarrow & H_{n+1}(A,B) &\stackrel{i_{\ast}}{\longrightarrow}&H_{n+1}(X,B)& \stackrel{j_{\ast}}{\longrightarrow}& H_{n+1}(X,A)&\stackrel{\partial_{\ast}}{\longrightarrow}& H_{n}(A,B) & \rightarrow \\ &\downarrow{f'_{\ast}}&&\downarrow{f''_{\ast}}&&\downarrow{f'''_{\ast}}&&\downarrow{f'_{\ast}}&\\ \rightarrow & H_{n+1}(G,H)&\stackrel{i_{\ast}}{\longrightarrow}&H_{n+1}(Y,H)& \stackrel{j_{\ast}}{\longrightarrow}&H_{n+1}(Y,G)& \stackrel{\partial_{\ast}}{\longrightarrow}&H_{n}(G,H) & \rightarrow \end{array}$$

Question 3: Show that if the inclusion $B \rightarrow A$ induces an isomorphism $H_{n}(A) \rightarrow H_{n}(X,A)$ for all $n$, then $j_{\ast} \colon H_{n}(X,B) \rightarrow H_{n} (X,A)$ is also an isomorphism for every $n$.

What I need help with : So I'm working out of Dold and have just started working through singular homology. These questions seem like basic diagram chasing but I don't know what definitions I need and where to start.