Consider a map of topological pairs $(F,f): (X,A) \rightarrow (Y,B)$ consisting of a map $F: X \rightarrow Y$ and a map $f: A \rightarrow B$ such that $f$ is the restriction of $F$ from $A$ to $B$. We assume that $F$ and $f$ induce isomorphisms on all homology-modules. How can we show that also $(F,f)$ induces isomorphisms on all homology-modules? I guess one might be able to use functorial properties of the homology functor or some axioms like long exact sequence of a pair.
2026-04-01 03:06:44.1775012804
When does $(F,f): (X,A) \rightarrow (Y,B)$ induce isomorphisms on homology?
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Hint: Use the long exact sequence induced on homology and the 5-lemma.