----- Let $X,Y$ be [[topological space|topological spaces]]; $A,B$ respective [[subspace topology|subspaces]]. > [!proposition] Proposition. ([[detecting isomorphisms with relative homology]]) > > The [[long exact sequence for relative singular homology]], together with the [[five lemma]], yields the following. Let $f:(X,A) \to (Y,B)$ be a [[topological pair|map of pairs]]. $f$ induces: > 1. A map $f_{*}:H_{*}(X,A) \to H_{*}(Y,B)$ on [[relative singular homology|relative homology]], > 2. A map $H_{*}(X) \to H_{*}(Y)$ on [[singular homology|homology]], and > 3. A map $H_{*}(A) \to H_{*}(B)$ on[^1] [[singular homology|homology]]. > > If any two of these maps are [[isomorphism|isomorphisms]], then so is the third. > (The most useful version is $(1)+(3) \implies (2)$.) > [!proof]- Proof. ([[detecting isomorphisms with relative homology]]) > We'll prove $(2)+(3) \implies (1)$. The other two will be good to prove here when studying. > > A [[topological pair|map of pairs]] $f:(X,A) \to (Y,B)$ induces a [[chain map]] of [[exact sequence|exact sequences]] (by [[long exact sequence for relative singular homology|naturality]], I think) > > [[detecting isomorphisms with relative homology]] (basically five lemma)) > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BjKCHBAL6l0mXPkIoAjOSq1GLNgAkA+lgAUAQQCUIISOx4CRAEwzq9Zq0QgV6gBo69IDAfFEAzGbmWlqtXdJtXWFnUUMJZAAWLwsFaxVgLABaSQFNRxCXMSMUAFYY+SsbZUSUtIdg-WyIgDYCn2tOHj5BTLC3KVJJWVii2zUAIQyq8JMunsLfdQBNYdDXHORPbvNJ+L9p0iHK+eqiaJXvOOLS1MG5rNG88dWGk+Sz2Z3LjuQ6w962Jt5+Z-bFsgfNYcbg-QSyGBQADm8CIoAAZgAnCAAWyQZBAOAgSFSISRqJx1CxSGMTnxaMQpkx2MQ7jJyIpnmpSGiRyKnDQdEReEYO3JLKJNNy9IJiHyzMQNRFFMkTOJiAA7NKkArBUgABzKxDqtWIACcWr1uskGM+jXYnO5WF51AYdAARjAGAAFf4SEAMGDwnB8hk4jHyyS4hF+xDSCVKvGhqnyzVR0VymkG+MU1mB4ha8WB4Mgflh8OB0kUARAA > \begin{tikzcd} > \cdots \arrow[r] & H_i(A) \arrow[r] \arrow[d] & H_i(X) \arrow[r] \arrow[d] & {H_i(X,A)} \arrow[r, "\partial"] \arrow[d] & H_{i-1}(A) \arrow[r] \arrow[d] & H_{i-1}(X) \arrow[r] \arrow[d] & \cdots \\ > \cdots \arrow[r] & H_i(B) \arrow[r] & H_i(Y) \arrow[r] & {H_i(Y,B)} \arrow[r, "\partial"] & H_{i-1}(B) \arrow[r] & H_{i-1}(Y) \arrow[r] & \cdots > \end{tikzcd} > \end{document} > ``` > > $(2)$ and $(3)$ imply that all the vertical arrows are [[isomorphism|isomorphisms]] except the middle. The [[five lemma]] then says the middle map is an [[isomorphism]], so that $(1)$ holds once we consider this diagram for all $i$. > > > The other two implications are similar. ----- #### [^1]: Because it is a [[topological pair|map of pairs]]: $f(A) \subset B$, so $f:X\to Y$ restricts to a map $A \to B$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```