---- > [!theorem] Theorem. ([[relative homology for a good pair is reduced homology of the quotient]]) > Let $X$ be a [[topological space]] and $A \subset X$ a [[subspace topology|subspace]]. > > There is a natural [[topological pair|map of pairs]] $(X, A) \to (X / A, A /A).$If the [[topological pair|pair]] $(X,A)$ is a [[good pair|good]], then the [[relative singular homology|induced map]][^1] > >$H_{n}(X,A) \to H_{n}(X / A, A /A )=\widetilde{H}_{n}(X / A)$ >is an [[isomorphism]]. ^theorem > [!proof]- Proof. ([[relative homology for a good pair is reduced homology of the quotient]]) > Summary: > - Show $H_{n}(X,A) \to H_{n}(X,U)$ is an isomorphism. Say $H_{n}(X / A, A / A) \to H_{n}(X / A, U/A)$ is an isomorphism. > - Make a commutative square with these maps being the two rows. Then excise $A$ and $A / A$ respectively. > - The map of pairs $(X-A, U-A) \to (X / A - A / A, U/A - A / A)$ is a [[homeomorphism]]. Then finish. > > By assumption, the pair $(X,A)$ is [[good pair|good]]; fix $U \supset \overline{A}$ such that the [[inclusion map]] $\iota:A \hookrightarrow U$ is a [[deformation retract]]. Since [[deformation retract|deformation retracts]] are [[homotopy equivalent|homotopy equivalences]], [[homotopy invariance of singular homology|the map]] $H_{*}(A) \to H_{*}(U)$is an [[isomorphism]]. (Naturality of?) the [[long exact sequence for relative singular homology|LES]] on [[singular homology|homology]] gives a [[chain map]] of [[exact sequence|exact sequences]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > \begin{tikzcd}[column sep=large, row sep=huge] > H_n(A) \arrow[r] \arrow[d] & > H_n(X) \arrow[r] \arrow[d, equal] & > H_n(X, A) \arrow[r] \arrow[d] & > H_{n-1}(A) \arrow[r] \arrow[d] & > H_{n-1}(X) \arrow[d, equal] \\ > H_n(U) \arrow[r] & > H_n(X) \arrow[r] & > H_n(X, U) \arrow[r] & > H_{n-1}(U) \arrow[r] & > H_{n-1}(X) > \end{tikzcd} > \end{document} > ``` > > wherein all maps but the middle are [[isomorphism|isomorphisms]]; by the [[five lemma]], $H_{n}(X,A) \to H_{n}(X,U)$ > is an [[isomorphism]]. > > Point-set topology shows $\iota |_{A}: A / A \to U/A$ is also a [[homotopy equivalent|homotopy equivalence]]. A rehash above the argument says $H_{n}(X / A, A /A) \to H_{n}(X / A, U / A)$ > is an [[isomorphism]]. Then: > > ![[Pasted image 20250407222609.png]] > > where the pink vertical map is an [[isomorphism]] because the map of pairs $(X-A, U-A) \to (X/A - A / A, U / A- A / A)$ is a [[homeomorphism]]. ---- #### [^1]: Notation: Here, $X / A$ denotes the [[quotient space|quotient]] of $X$ the [[equivalence relation]] $x_{1} \sim x_{2} \iff x_{1},x_{2} \in A$, $H_{n}(X,A)$ denotes the [[singular homology|homology]] of $X$ [[relative singular homology|relative]] to $A$, and $\widetilde{H}_{n}(X / A)$ is the [[reduced homology]] of $X / A$. Of course, $A/A$ is just a point. ----- #### 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 > ```