---- > [!definition] Definition. ([[relative cup product]]) > ~ ^definition Let $X$ be a [[topological space]], $A$ a [[subspace topology|subspace]] (so that $(X,A)$ is a [[topological pair|pair]]), and let $R$ be a [[ring]]. We can form a **relative [[cup product]]** on ([[relative singular homology|relative]]) [[singular cohomology|cohomology]] as $\begin{align}- \smile - : H^{k}(X, A;R) \otimes H^{\ell}(X ; R)& \to H^{k+\ell}(X, A; R) \\ \big[[\varphi] \big] \otimes [\psi] & \mapsto \big[[\varphi \smile \psi]\big], \end{align}$ which works because $(\varphi \smile \psi)(\sigma)=\varphi(\sigma |_{[e_{0},\dots,e_{k}]}) \psi(\sigma |_{[e_{k},\dots,e_{k+\ell}]})$ If $\varphi \in C^{k}(X, A; R)$, then $\varphi$ is a map $\varphi: C_{k}(X, A)=\frac{C_{k}(X)}{C_{k}(A)} \to R.$ [[characterization of quotienting a module|In other words]], it is a map $C_{k}(X) \to A$ that vanishes on $A$. Then if $\sigma \in C_{k+\ell}(A)$ and $\psi \in C^{\ell}(X; R)$, we have $(\varphi \smile \psi)(\sigma)=\varphi(\sigma |_{[e_{0},\dots,e_{k}]}) \psi(\sigma |_{[e_{k},\dots,e_{k+\ell}]}).$ As $\sigma |_{[e_{0},\dots,e_{k}]} \in C_{k}(A)$, $\varphi$ kills it, and this vanishes. ---- #### (seems very similar to [[cross product on cohomology]]) [[(co)homology with coefficients|homology with coefficients]] ---- #### 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 > ```