---- > [!definition] Definition. ([[relative cap product]]) > Let $R$ be a ([[commutative ring|say, commutative]]) [[ring]]. Let $X$ be a [[topological space]] and $A$ a [[subspace topology|subspace]]. > > As with the [[cup product]], there are [[relative cup product|relative versions]] of the [[cap product]] for $(X,A)$: > $\begin{align}- \frown - : H_{k}(X, A;R) \otimes H^{\ell} (X ;R) \to H_{k-\ell}(X, A ;R) \end{align}$ and $- \frown - : H_{k}(X, A; R) \otimes H^{\ell}(X, A; R) \to H_{k-\ell}(X; R).$ > Note that the cap product of two [[relative singular homology|relative classes]] is an [[singular homology|absolute class]]. ---- #### ---- #### 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 > ```