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> [!definition] Definition. ([[cross product on cohomology]])
>
> Let $X,Y$ be [[topological space|topological spaces]]. Let $\pi_{X}:X \times Y \to X$ and $\pi_{Y}:X \times Y \to Y$ be the natural projections, [[singular (co)chain map and homomorphism induced by a continuous map|inducing]] maps $\pi_{X}^{*}:H^{k}(X ; R) \to H^{k}(X \times Y;R)$ and $\pi _Y^{*}:H^{\ell}(Y;R) \to H^{\ell}(X \times Y;R)$ on [[singular cohomology]] respectively.
>
> Then we have a **cross product** $\begin{align}
> \times: H^{k}(X; R) \otimes_{R} H^{\ell}(Y; R) &\to H^{k+\ell}(X \times Y; R)
> \end{align}$
> defined on [[tensor product of modules|pure tensors]] as $\alpha \otimes \beta \xmapsto{\times} ( \pi_{X}^{*} \alpha ) \smile (\pi_{Y}^{*} \beta)$
> and [[bilinear map|bilinearly]] extending. We denote the image of a [[tensor product of modules|pure tensor]] $\alpha \otimes \beta$ under $\times$ as $\alpha \times \beta$.
>
> Note that the diagonal map $\Delta:X \to X \times X$ given by $\Delta(x):=(x,x)$ satisfies $\Delta^{*}(\alpha \times \beta)=\alpha \smile \beta$
> for all $\alpha, \beta \in H^{*}(X; R)$.[^1] In other words, the [[cup product]] arises as the composition $- \smile - : H^{k}(X ; R) \otimes H^{\ell}(X ; R) \xrightarrow{\times} H^{k+\ell}(X \times X;R) \xrightarrow{\Delta_{X}}H^{k+\ell}(X ; R).$
> So one product determines the other.
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####
[^1]: Here is the computation. We remove the "$;R
quot; for readability. $\Delta^{*}: H^{k}(X \times X) \to H^{k}(X)$ is given $\Delta^{*}([\varphi])=[\varphi \circ \Delta_{k}]$, where $\Delta_{k}:H_{k}(X) \to H_{k}(X \times X)$ is given by $\sigma \mapsto (\sigma, \sigma)$. The maps $\pi_{(1)}^{*},\pi_{(2)}^{*}:H^{k}(X) \to H^{k}(X \times X)$ induced respectively by the projections $\pi _{(1)}, \pi _{(2)}$ are given by $\pi_{(1)}^{*}([\varphi])=[\varphi \circ \pi^{(1)}_{*}]$, where $\pi_{*}^{(1)}:H_{k}(X \times X) \to H_{k}(X)$ is given by $(\sigma_{(1)}, \sigma_{(2)})\mapsto \sigma_{(1)}$. Analogous behavior for $\pi_{(2)}$. With these in hand, we can compute $\begin{align}
\big(\Delta^{*}(\alpha \times \beta) \big) \ (\sigma) &= \big((\alpha \times \beta) \circ \Delta_{k} \big) \ (\sigma) \\
&= \big((\pi_{(1)}^{*} \alpha \smile \pi_{(2)}^{*} \beta ) \circ \Delta_{k} \big) \ (\sigma) \\
&=\big(\big( ( \alpha \circ \pi_{*}^{(1)}) \smile (\beta \circ \pi_{*}^{(2)}) \big) \circ \Delta_{k}\big) \ (\sigma) \\
&= \big( ( \alpha \circ \pi_{*}^{(1)}) \smile (\beta \circ \pi_{*}^{(2)}) \big) \ (\sigma, \sigma) \\
& = ( \alpha \circ \pi_{*}^{(1)}) \big( (\sigma, \sigma) |_{e_{0},\dots,e_{k}} \big) \cdot (\beta \circ \pi_{*}^{(2)})\big( (\sigma, \sigma) |_{e_{k},\dots,e_{k+k}} \big) \\
&= \alpha ( \sigma |_{e_{0},\dots,e_{k}}) \cdot \beta(\sigma |_{e_{k},\dots, e_{k+k}}) \\
&= (\alpha \smile \beta)(\sigma).
\end{align}$
(Maybe there is a cleaner way.)
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#### 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
> ```