Let $X,Y,Z$ be smooth connected manifolds and $f \colon X \times Y \rightarrow Z$ a smooth map. Suppose that we have $H_{*}(X \times Y; \mathbb{Z})$ is isomorphic to $\bigoplus_{p+q=*}(H_{p}(X; \mathbb{Z}) \otimes H_{q}(Y; \mathbb{Z}))$ by the Künneth theorem, which can be assumed if $H_{i}(Y; \mathbb{Z})$ has no torsion for any $i$.The map $f$ induces a map on homologies, $f_* \colon H_{*}(X \times Y; \mathbb{Z}) \rightarrow H_{*}(Z; \mathbb{Z})$.

My question is whether one can interpret the image of $f_{*}$ in terms of $\bigoplus_{p+q=*}(H_{p}(X; \mathbb{Z}) \otimes H_{q}(Y; \mathbb{Z}))$. More precisely, for $a \otimes b \in H_{p}(X; \mathbb{Z}) \otimes H_{q}(Y; \mathbb{Z})$, we have $a \times b \in H_{p+q}(X \times Y; \mathbb{Z})$, where $\times$ denotes the cross product. Then is it possible to describe $f_{*}(a \times b)$ in terms of $f_{*}i_{*}(a) \in H_{p}(X \times Y; \mathbb{Z})$ and $f_{*}j_{*}(b) \in H_{q}(X \times Y; \mathbb{Z})$. Here $i,j$ are canonical inclusions of $X, Y$ into $X \times Y$, respectively. In particular, I would be very happy if I can say $f(a \times b)=0$ for any $a$ and $b$ with $f_{*}j_{*}(b)=0$.

Please let me know anything you know about this question.Thank you in advance.