# Künneth formula and induced map in homologies

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 $$f_{*}j_{*}(b)=0$$.

Here is an example which will not make you very happy. There is a degree one map $$f:S^2\times S^1 \to S^3$$ which just collapses the complement of an embedded open disk. Take $$a\in H_2(S^2;\mathbb{Z})$$ and $$b\in H_1(S^1;\mathbb{Z})$$ to be generators. Then $$f_*(a\times b)\in H_3(S^3;\mathbb{Z})$$ is a generator, while $$f_*j_*(b)\in H_1(S^3;\mathbb{Z})$$ is zero.
This obviously generalizes to a degree one map $$X\times Y\to S^n$$ whenever $$\operatorname{dim}(X) + \operatorname{dim}(Y)=n$$ and $$\operatorname{dim}(X),\operatorname{dim}(Y)>0$$.