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.



Künneth formula and induced map in homologies
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 $\operatorname{dim}(X),\operatorname{dim}(Y)>0$.


