Künneth formula and induced map in homologies

Question

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.

Answer 1

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 $\operatorname{dim}(X),\operatorname{dim}(Y)>0$.