I am currently working on "The topology of discrete groups" by Baumslag, Dyer and Heller, in particular chapter 4, where the implication Group G is mitotic $\Rightarrow$ G is acyclic is proven.
We are in the following situation:
A,B groups, k a field (with trivial A- and B-action), $\varphi \colon A \to B$ group hom
$\lambda \colon A \to A \times A$; $a \mapsto (a,1)$
$(\varphi \times \varphi) \colon A \times A \to B \times B$ ; $(a,a) \mapsto (\varphi(a),\varphi(a))$
$\lambda_{*} = H_n(\lambda;k)$ and $(\varphi \times \varphi)_{*}= H_n(\varphi \times \varphi;k)$
The authors then just state that for $\alpha \in H_n(A \times A ; k)$ we have
$(\varphi \times \varphi)_{*} \circ \lambda_{*} (\alpha) = \varphi_{*}(\alpha) \otimes 1$
due to the Künneth formula
$H_n(B \times B; k) \cong \bigoplus_{k=0}^{n} H_k(B;k) \otimes H_{n-k}(B;k)$
I understand the application of the Künneth theorem but i fail to see the explicit description they give for the image of $(\varphi \times \varphi)_{*} \circ \lambda_{*}$.
I tried factoring like this: $A \to A \times 1 \to A \times A \to B \times B$ but i am not convinced at all.
Any help is greatly appreciated!