I am wondering is there a formula for the whitehead group of product of groups. In other words, if we know the whitehead group of two groups, are we able to calculate the whitehead group of their products. If not, is there a example such that the whitehead group of the product is nontrivial while the whitehead group of both groups vanishes.



whitehead group of product of groups
This is a partial answer. Let $G$ be a group. Then by identifying $\mathbb{Z}G[t,t^{-1}]$ with $\mathbb{Z}[G\times\mathbb{Z}]$ and using Bass-Heller-Swan decomposition you will get
$K\_1(\mathbb{Z}[G\times\mathbb{Z}])=K\_1(\mathbb{Z}G)\oplus\tilde{K}\_0(\mathbb{Z}G)\oplus NK\_1(\mathbb{Z}G)^2$. Therefore


$Wh(G\times\mathbb{Z})=Wh(G)\oplus\tilde{K}\_0(\mathbb{Z}G)\oplus NK\_1(\mathbb{Z}G)^2$.


As for the example you are looking for, take $G=\mathbb{Z}/4\mathbb{Z}$ and $H=\mathbb{Z}^{3}$. The Whitehead group of these two groups vanishes but the Whitehead group of its product does not. The reason is that $G\times H$ is the fundamental group of $L\times T\ ^3$ where $L$ is a 3-dimensional lens space and $T\ ^3$ is a 3-dimensional torus. Farrell and Hsiang (F. T. Farrell and W. C. Hsiang, Bull. Amer. Math. Soc. Volume 73, Number 5 (1967), 741-744) proved that there is a non trivial h-cobordism with that space as one of its boundary components.


