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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

$Wh(G\times\mathbb{Z})=Wh(G)\oplus\tilde{K}_0(\mathbb{Z}G)\oplus NK_1(\mathbb{Z}G)^2$.

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.

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.

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

$Wh(G\times\mathbb{Z})=Wh(G)\oplus\tilde{K}_0(\mathbb{Z}G)\oplus NK_1(\mathbb{Z}G)^2$.

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.

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

$Wh(G\times\mathbb{Z})=Wh(G)\oplus\tilde{K}_0(\mathbb{Z}G)\oplus NK_1(\mathbb{Z}G)^2$.

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.

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 of 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

$Wh(G\times\mathbb{Z})=Wh(G)\oplus\tilde{K}_0(\mathbb{Z}G)\oplus NK_1(\mathbb{Z}G)^2$.

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.

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

$Wh(G\times\mathbb{Z})=Wh(G)\oplus\tilde{K}_0(\mathbb{Z}G)\oplus NK_1(\mathbb{Z}G)^2$.

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.