2 problems

Let $X$ be a 4-manifold. The involution on $\operatorname{Wh}_1(\pi_1 X;\mathbb{Z}_2\times\pi_2 X)$ is induced by the involution on $(\mathbb{Z}_2\times\pi_2 X)[\pi_1 X]$, and let…

Let $X$ be a 4-manifold with $k_1 X=0$. The subgroups $\Theta(\mathcal{J}(X))$ and $\Theta(\mathcal{J}(X)\cap\ker\Sigma)$ are defined in the target of the pseudo-isotopy invariant…