Obstructions to splitting manifolds with infinite fundamental group

In this paper we calculate the obstruction groups to splitting along one-sided submanifolds when the fundamental group of the submanifold is isomorphic to $\mathbb Z$ or $\mathbb Z\oplus\mathbb Z/2$. We also consider the case where the obstruction group is not a Browder–Livesey group. We construct a new Levine braid that connects the Wall groups to the obstruction group for splitting. We solve the problem of the mutual disposition of images of several natural maps in Wall groups for finite 2-groups with exceptional orientation character.