A decomposition of a manifold, obtained by cutting along a two-sided codimension-one connected incompressible submanifold, induces a decomposition of the fundamental group as an injective amalgamated product or as an HNN extension.
The converse of this fact, called the realization problem, is discussed. In every dimension except 4, the realization problem is more-or-less always solvable. However, in dimension 4 there are splittings of fundamental groups which are not induced by splittings of 4-manifolds. Often by allowing for stabilization of the 4-manifolds, it is possible to realize splittings of their fundamental groups.
The obstruction to stably realizing a splitting is an equation in a 3-dimensional oriented or spin bordism group, depending on the second Stiefel--Whitney class of the universal cover. Simple examples of realizable and non-realizable splittings are described. The proofs of existence and uniqueness are outlined.