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.