The *Kneser Conjecture* says that if a closed connected 3-manifold M has π_{1}(M)=G_{1} * G_{2},
then there exist 3-manifolds M_{1} and M_{2} such that π_{1}(M_{i})=G_{i} and M=M_{1}#M_{2}. Hillman has generalized this to 4-manifolds at the expense of introducing stabilizing S^{2} × S^{2} factors to M.
More precisely, if a closed connected 4-manifold M has π_{1}(M)=G_{1} * G_{2}, then there exist 4-manifolds M_{1} and M_{2} such that π_{1}(M_{i})=G_{i} and M#n(S^{2} × S^{2})=M_{
1}#M_{2} for some nonnegative integer n.

This talk will review these results, and then consider a generalization of these theorems to 3- and 4-manifolds with fundamental groups which are amalgamated free products. The
3-dimensional case for amalgamated products has been done by Feustel. The talk will end with a discussion of progress made in the case that the amalgamation is over a cyclic group of prime order.