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Let $M$ be a 3-manifold. Every knotted (embedded) surface in $M \times \R$can be moved via an ambient isotopy in such a way that its projection into $M$is a generic surface. A surface is generic if every point on it is either aregular, double or triple value - the transversal intersection of 1, 2 or 3embedded surface sheets, or a "branch value" that look like Whitney's umbrella.We elaborate on this in Definition 3.1.1. The double values form arcs, andalong each arc two long strips of surface intersect. In a knotted surface, theadditional $\R$ coordinate distinguishes between the two strips. One of themmust be "higher" than the other. We elaborate on this in Definition 3.1.3.  The lifting problem is the problem of determining if a \gls{genericsurface}in $M$ can occur as the $M$-projection of a knotted surface in 4-space in $M\times \R$. The main purpose of this thesis is to study the computationalaspects of the lifting problem. We will prove that the problem is NP-complete,and devise an efficient algorithm that determines if a generic surface isliftable.

The Lifting Problem is NP Complete