Haken normal essential-disk existence theorem

Given: The compact triangulated PL three-manifold with nonempty boundary, no embedded projective plane, and the stated essential properly embedded disk.

An essential proper PL disk has an essential normal replacement in the same triangulation. Essential PL disks admit normal disk replacements

Among essential normal disks, a least-weight disk with lexicographically least vector is fundamental in the original seven-coordinate system. This supplier does not require irreducibility, sphere separation, orientability or projective-plane exclusion. Least-weight lexicographic essential normal disks are fundamental

External input (PL Loop Theorem). Let $N$ be a triangulated PL $3$-manifold, not necessarily orientable. A continuous map $(D^2,\mathrm{\partial }{D}^2)\to (N,\mathrm{\partial }N)$ whose boundary is not null-homotopic in $\mathrm{\partial }N$ implies existence of a properly embedded PL disk with essential boundary. We use the PL form in the supplied triangulation, not approximation of the original possibly wild embedding. This is the established Loop Theorem, as stated without orientability in Hatcher, Notes on Basic 3-Manifold Topology, Theorem3.1, §3.1, printed pp.56–59. Its tower starts in the PL category; the proof then presents the disk surgeries smoothly. The theorem is used here as an explicit external input, not as a locally reconstructed PL-category proof.