math.GT — Geometric Topology

Marked Positive Handleslide Pentagons in Compression Bodies: Relative Coherence

Contributed by Hao-Yu Sun

Let $W:X\to Y_{\mathrm{geom}}$ be a completed oriented compression body with $N\geq 3$ ordered product $1$-handles and a supplied protected three-handle sector. We realize one positive handleslide pentagon by five diffeomorphisms with compatible complete endpoint markings. An isotopy fixed on the entire protected boundary collar identifies the two-edge and three-edge route composites, so the comparison takes place at the level of marked maps rather than homology matrices. The construction fixes the marked disk systems at all five vertices before choosing any edge map, so incident edges use the same endpoint data. The cut-ball maps intertwine all fifteen disk pairings on common full-normal intervals and therefore descend smoothly through the paired faces. Extending these quotient maps by the identity gives global representatives. The same maps satisfy the finite compatibility conditions, including at $N=3$, when the unused-handle domains are empty but remain part of the typed construction. The resulting boundary maps preserve the embedded meridian and product-framed dual-longitude rows, so their homological actions have blocks $M_e$ and $M_e^{-T}$ with no mixed shear. They also preserve the same rational disk-kernel Lagrangian. Walker's forward multiplication law lifts the relative isotopy to a literal weight-zero equality of mapping cylinders, including the seam and route weights.