math.GT — Geometric Topology

Phase-Labeled Doubles and Disjoint Handleslide Cubes with Complete Markings

Contributed by Hao-Yu Sun

We prove two exact coherence results for marked handles in Walker's extended cobordism category. At fixed disjoint sites, a matching pair of forward and reverse phases contributes one local $S^1\times S^2$ summand, whereas a mismatch contributes an identity cylinder. Thus the resulting double depends only on the total Walker weight and the number of phase matches, not on the orders of attachment. For positive handleslides supported in pairwise-disjoint protected sectors, we construct the complete marked Boolean cube and prove that every two-face commutes with weight zero; consequently, every finite route has the same composite and the endpoint identities telescope. An optional odd-prime $\mathrm{SO}(3)$-TQFT corollary evaluates the fixed-site double as $\kappa_p^N(D_p^{\mathrm{GM}})^b$ times the identity. Overlapping slides and their pentagon are treated by a separate companion theorem.