Hexagon

math.GT — Geometric Topology

Marked Heegaard Boundary Modules: Filling, Joint Completion, and Integral Lattices

Contributed by Hao-Yu Sun

A marked handlebody state must retain its filling relations and coefficient control before complementary capping or root specialization. We construct boundary modules and regulated integral lattices that keep the disk and cyclotomic residues of one finite-stage representative together. Standard handle-attachment and recoupling results supply the geometric inputs; the comparison constructs endpoint lattices, separating transport, raw and saturated quotient towers, and exact cap domains. These objects supply the companion rootwise theory, while the two completed topologies remain distinct and root specialization remains restricted to its declared domain.

Primarily AI-generated textHuman understanding: some partsHeegaard splittingcapintegral latticejoint completionskein module

Oct 18, 2026PDF ↗View work →

hexagon:2610.00340v1

math.GT — Geometric Topology

Integral root fibers and closed factorizations: Pairing maps, Rees covariance, caps, and sewing

Contributed by Hao-Yu Sun

We construct rootwise and formal Rees boundary data from the marked modules and lattices of the companion boundary-module construction, keeping their distinct coefficient and completion domains. Using a Newton interpolation basis, we determine the Smith invariants and the integral image of the map induced by the Hopf pairing. Factorial/Rees covariance and compatible finite-stage representatives produce a strict rootwise/Rees module with its theta endpoint. At an admissible odd prime, no continuous homomorphism extends algebraic root specialization to the entire completed Rees source; further obstructions exclude unrestricted Newton padding and a state-natural lift of a contracted scalar identity. Established perturbative, Habiro, and stated-skein/Morita constructions give closed comparisons after the declared cap or sewing map, with their boundary state objects kept separate.

Primarily AI-generated textHuman understanding: some partsHabiro capHopf pairingMorita sewingRees completionroot of unity

Oct 18, 2026PDF ↗View work →

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Marked Heegaard Boundary Modules: Filling, Joint Completion, and Integral Lattices

Integral root fibers and closed factorizations: Pairing maps, Rees covariance, caps, and sewing

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