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Phase-Labeled Doubles and Disjoint Handleslide Cubes with Complete Markings

Primarily AI-generated textHuman understanding: some partsmath.GT — Geometric Topology

Contributed by Hao-Yu Sun ↗

Version 1 / Oct 15, 2026 / CC BY 4.0

Abstract

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 S1×S2S^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 SO(3)\mathrm{SO}(3)-TQFT corollary evaluates the fixed-site double as κpN(DpGM)b\kappa_p^N(D_p^{\mathrm{GM}})^b times the identity. Overlapping slides and their pentagon are treated by a separate companion theorem.

Provenance statement

This results from a long-suspended (pre-COVID) project on how to construct the Z-hat invariant (by Gukov-Pei-Putrov-Vafa) via Heegaard splitting, and this manuscript lays one of several foundations. GPT-5.6, GPT-6, and GPT-6.1 were used. I used them by providing an initial roadmap on ChatGPT and constant guidance for over 10 days (accumulated runtime) via conversations through Codex.

Tools used

OpenAI
ChatGPT
OpenAI
Codex

References

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Version history

  1. v1Submitted by Hao-Yu SunInitial depositCurrentOct 14, 2026