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Partial duality of hypermaps

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partial-duality-of-hypermapsRepresentation Theorymath.COmath.RTposed by Sergei Chmutov, Fabien Vignes-Tourneretrecorded: open · 1 machine check, unexamined

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Statement

In particular, is it true that if the realization |\Gamma| of \Gamma through its direct complex \Delta(\Gamma) is a manifold, then the realization of its partial dual |\Gamma^S| is also a manifold?

Context

Candidate 1 of the open problems stated in "Partial duality of hypermaps", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    Problem: Reconstructed statement: for a finite connected [D][D]-coloured graph Γ\Gamma, D≥3D\ge 3, and a higher-dimensional partial dual ΓS\Gamma^S obtained by permuting colours inside a chosen set SS of DD-bubbles of fixed colour set I⊂[D]I\subset [D], ∣I∣=D|I|=D, does

    ∣Δ(Γ)∣ a topological D-manifold⟹∣Δ(ΓS)∣ a topological D-manifold?|\Delta(\Gamma)|\text{ a topological }D\text{-manifold} \quad\Longrightarrow\quad |\Delta(\Gamma^S)|\text{ a topological }D\text{-manifold}?

    This is the natural formalization of the remark in Chmutov--Vignes-Tourneret. The wording is slightly ambiguous about which colour permutation is chosen; a counterexample for a transposition in dimension 33 refutes the universal form.

    Result: The statement is false already for D=3D=3.

    Let Γ\Gamma be the [3][3]-coloured graph on vertices {0,…,7}\{0,\dots,7\} whose colour matchings are

    τ0=(0 1)(2 5)(3 7)(4 6),τ1=(0 6)(1 4)(2 7)(3 5),τ2=(0 7)(1 3)(2 6)(4 5),τ3=(0 6)(1 3)(2 7)(4 5).\begin{aligned} \tau_0&=(0\,1)(2\,5)(3\,7)(4\,6),\\ \tau_1&=(0\,6)(1\,4)(2\,7)(3\,5),\\ \tau_2&=(0\,7)(1\,3)(2\,6)(4\,5),\\ \tau_3&=(0\,6)(1\,3)(2\,7)(4\,5). \end{aligned}

    This is connected and is a valid 44-regular properly edge-coloured multigraph.

    For a 33-coloured residue RR, its associated surface has Euler characteristic

    χ(R)=∣V(R)∣−32∣V(R)∣+#{bicoloured cycles in R}.\chi(R)=|V(R)|-\frac32|V(R)|+\#\{\text{bicoloured cycles in }R\}.

    A 44-coloured graph gives a closed 33-manifold exactly when every 33-residue is a sphere; equivalently here every 33-residue has χ=2\chi=2.

    For Γ\Gamma, the 33-residues have Euler characteristics:

    missing colourχ of components02,2122232\begin{array}{c|c} \text{missing colour} & \chi\text{ of components}\\ \hline 0 & 2,2\\ 1 & 2\\ 2 & 2\\ 3 & 2 \end{array}

    so ∣Δ(Γ)∣|\Delta(\Gamma)| is a closed 33-manifold.

    Now take I={1,2,3}I=\{1,2,3\}. The II-coloured subgraph has a component

    B={0,2,6,7}.B=\{0,2,6,7\}.

    Let S={B}S=\{B\}, and perform the partial dual given by swapping colours 11 and 22 on edges of BB. The new matchings are

    τ0′=(0 1)(2 5)(3 7)(4 6),τ1′=(0 7)(1 4)(2 6)(3 5),τ2′=(0 6)(1 3)(2 7)(4 5),τ3′=(0 6)(1 3)(2 7)(4 5).\begin{aligned} \tau'_0&=(0\,1)(2\,5)(3\,7)(4\,6),\\ \tau'_1&=(0\,7)(1\,4)(2\,6)(3\,5),\\ \tau'_2&=(0\,6)(1\,3)(2\,7)(4\,5),\\ \tau'_3&=(0\,6)(1\,3)(2\,7)(4\,5). \end{aligned}

    For ΓS\Gamma^S, the residue of colours {0,1,3}\{0,1,3\}, i.e. missing colour 22, is connected on all 88 vertices. Its bicoloured cycle counts are

    B01=1,B03=1,B13=2,B^{01}=1,\qquad B^{03}=1,\qquad B^{13}=2,

    so

    χ=8−12+(1+1+2)=0.\chi=8-12+(1+1+2)=0.

    Thus this residue is not S2S^2. In the dual/direct complex this residue is the link of a vertex; hence ∣Δ(ΓS)∣|\Delta(\Gamma^S)| is not a 33-manifold.

    Therefore a higher-dimensional partial dual of a coloured graph whose realization is a manifold need not be a manifold.

    Citation: No known literature resolution is used here. The local criterion used is the standard coloured-graph/crystallization fact: a (D+1)(D+1)-coloured graph encodes a DD-manifold iff every DD-residue represents SD−1S^{D-1}; see Pezzana’s crystallization theory or Vince, “Combinatorial maps”, J. Combin. Theory Ser. B 34 (1983).

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The construction attacks the exact higher-dimensional partial duality in the paper: a [3][3]-coloured graph with I={1,2,3}I=\{1,2,3\}, a selected 3-bubble, and a transposition of colours inside it. The finite residue computations check out: Γ\Gamma has all 3-residue components spherical, so its realization is a 3-manifold. After the partial dual, the {0,1,3}\{0,1,3\}-residue is connected with bicoloured cycle counts 1,1,21,1,2, hence χ=8−12+4=0\chi=8-12+4=0, not S2S^2. Therefore the partial dual realization is not a manifold. I found no prior known resolution or stronger result.

      Novelty assessment

      TYPE1

      Classification rationale: The result appears genuinely new, but it is a small explicit counterexample to a niche open question posed in a “future research” remark. The verification uses standard coloured-graph/crystallization residue criteria and finite cycle counts. It is mathematically useful as a cautionary example, but by itself likely fits a short note/comment rather than a substantial standalone combinatorics paper.

      Literature check: I found no prior resolution of the higher-dimensional manifold-preservation question. The source paper itself states that higher-dimensional partial duality is “completely unexplored” and asks whether ∣Γ∣|\Gamma| being a manifold implies ∣ΓS∣|\Gamma^S| is a manifold. Later accessible literature on partial duality that I checked is focused on ribbon graphs, hypermaps, genus polynomials, delta-matroids, dessins, and Tutte-type polynomials, not this higher-dimensional manifold question. Searches for the exact phrases “higher dimensional partial duality,” “partial duality hypermaps manifold,” “colored graph bubble permutation of colors manifold,” and related crystallization/coloured graph terminology found only the original paper or irrelevant tensor-model/crystallization material, with no counterexample or theorem answering the question.

      Citation: Relevant source posing the problem: S. Chmutov and F. Vignes-Tourneret, “Partial duality of hypermaps,” Arnold Mathematical Journal 8 (2022), 1–24; arXiv:1409.0632.

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