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Cobordism of domes over curves

Robert Miranda

TL;DR

This work studies the cobordism of domes over integral curves, showing that while no universal orientable cobordism exists to a single unit rhombus, every integral curve is orientably cobordant to a finite union of unit rhombi. The authors develop a rhombus-equivalence framework and leverage planar reductions via a Steinitz-type packing to obtain an explicit bound: for a curve with length $n$ (i.e., $|\\gamma|=n$), at most $k = n^2 + 2n - 12$ rhombi are needed to realize an orientable cobordism. They generalize the moduli-space setup to genus-$n$ graph surfaces with multiple boundary components, introduce boundary maps linking polyhedra to boundary polygons, and analyze the resulting symplectic structure to study isotropy and cobordisms. A key result is that not every integral curve is orientably cobordant to a unit rhombus, demonstrated via a measure-theoretic obstruction and an explicit length-$5$ counterexample, highlighting limitations of cobordism strategies based solely on single rhombi. The work provides a robust framework for multi-boundary cobordisms and raises questions about potential linear bounds and nonorientable cases, with implications for the geometry of domes over curves.

Abstract

An integral curve is a closed piecewise linear curve comprised of unit intervals. A dome is a polyhedral surface whose faces are equilateral triangles and whose boundary is an integral curve. Glazyrin and Pak showed that not every integral curve can be domed by analyzing the case of unit rhombi, and conjectured that every integral curve is cobordant to a unit rhombus. We show that this is false for oriented domes, but that every integral curve is orientably cobordant to the union of finitely many unit rhombi.

Cobordism of domes over curves

TL;DR

This work studies the cobordism of domes over integral curves, showing that while no universal orientable cobordism exists to a single unit rhombus, every integral curve is orientably cobordant to a finite union of unit rhombi. The authors develop a rhombus-equivalence framework and leverage planar reductions via a Steinitz-type packing to obtain an explicit bound: for a curve with length (i.e., ), at most rhombi are needed to realize an orientable cobordism. They generalize the moduli-space setup to genus- graph surfaces with multiple boundary components, introduce boundary maps linking polyhedra to boundary polygons, and analyze the resulting symplectic structure to study isotropy and cobordisms. A key result is that not every integral curve is orientably cobordant to a unit rhombus, demonstrated via a measure-theoretic obstruction and an explicit length- counterexample, highlighting limitations of cobordism strategies based solely on single rhombi. The work provides a robust framework for multi-boundary cobordisms and raises questions about potential linear bounds and nonorientable cases, with implications for the geometry of domes over curves.

Abstract

An integral curve is a closed piecewise linear curve comprised of unit intervals. A dome is a polyhedral surface whose faces are equilateral triangles and whose boundary is an integral curve. Glazyrin and Pak showed that not every integral curve can be domed by analyzing the case of unit rhombi, and conjectured that every integral curve is cobordant to a unit rhombus. We show that this is false for oriented domes, but that every integral curve is orientably cobordant to the union of finitely many unit rhombi.
Paper Structure (5 sections, 13 theorems, 6 equations, 3 figures)

This paper contains 5 sections, 13 theorems, 6 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Every integral curve is cobordant to finitely many unit rhombi
  3. Spaces of Polygons and Polyhedra
  4. Not Every Integral Curve is Cobordant to a Unit Rhombus
  5. Final remarks

Key Result

Theorem 1.2

There exists an integral curve $\gamma$ of length $5$ which is not orientably cobordant to any unit rhombus $\rho$.

Figures (3)

  • Figure 1: An integral path $\eta$ above a plane $H$ with heights $h_i$.
  • Figure 2: Adding a rhombus $[v_{i-1} v_i v_{i+1} v_i']$.
  • Figure 3: The $n = 5$ case and the general case.

Theorems & Definitions (25)

  • Conjecture 1.1: Gla
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 1.3
  • proof
  • Lemma 3.1
  • ...and 15 more