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Homological methods in rigidity theory using graphs of groups

Joannes Vermant

Abstract

In recent work, Stokes and Vermant considered graph-of-groups realisations of hypergraphs as a new description of rigidity-theoretic problems. In this paper, we show that the infinitesimal aspects of graph-of-groups realisations can be analysed using cellular sheaves and their cohomology. Using these tools, we give an algebraic condition for Henneberg moves to preserve independence, and we prove that the infinitesimal rigidity and flexibility of certain graph-of-groups realisations are generic properties. We use these results to show that whenever a rigidity-theoretic problem is defined in a real algebraic group $G$ using a $1$-dimensional connected subgroup $H$ with $N_{G}(H)/H$ finite, then the so-called Maxwell-count leads to a necessary and sufficient condition for minimal rigidity, generalising various known results in the literature.

Homological methods in rigidity theory using graphs of groups

Abstract

In recent work, Stokes and Vermant considered graph-of-groups realisations of hypergraphs as a new description of rigidity-theoretic problems. In this paper, we show that the infinitesimal aspects of graph-of-groups realisations can be analysed using cellular sheaves and their cohomology. Using these tools, we give an algebraic condition for Henneberg moves to preserve independence, and we prove that the infinitesimal rigidity and flexibility of certain graph-of-groups realisations are generic properties. We use these results to show that whenever a rigidity-theoretic problem is defined in a real algebraic group using a -dimensional connected subgroup with finite, then the so-called Maxwell-count leads to a necessary and sufficient condition for minimal rigidity, generalising various known results in the literature.
Paper Structure (19 sections, 30 theorems, 156 equations, 6 figures)

This paper contains 19 sections, 30 theorems, 156 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motion sheaves
  3. Background on cellular sheaves and their cohomology
  4. Restricting to a sub-graph
  5. Definitions and statement of main theorem.
  6. Basic observations
  7. Genericity for motion sheaves
  8. Associated sheaves on multigraphs
  9. Inductive constructions for associated sheaves
  10. Proof of \ref{['thm: main theorem']}
  11. Inductive constructions for $M_{s, n}(\Gamma)$.
  12. Applications to graph-of-groups realisations
  13. Graph-of-groups realisations of hypergraphs
  14. Motion sheaves from graphs of groups
  15. Genericity of infinitesimal rigidity for graph-of-groups realisations.
  16. ...and 4 more sections

Key Result

Lemma 2.3

The map $H^{1}(\Gamma, \mathcal{F}) \rightarrow H^{1}(\Gamma, \mathcal{F}\vert_{\Gamma'}^{(2)})$ induced by eq: les_restriction is surjective. In particular, if $H^{1}(\Gamma, \mathcal{F})= 0$, then $H^{1}(\Gamma, \mathcal{F}\vert_{\Gamma'}^{(2)})=0$.

Figures (6)

  • Figure 1: Two cochains defining the same element of $H^{1}(\Gamma, \underline{\mathbb{V}})$. They are the same by equation \ref{['eq: rule for h1']}
  • Figure 2: An illustration of the cochain for the first case for the $4$-dimensional $2$-extension. Edges not involved in the $2$-extension are not pictured. The edges $e_1$ and $e_2$ are replaced, and edges $f_1,\dots, f_6$ are added. The picture illustrates the construction of the cochain equal to $W_{f_{6}}$, where $f_6$ is the edge between $v_2$ and $v_*$.
  • Figure 3: An illustration of the cochain for the second case for the $4$-dimensional $2$-extension. The picture illustrates the construction of the cochain equal to $W_{f_1} + W_{f_2}$
  • Figure 4: An illustration of the cochain for the first case for the $4$-dimensional $2$-extension. Edges not involved in the $2$-extension are not pictured. The edges $e_1$ and $e_2$ are replaced, and edges $f_1,\dots, f_6$ are added. The picture illustrates the construction of the cochain equal to $W_{v_* \sim f_6}$, where $f_6$ is the edge between $v_2$ and $v_*$.
  • Figure 5: An illustration of the cochain for the second case for the $4$-dimensional $2$-extension. The picture illustrates the construction of the cochain equal to $W_{u_1, u_1 \sim f_1} + W_{u_1', u_1' \sim f_2}$
  • ...and 1 more figures

Theorems & Definitions (72)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • proof
  • Definition 2.8
  • Theorem 2.9
  • ...and 62 more