Carpentry, geometry, and category theory
(Notes for the talk at CMI in 2022).
Carpentry
The carpenter’s rule problem
Definition of carpenter’s rule
A chain \(C\) with edges of specified length.
The carpenter’s rule problem
Can a planar embedding of \(C\) be deformed to be linear?
Generalisations
- Consider embeddings in \(\mathbf{R}^{n}\) for \(n = 2, 3, 4, \dots\).
- Consider metric graphs \(G\) other than a chain.
- Results
- Chains can be straightened in \(\mathbf{R^n}\) for \(n \geq 4\) (Cocan, Roxana and O'Rourke, Joseph, 2001).
- There exist locked graphs in \(\mathbf{R}^2\).
- There exist locked chains in \(\mathbf{R}^3\).
- Chains can be straightened in \(\mathbf{R}^2\) (Connelly, Robert and Demaine, Erik D. and Rote, G{\"u}nter, 2003, Streinu, Ileana, 2000)
Pointed pseudo-triangulations
Definition
Given a collection of \(n\) points in general position, a ppt is a maximal collection of edges such that
- No two edges cross.
- At every vertex, all incident edges lie in a half-plane.
Properties
- Each region is a pseudo-triangle.
- There are \(2n-3\) edges.
- Each edge is uniquely flippable. The flip graph is connected.
Chains and motions
- A ppt is minimally rigid.
- A ppt minus an external edge has one direction of freedom. Furthermore, along one of the two rays, the motion is expansive.
- A polygonal chain lies on a ppt minus an external edge.
Expansive motions solve the Carpenter’s rule problem
Start with a chain \(C\). Complete it to a ppt minus an external edge. Move in the unique expansive direction. Expansivity guarrantees that the configuration remains non-crossing. Eventually the chain unfolds.
Geometry
The configuration space is a \(K(\pi, 1)\)
(Fox, R. and Neuwirth, L., 1962, Brieskorn, Egbert, 1973, Deligne, Pierre, 1972)
New proof that exhibits an explicit contraction of the universal cover.
Consider \(\widetilde {\operatorname{Conf}_n}\), the space of \(n\) ordered points along with an \(n\)-chain up to isotopy. Then \(\widetilde {\operatorname{Conf}_n} \to \operatorname{Conf}_{n}\) is a covering map. Given a point of \(\widetilde {\operatorname{Conf}_{n}}\), the support of the chain lies on a ppt*. Move in an expansive direction. Eventually, the chain will unfold, giving a deformation retraction of \(\widetilde {\operatorname{Conf}_{n}}\) to a subspace where the chain is linear. It is easy to see directly that the latter is contractible.
Category theory
The stability manifold
The space \(\operatorname{Conf}_n\) arises in a seemingly unrelated setting. To every triangulated category \(C\), we can associate a manifold \(\operatorname{Stab} C\). This manifold is a parameter space for Bridgeland stability conditions on \(C\) (Bridgeland, Tom, 2007). For every \(n\), there is a particular category \(C_n\) of interest in algebraic geometry, symplectic geometry, and representation theory such that we have an isomorphism \[ \operatorname{Stab} C_n = \widetilde{\operatorname{Conf}}_n.\] We deduce that \(\operatorname{Stab} C_n\) is contractible.
Categorical representations
Why do we care about categories like \(C_n\) and their stability manifolds? There is a class of groups \(G\) called Artin-Tits Braid groups that are quite poorly understood. One of the reasons that they are poorly understood is that we do not know any good linear representations of \(G\). In fact, it is unknown whether \(G\) has any faithful finite dimensional representations!
Although it is hard to construct a vector space \(V\) on which \(G\) acts, it is much more natural to construct categories \(C\) on which \(G\) acts. Contractibility of \(\operatorname{Stab} C\) implies that this action is faithful. As a result, we can deduce many of the consequences of having a faithful linear representation.