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This post is under construction. See the previous post in the series for context. This post serves to generalize the formalism of this post concerning linear weaves. See the bottom of the first post in the series on chain mail mathematics for a table of contents.

Let \(G\) be a dicsrete group acting on \(\RR^3\) through isometries. The purpose today is to define the notion of weaves which are preserved by \(G\) (i.e. weaves which are equivariant), and handwave some formal nonsense to upgrade non-equivariant invariants to invariant ones. We’ll work this on the level of equivariance under \(\GG := \Isom(\RR^3)\), then recover the discrete case by looking at fixed points.

For this definition to recover the linear case, we’ll need to be able to fix a particular translaitonal symmetry. We argue for this in the following section.

A word on the dilation trick for linear weaves

When proving that linking detects the unweave, an important trick was to note that the space of weaves with all components the same radius deformation retracts onto the subspace with one fixed radius. This is true for infinite unit weaves, and of linear weaves as well; if \(\tau\) is a periodicity of \(w\), then \(r \tau\) is a periodicidy of \(r w\).

We could alternatively kill off this extra degree of freedom not by fixing ring sizes, but instead by fixing the periodicity: let \(\cW'_{n,\per}\) be the \(n\)-periodic infinite weaves of shared radius \(r\) and let \(\cW'^{\tau}_{n}\) be the infinite weave which are rendered \(n\)-periodic by \(\tau\) a translational symmetry.

Proposition 1. \(\cW'_{n,\per} \simeq \cW'^\tau_{n} \times \RRP^2\) non-canonically; in particular, they have the same \(\pi_0\).

The element \(\tau \in \GG\) constitutes translation along a line, i.e. an element \(t \in \RRP^2\). Let \(\rho:\RRP^2 \rightarrow \GG\) be a continuous map such that \(\rho(x)x = t\) for all \(x\), and let \(\pi:\cW'_{n,\per} \rightarrow \RRP^2\) be the map taking the underlying line of the translational symmetry. Then, action by \(\rho\) together with scaled division by the length of an isometry and the “length divided by \(\tau\)” map yields a map

$$\varphi:\cW'_{n,\per} \rightarrow \cW'^\tau_n \times \RRP^2 \times \RR_{>0};$$

we would like to prove that this is an equivalence.

In fact, taking the pointwise inverse map \(\rho^{-1}:\RRP^2 \rightarrow \GG\), we attain a composition in the other direction

$$\psi:\cW'^\tau_n \times \RRP^2 \times \RR_{>0} \xrightarrow{\id \times \rho^{-1}(t) \cdot r} \cW'^{\tau_n} \times \GG \hookrightarrow \cW'_{n,\per} \times \GG \xrightarrow{\;\;\mu\;\;} \cW'_{n,\per}$$

In fact, it is not too hard to see that \(\varphi\) and \(\psi\) are inverse homeomorphisms, so in particular, they are homotopy equivalences, and we are done.

This theorem allows us to talk about linear weaves by fixing a particular symmetry, and hence we are talking about equivariant isometric embeddings. We generalize this greatly in the following sections.

An action of \(\GG = \Isom(\RR^3)\) on weaves

For \(n \in \NN \cup \cbr{\infty}\), we write \(H_n := \Aut(S^1)^n \simeq \prn{S^1}^{\times n}\) for the group of \(n\) reparameterizations of circles circles. There is an evident embedding \(\cW_n \hookrightarrow \mathrm{Emb}(n \cdot S^1, \RR^3)_{H_n}\) of weaves into circle embeddings modulo reparameterization.

Proposition 2. The image of \(\iota:\cW_n^\Sym \hookrightarrow \Map(n \cdot S^1, \RR^3)_{H_n}\) is preserved by the evident \(\GG\) action; hence \(\cW_n^\sym\) is a \(\GG\)-space.

To see this, note that the action on \(\RR^3\) restricts to isometries between totally geodesic surfaces; in particular, it takes every linearly embedded plane isometrically into another linearly embedded plane. This implies that, for \(\varphi \in \GG\) and \(w \in \cW_n\), the embedding \(\varphi w\) consists of linearly embedded circles. This is precisely the image of \(\iota\), so we are done.

\(G\)-weaves, a definition

Let \(w \in \cW_n^\Sym\) be an ordered weave realization, and let \(H = \operatorname{Stab}(w) \subset \GG\) be the stabilizer of \(w\). Then, the set of components \(\abs{w}\) naturally inherets the structure of an \(H\)-set. We will use this to decompose the set of \(G\) weaves in the following definition:

Definition 3. Let \(G\) be a discrete group acting on \(\RR^3\) by isometries, and let \(S\) be a \(G\)-set. Then, the space of \(G\)-weaves with \(S\) components, denoted \(\cW_S^G\), is the space of weaves fixed by \(G\) whose unit \(G\)-set is \(S\):

$$\cW_S^G := \prn{\cW_{\abs{S}}^{\Sym}}^G \times_{\Set^{\mathrm{B}G}} \cbr{S}.$$

The space of \(G\) weaves of size \(n\) is the coproduct

$$\cW_n^G := \coprod_{|S_G| = n} \prn{\cW_S^G}_{\Aut_G(S)}.$$

The space of \(G\) weaves is the coproduct

$$\cW^G := \coprod_n \cW_n^G$$

Some examples in chains, sheets, volumes

For \(g \in \GG\) we write \(\cW_n^g\) to mean \(\cW_n^{\langle g \rangle}\), where \(\langle g \rangle \subset \GG\) is the (possibly infinite) subgroup generated by \(g\). The following is proposition is easy to prove, and recognizes some familiar spaces of weaves as \(G\)-weaves.

Proposition 5.

  1. Let \(e\) denote the identity automorphism. Then, \(\cW_n^e \simeq \cW_n.\)
  2. Let \(\tau\) be a translational symmetry of \(\RR^3\). Then, \(\cW_{n,\per}^\tau \simeq \cW_n \times \RRP^2\).
  3. Let \(s\) be a reflective symmetry of \(\RR^3\). Then, \(\cW^s \subset \cW\) is the space of chiral unit weaves.

The reason I want to move to \(G\) invariant things is mostly bookkeeping for higher dimensional arrangements. Hence we may set the following notation:

Notation.

  1. Let \(P \subset \RR^2 \subset \RR^3\) be a parallelogram. Let \(\tau_1,\tau_2\) be translation along its edges. Then, the space of \(P\) weaves of size \(n\) is the space
    $$\cW_n^P := \cW_n^{\langle \tau_1,\tau_2 \rangle}.$$

    a parallelogram weave is an element of \(\pi_0 \cW_n^P\) for some \(P\).

  2. Let \(C \subset \RR^3\) be a parallelopiped. Let \(\tau_1,\tau_2,\tau_3\) be translation along its edges. Then, the space of \(C\)-weaves of size n is the space
    $$\cW_n^C := \cW_n^{\langle \tau_1,\tau_2 \rangle}.$$

    a parallelopiped weave is an element of \(\pi_0 \cW_n^C\) for some \(C\).

  3. Let \(T \subset \RR^2 \subset \RR^3\) be a triangle. Let \(s_1,s_2,s_3 \in O(3)\) be reflective symmetries about the hyperplanes intersecting \(\RR^2\) orthogonally, with intersection parallel to the three edges of \(T\). Let \(\tau_1,\tau_2,\tau_3 \in O(3)\) be the translation symmetries corresponding with the edges of \(T\). Then, the space of \(T\) weaves of size \(n\) is the space
    $$\cW_n^T := \cW_n^{\langle s_i,\tau_i \rangle_{1 \leq i \leq 3}}.$$

    a triangular weave is an element of \(\cW_n^T\) for some \(T\).

  4. Let \(H \subset \RR^2 \subset \RR^3\) be a hexagon with opposite edges parallel. Let \(\tau_1,\tau_2,\tau_3\) be translation along its edges. Then, the space of \(H\) weaves of size \(n\) is the space
    $$\cW_n^H := \cW_n^{\langle s_i,\tau_i \rangle_{1 \leq i \leq 3}}.$$

    a hexagonal weave is an element of \(\cW_n^H\) for some \(T\).

My point here is that this covers nearly every chain mail weave in existence; there is an evident sense in which in can only handle fundamental domains for a \(G\)-action, however, so I’m skeptical that it can handle the correct notion of compounding for sheet weaves.

A general strategy to define \(G\)-invariants

An invariant of \(G\)-weaves is a continuous map \(\cW^G \rightarrow X\), where \(X\) is a space of values for the invariant. Gradings of \(G\)-weaves correspond with discrete-valued invariants; for instance, the grading by underlying \(G\)-set corresponds with a map \(\cW^G \rightarrow \pi_0\prn{\Set^{BG}}\).

One strategy to create these is to define a \(G\)-equivariant invariant, i.e. a map of \(G\)-spaces \(\cW \rightarrow X\); this then induces maps \(\cW^G \rightarrow X^G\) for each \(G \subset \GG\) (hence each action by isometries), compatible with restrictions and conjugation. This is how we will define the linking graph of \(G\) weaves.

We will want to define the linking graph not just as an invariant, but as a graded one. In order to do so, we will define the \(G\)-set grading as an \(\GG\)-invariant \(\cW \rightarrow \underline{ \pi_0 \Set_\GG}\) for some \(\GG\)-set satisfying \(\prn{\underline{\pi_0 \Set_{\GG}}}^G \simeq \pi_0 \Set^{BG}\) and demand that the linking graph is an equivariant map over \(\underline{\pi_0 \Set_{\GG}}\).

The \(G\)-set grading

This section takes inspiration from the parameterized higher category theory of Barwick, Dotto, Glasman, Nardin, and Shah. We will be much more simple minded than them, while still indulging in a bit of nonsense.

By Elmendorf’s theorem, the data of \(\cW\), up to map inducing a weak equivalence on the space of \(G\)-weaves for all \(G \subset \GG\), is equivalent to the data of a functor of \(\infty\)-categories \(\cW^{(-)}:\cO_{\GG}^\op \rightarrow \cS\), where \(\cO_\GG\) is the orbit category and \(\cS\) is the \(\infty\)-category of spaces. This, by the Grothendieck construction, is equivalent to a cocartesian fibration \(\underline{\cW} \rightarrow \cO_\GG\) with groupoid fibers; Elmendorf’s theorem tells us that \(\GG\)-equivariant homotopy theory is equivalent to parameterized homotopy theory over \(\cO_\GG\).

We may immediately, and temporarily, generalize to parmeterized (higher) category theory. For instance, the category of \(G\)-sets provides a functor \(\cO_{\GG}^{\op} \rightarrow \Cat\) (for instance, by composing the parameterized category of \(G\)-spaces with the category of discrete objects), which then corresponds with a cocartesian fibration \(\underline{\Set_{\GG}} \rightarrow \cO_\GG\) having category fibers.

Via the functor \(\pi_0:\Cat \rightarrow \Set\), we may further restrict this to a cocartesian fibration \(\underline{\pi_0 \Set_{\GG}} \rightarrow \cO_G\) having discrete fibers. By Elmendorf’s theorem, this yields a \(\GG\)-space.

To lift the \(G\)-weave grading \(\cW^G \rightarrow \pi_0\Set^{BG}\) to the equivariant world, we must provide a \(\GG\)-equivariant map \(\underline{\cW} \rightarrow \underline{\pi_0 \Set^{\GG}}\), i.e. a natural transformation \(\alpha:\cW^{(-)} \implies \pi_0 \Set^{B(-)}:\cO_{\GG}^\op \rightarrow \cS\).

Since \(\pi_0 \Set^{BG}\) is discrete, the data of a map \(W^G \rightarrow \pi_0 \Set^{BG}\) is the same as supplying a functor \(\cW_{(-)}^G:\pi_0 \Set^{BG} \rightarrow \cS\) whose underlying object (i.e. colimit) is \(\cW^G\), which we have already supplied. Naturality follows from functoriality of \(\cW_{(-)}^G\) under restrictions and conjugacy, i.e. from the evident maps \(\cW_{S}^G \rightarrow \cW_{\Res_H^G S}^H\) for \(H \leq G\) by inclusion and \(\cW_S^G \rightarrow \cW_{hSh^{-1}}^{hGh^{-1}}\) for \(h \in \GG\) by the conjugation action on \(\cW^\Sym_n\).