Equivariant chamber-point and path assignment

PDF p. 25: Lemma 6.6, proof, construction of p; PDF p. 26: Lemma 6.6, proof, construction of p

Let $(W,S)$ be a Coxeter system with finite $W$, let $J\subseteq S$, and put $G=N(W,J)={\mathrm{Norm}}_W(W_J)\mathrm{/}{W}_J$, where $W_J$ is generated by $J$. Use the relative real space $E$, its finite hyperplane arrangement ${\mathcal{H}}^J$, and its complexified complement ${\mathcal{X}}_J^{\mathbb{C}}$ from Relative complexified hyperplane complement. A chamber is a connected component of $E\setminus {\bigcup }_{H\in {\mathcal{H}}^J}H$; two chambers are adjacent if they share a wall. For each chamber $C$ use its open set $X_C$, and for an ordered adjacent pair $(C,C^{\mathrm{\prime }})$ use the specified component $X_{C,C^{\mathrm{\prime }}}\subseteq X_C\cap X_{C^{\mathrm{\prime }}}$, as defined in Chamber open sets used for the topological equivalence.

Choose points $a_C\in X_C$ satisfying $a_{gC}=g{a}_C$ for every $g\in G$. Assign $C$ to $a_C$. For a directed arrow $C\to C^{\mathrm{\prime }}$ between adjacent chambers, choose $d\in X_{C,C^{\mathrm{\prime }}}$, a continuous path ${\gamma }_1:[0,1]\to X_C$ from $a_C$ to $d$, and a continuous path ${\gamma }_2:[0,1]\to X_{C^{\mathrm{\prime }}}$ from $d$ to $a_{C^{\mathrm{\prime }}}$. Assign the arrow the endpoint-fixed homotopy class $[{\gamma }_1{\gamma }_2]$ of their concatenation in ${\mathcal{X}}_J^{\mathbb{C}}$, traversing ${\gamma }_1$ first. An endpoint-fixed homotopy is a continuous deformation through paths with the same two endpoints; these classes are the arrows of the fundamental groupoid ${\pi }_1({\mathcal{X}}_J^{\mathbb{C}})$.

The assignment on directed arrows is independent of the choices of $d,{\gamma }_1,{\gamma }_2$ once the $a_C$ are fixed, and it is $G$-equivariant. It is an assignment on the directed chamber graph; no claim that it respects the positive category's relations or gives an equivalence is made in this definition.

Real Coxeter representation and contragredient notation

Relative complexified hyperplane complement

Chamber open sets used for the topological equivalence

Contractibility and intersection components of chamber open sets

Faithful normaliser-quotient action on the relative cone

Normaliser action on labels, orbits and adjacency

Positive arrangement category and its groupoid completion

Fundamental groupoid of a space

Paths, path-connected spaces and path components

Homotopy equivalences, homotopy inverses and spaces of the same homotopy type

Given: The finite relative arrangement, chamber charts, ordered components and directed chamber graph in the Definition, with the topology and path conventions of the declared suppliers.

1.1 The finite arrangement has finitely many chambers: each chamber is determined by its side of each hyperplane. The action of $G$ on chambers is free and preserves adjacency by Normaliser action on labels, orbits and adjacency. Each $X_C$ is nonempty and contractible by Contractibility and intersection components of chamber open sets. Choose one chamber and one point in its chart for each of the finitely many $G$-orbits. The normaliser acts linearly on $E$ by Faithful normaliser-quotient action on the relative cone and permutes the restricted root hyperplanes: a normaliser lift permutes roots under the canonical representation and preserves $E$. Its induced homeomorphism of the complexified complement sends $X_C$ to $X_{gC}$, since it sends each chamber side of a hyperplane to the corresponding side of the image hyperplane. Extend the chosen points by $a_{gC}=g{a}_C$. Freeness makes the extending group element unique for a fixed orbit representative, so these points are well-defined and equivariant. Only finitely many choices are used. [given]

1.2 Fix adjacent $C,C^{\mathrm{\prime }}$ with unique separating hyperplane $H_0$, and choose $c\in C$. For $H\mathrm{\ne }{H}_0$, the half-spaces $D_H(C)$ and $D_H(C^{\mathrm{\prime }})$ agree: otherwise $H$ would also separate the chambers. The point $c+ic$ belongs to $X_{C,C^{\mathrm{\prime }}}$. For any $z=x+iy\in X_{C,C^{\mathrm{\prime }}}$, first follow $u_z(t)=x+i((1-t)y+tc)$, and then $v_z(t)=((1-t)x+tc)+ic$, with $0\le t\le 1$. During $u_z$, the real coordinate stays in $D_{H_0}(C)$; for any $H$ containing $x$, necessarily $H\mathrm{\ne }{H}_0$, and both $y$ and $c$ lie in the common required open half-space. A half-space is convex because its defining strict linear inequality is preserved by convex combinations. Thus the moving imaginary coordinate satisfies the defining conditions. During $v_z$, the real coordinate remains in $D_{H_0}(C)$ by the same convexity, while $c$ lies in every common required half-space for $H\mathrm{\ne }{H}_0$. Neither path can meet any $H\times H$. Both paths therefore stay in $X_{C,C^{\mathrm{\prime }}}$, by Chamber open sets used for the topological equivalence. Their coordinate formulas are continuous in the product subspace topology of Relative complexified hyperplane complement, and they join at $x+ic$. Concatenating gives a path from any $z$ to $c+ic$; following such a path and the reverse of another joins any two points. Consequently $X_{C,C^{\mathrm{\prime }}}$ is nonempty and path-connected in the sense of Paths, path-connected spaces and path components. This conclusion is proved here, rather than imported from the component proposition. [given]

1.3 By Nullhomotopic maps and contractible spaces, applying contractibility to the identity map of a chart gives a homotopy from that identity to a constant map, that is, a contraction. Each nonempty contractible chart is path-connected: a contraction supplies a path from any point to its constant value, and concatenating one such path with the reverse of another joins any two points. By Contractibility and intersection components of chamber open sets, $X_{C,C^{\mathrm{\prime }}}$ lies in both charts, so the required point and paths exist. Their concatenation is a path in the complement, so its endpoint-fixed class is an arrow of the fundamental groupoid as defined in Fundamental groupoid of a space. [given, step 1.1, step 1.2]

2.1 To see independence, take a second choice $d^{\mathrm{\prime }},{\gamma }_1^{\mathrm{\prime }},{\gamma }_2^{\mathrm{\prime }}$ and a path $\delta$ in $X_{C,C^{\mathrm{\prime }}}$ from $d$ to $d^{\mathrm{\prime }}$. In a contractible space any two paths with the same endpoints are homotopic relative to endpoints: a contraction fills their concatenated loop, and cutting the filling gives the relative homotopy. Apply this first in $X_C$ to ${\gamma }_1\delta$ and ${\gamma }_1^{\mathrm{\prime }}$, and then in $X_{C^{\mathrm{\prime }}}$ to ${\gamma }_2$ and $\delta {\gamma }_2^{\mathrm{\prime }}$. It follows, after cancelling the path $\delta$ against its reverse up to endpoint-fixed homotopy, that $[{\gamma }_1{\gamma }_2]=[{\gamma }_1^{\mathrm{\prime }}{\gamma }_2^{\mathrm{\prime }}]$. These path and homotopy conventions are those of Paths, path-connected spaces and path components, Homotopy equivalences, homotopy inverses and spaces of the same homotopy type, and Fundamental groupoid of a space. [given, step 1.1, step 1.2, step 1.3]

2.2 The same linear action sends the unique separating hyperplane of adjacent $C,C^{\mathrm{\prime }}$ and its $C$-side to those of $gC,g{C}^{\mathrm{\prime }}$. The defining conditions therefore give $g{X}_{C,C^{\mathrm{\prime }}}=X_{gC,g{C}^{\mathrm{\prime }}}$. Applying $g$ to $d,{\gamma }_1,{\gamma }_2$ produces valid choices for the image arrow, with endpoints $a_{gC},a_{g{C}^{\mathrm{\prime }}}$. Independence of choices shows that the assigned class for that arrow is the image of $[{\gamma }_1{\gamma }_2]$. Hence the assignment is equivariant. [given, step 1.1, step 2.1] ∎

The article reference attributes the construction. The existence, independence and equivariance arguments above use the declared suppliers and elementary path manipulations; the article's construction is not used as its own proof.