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Spherical normaliser quotient classifying spaces

PDF p. 3: Theorem 1.8; PDF p. 27: Corollary 6.14

Statement

Assume the Axiom of Choice. Let $(W, S)$ be a finite-rank spherical Coxeter system, meaning that $W$ is finite, and let $J \subseteq S$ . Let $A$ be the associated Artin–Tits group and $P$ the kernel of its standard projection to $W$ . For $G = W, A$ , let $G_{J}$ be the standard parabolic subgroup generated by the generators indexed by $J$ , and put $N (G, J) = {Norm}_{G} (G_{J}) / G_{J}$ , where the normaliser consists of the elements conjugating $G_{J}$ to itself. Let $N (P, J)$ be the image of $P \cap {Norm}_{A} (A_{J})$ in $N (A, J)$ .

Let $X = X_{J}^{C}$ be the relative complexified hyperplane complement. The induced action of $N (W, J)$ gives a quotient $X / N (W, J)$ . Then $X$ is a $K (N (P, J), 1)$ and $X / N (W, J)$ is a $K (N (A, J), 1)$ . Here $K (G, 1)$ means a path-connected space whose fundamental group is isomorphic to $G$ and whose homotopy groups in all degrees at least two vanish.

Statement references

The Axiom of Choice

Coxeter-system, roots, length and weak-order conventions

Artin–Tits group, pure subgroup and positive lifts

Normalisers and their quotients

Relative complexified hyperplane complement

Relative free action, normal cover and exact sequence

Paths, path-connected spaces and path components

Based loops and the fundamental group

Higher homotopy group by based cubes

Facts & Assumptions

[F1]

The normaliser quotients and $N (P, J)$ are the groups defined in the Statement, and $X$ is the relative complement defined there. Normalisers and their quotients Relative complexified hyperplane complement

[F2]

In spherical type, $X$ is path-connected and its homotopy groups in degrees at least two vanish. Deligne asphericity used for the relative complement

[F3]

The orbit map $q : X \to X / N (W, J)$ is a covering map. Relative free action, normal cover and exact sequence

[F4]

For a covering of nonempty path-connected spaces, the induced maps on homotopy groups in degrees at least two are isomorphisms. Coverings preserve the higher-homotopy obstruction to K(pi,1)

[F5]

Assuming the Axiom of Choice, the algebraic and topological normaliser exact sequences are isomorphic. In particular, for $x \in X$ and $\overset{ˉ}{x} = q (x)$ , $π_{1} (X, x) ≅ N (P, J)$ and $π_{1} (X / N (W, J), \overset{ˉ}{x}) ≅ N (A, J)$ . Isomorphism of algebraic and topological normaliser sequences The Axiom of Choice

[F6]

A path is a continuous map from $[0, 1]$ , and path-connectedness means that any two points can be joined by a path. Fundamental groups consist of based-loop homotopy classes, and higher homotopy groups are defined by based cubes. Paths, path-connected spaces and path components Based loops and the fundamental group Higher homotopy group by based cubes

[F7]

A map of topological spaces is continuous when it is continuous at every point; equivalently, the inverse image of every open set is open. Continuity of a map of topological spaces at a point and globally For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A)

Proof

Given: The spherical Coxeter system, subset $J$ , associated groups and relative complement in the Statement, and the Axiom of Choice.

Goal: Identify both fundamental groups and prove that both spaces have vanishing higher homotopy groups.

Given: The data and assumption in the Statement.

Goal: Establish the two stated $K (G, 1)$ assertions.

Proof technique: direct.

1.1F2F5choose

By [F2], $X$ is nonempty and path-connected and has trivial homotopy groups in degrees at least two. Choose $x \in X$ . By [F5], its fundamental group at $x$ is isomorphic to $N (P, J)$ . Thus $X$ is a $K (N (P, J), 1)$ under the convention in the Statement.

2.1F2F3F4F5F6F7step 1.1chooseconstruct∎

For any $b_{0}, b_{1} \in X / N (W, J)$ , surjectivity of $q$ gives $x_{0}, x_{1} \in X$ with $q (x_{i}) = b_{i}$ . By [F2] and [F6], choose a path $γ : [0, 1] \to X$ from $x_{0}$ to $x_{1}$ . Then $q \circ γ$ is a path from $b_{0}$ to $b_{1}$ : it is continuous since for every open $U$ in the quotient, $(q \circ γ)^{- 1} (U) = γ^{- 1} (q^{- 1} (U))$ is open, by continuity of $q$ and $γ$ and [F7]. Thus the quotient is path-connected and is nonempty because $X$ is nonempty. By [F3] and [F4], $q$ induces isomorphisms on all homotopy groups in degrees at least two, so those groups vanish for the quotient by step 1.1. By [F5], its fundamental group at $q (x)$ is isomorphic to $N (A, J)$ . Hence the quotient is a $K (N (A, J), 1)$ .

Source notes

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 3, Theorem 1.8; p. 27, Corollary 6.14 and its proof