Spherical relative arrangements and Deligne topology

The relative hyperplanes are the distinct nontrivial traces of root hyperplanes on ${\mathrm{\Theta }}_J$. In finite type the relative cone interior is ${\mathrm{\Theta }}_J$, and ${\mathcal{X}}_J^{\mathbb{C}}$ is the complexified complement of these linear hyperplanes. Tits cone intersections, relative roots and arrangements Relative complexified hyperplane complement

Each chamber of the relative arrangement has the form $x{C}_I$, where $x\in W$, $I\subseteq S$, and $C_I$ consists of dual functionals with zero simple-root coordinates on $I$ and strictly positive coordinates on $S\setminus I$. Independent relative chamber labelling input

The set $C_I$ spans the coordinate subspace $E_I$ on which the coordinates indexed by $I$ vanish; it is an open positive orthant in that subspace. The action of $x$ is a linear automorphism. Coordinate faces span their annihilator subspaces Real Coxeter representation and contragredient notation

For a finite central real arrangement with open simplicial chambers, the positive gallery category has a finite-type atomic categorical Garside structure, and its groupoid completion is equivalent to the fundamental groupoid on chamber basepoints of the complexified complement. Independent finite simplicial arrangement theorem Categorical Garside structures and simple arrows

The positive arrangement category is the category of directed chamber paths modulo identification of geodesics with common endpoints, and $\mathcal{D}(J)$ is its groupoid completion. Positive arrangement category and its groupoid completion

The roots are $\mathrm{\Phi }=\{w{\alpha }_s:w\in W,s\in S\}$, and $H_{\alpha }=\{f\in \mathrm{\Theta }:f(\alpha )=0\}$ is a linear hyperplane. The contragredient action of each $w\in W$ is a linear automorphism of $\mathrm{\Theta }$. Real Coxeter representation and contragredient notation

For a topological space $X$, its fundamental groupoid has all points of $X$ as objects and endpoint-fixed path-homotopy classes as morphisms. Composition traverses consecutive paths in order, identities are constant path classes, and inverses are reversed path classes. Fundamental groupoid of a space

For the same finite central simplicial arrangement, the gallery-cover lemma gives an equivalence from the gallery completion to the all-points fundamental groupoid. Its proof constructs the quasi-inverse using a finite ordering of the chambers and explicit paths in chamber charts, without an arbitrary indexed selection of paths. Gallery fractions, fundamental groupoid and asphericity