17 problems

Let $G$ be the fundamental group of an aspherical two-complex, and let $\mathcal{P}=\langle G,x\mid R\rangle$ be a one-relator relative presentation whose natural homomorphism…

Brieskorn's conjecture. The complexified reflection arrangement of every finite Coxeter group is a $K(\pi,1)$-arrangement.

Let $W$ be an infinite Coxeter group, let $Y_W$ be the associated hyperplane complement in $\mathbb{C}\otimes V$ restricted to the Tits cone, and let $A$ be the associated Artin gr…

Relative asphericity conjecture. Under these hypotheses, $(L,K)$ is relatively aspherical.

Let $(\mathrm{W}[\Gamma],S)$ be a Coxeter system, let $\mathrm{A}[\Gamma]$ be its associated Artin group, and let $\overline{Sal}(\Gamma)$ be the Salvetti complex, whose fundamenta…

Non-positive surface curvature implies asphericity. If $\sigma_+(X)\leq 0$ then $X$ is aspherical.

Orbifold-asphericity conjecture. A simple orbifold $Q$ is orbifold-aspherical if and only if its underlying space $|Q|$ is aspherical as a manifold and $Q$ contains no $\Delta^k$-b…

Let $l>0$ and $k>0$ with $l\neq k$, and let $\mathcal{Q}$ be the relative presentation considered in the paper. Define condition (AEJ-E) by the following alternatives: (i) $g=h^2$,…

Let $\mathcal{Q}$ be the relative presentation considered in the paper, with parameters $l>0$, $k\neq 0$ and non-trivial elements $g,h\in G$. Define … Say that Case $(P)$ holds whe…

Let $\Phi$ be a reduced root system with Weyl arrangement $\mathscr A(\Phi)$. An arrangement of ideal type $\mathscr A_{\mathcal I}$ is a subarrangement of $\mathscr A(\Phi)$ deter…

Let $({\bf W},S)$ be a Coxeter system on a finite generating set $S$, let ${\bf G_W}$ be the corresponding Artin group, and let $\mathbf X_{\mathbf W}$ be the quotient of the Salve…

Let $T$ be a spherical subset of the standard generating set of an Artin group, let $W_T$ be the corresponding finite reflection group, and let $M_T$ be the complement of its refle…

Let $A$ be the Artin group associated to a Coxeter system $(W,S)$, and let $L$ be its nerve. For every spherical subset $T\subseteq S$, let $\operatorname{Sal} T$ be the Salvetti c…

Let $W$ be a Coxeter group acting on an open nonempty convex cone $I$, let ${\mathcal A}$ be the associated Coxeter arrangement of linear hyperplanes, and let … Let $A$ be the Arti…

Let $X$ be a homotopically trivial finite space of height $2$, and let $a\in X$ be a maximal point such that $X-a$ is connected. Whitehead's asphericity conjecture. The space $X-a$…

Let $(W,S)$ be a Coxeter system, let $A$ be the associated Artin group, and let $X$ be its Salvetti complex. The fundamental group of $X$ is $A$, and $BA=K(A,1)$ denotes a classify…

Let an arrangement be a collection of weighted lines in $\mathbb{CP}^2$ satisfying the conditions of Theorem existence. Its complement is the space obtained by removing the arrange…