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Let $\mathbb{H}^3:=\operatorname{SO}^\circ(3,1)/\operatorname{O(3)}$. Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that \begin{equation} X:=\mathbb{H}^3/\Gamma \end{equation} is contractible? Of course, we need to assume that $\Gamma$ has torsion.

Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$. Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that \begin{equation} X:=\mathbb{H}^3/\Gamma \end{equation} is contractible? Of course, we need to assume that $\Gamma$ has torsion.

Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{SO(3)}$. Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that \begin{equation} X:=\mathbb{H}^3/\Gamma \end{equation} is contractible? Of course, we need to assume that $\Gamma$ has torsion.

Let $\mathbb{H}^3:=\operatorname{SO}^\circ(3,1)/\operatorname{O(3)}$. Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that \begin{equation} X:=\mathbb{H}^3/\Gamma \end{equation} is contractible? Of course, we need to assume that $\Gamma$ has torsion.