4 problems

Let $(X,d)$ be a proper metric space and let $n\in\mathbb{N}_0$. Let $H\mathcal{X}_*(X)$ denote coarse homology. Coarse Hurewicz conjecture. There is a natural morphism … that make…

Coarse mapping-space conjecture. The coarse homotopy group $\pi_n^c(X,\omega)$ is isomorphic to the group, or pointed set when appropriate, of base-ray-preserving coarse homotopy c…

Let $M$ be a compact Kähler manifold with infinite fundamental group $\Gamma$. Strong Carlson–Toledo conjecture. There exists a finite étale cover $M'$ of $M$ such that the Hurewic…

Let $\Gamma$ be an infinite Kähler group. Carlson–Toledo conjecture. There exists a compact Kähler manifold $M$ with $\pi_1(M)$ a finite-index subgroup of $\Gamma$ such that the Hu…