3 problems

Torus and rationality conjecture. There exists a subcomplex $Y \subseteq X_A$ homotopic to $\mathbb{T}^n$. Furthermore, $\mathcal{X}$ is rational, with local systems on $Y$ corresp…

Torus conjecture. The complex $X_{A_{\mathcal{X}}}$ is homotopic to a real torus $\mathbb{T}$.

The projective cellular resolution conjecture. If $Y$ is a projective cellular resolution of $A$ as an $A \otimes_{\mathsf k} A^{\operatorname{op}}$-module, then $Y$ is homotopic t…