12 problems

Let $Y$ be a finite connected $2$-dimensional $CW$-complex. A connected tower with target $Y$ is a finite sequence of covering maps and embeddings … Call such a tower non-positive…

Let $Y$ be a $2$-complex. Write $b^{(2)}_2(\widetilde Y;\mathcal N(\pi_1(Y)))$ for the second $L^2$-Betti number of its universal covering, with coefficients in the von Neumann alg…

Q36 presentation conjecture. If $\mathcal{P}=\mathcal{P}_9(n_1,n_2;m_1,m_2)$ presents $Q_{36}$, then

Positive surface curvature implies no nonpositive-Euler-characteristic surface subgroups. If $\sigma_-(X)>0$ then no subgroup of $\pi_1(X)$ is isomorphic to $\pi_1(S)$.

Negative surface curvature implies hyperbolicity. If $\sigma_+(X)<0$ then $\pi_1(X)$ is word-hyperbolic.

Non-positive surface curvature implies solvable word problem. If $\sigma_+(X)\leq 0$ then the word problem is solvable in $\pi_1(X)$.

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

Non-positive irreducible curvature implies coherence. If $\rho_+(X)\leq 0$ then $\pi_1(X)$ is coherent.

Let $X$ be a contractible complex. Wise's conjecture. The complex $X$ has (weak) non-positive immersions. Here, non-positive immersions means that for every immersion…

Let $X$ be an aspherical 2-complex, and let $Y$ be a subcomplex of $X$. Whitehead's conjecture. The subcomplex $Y$ is aspherical. This is a famous conjecture about asphericity inhe…

Let $X$ be a compact $2$-complex with the generalized Dehn property, and let $\widetilde X$ denote its universal cover. Linear isoperimetric-function conjecture. Then…

Wise's conjecture. The fundamental group $\pi_1X$ is coherent.