18 problems

Let $Y$ be an aspherical finite $2$-dimensional $CW$-complex, and let $X$ be a connected $CW$-subcomplex of $Y$. Whitehead conjecture. The subcomplex $X$ is aspherical. This conjec…

Let $G$ be a finite group acting on a contractible $2$-complex. Casacuberta–Dicks conjecture. The action has a fixed point. Serre's fixed-point theorem establishes the analogous st…

Weak Andrews-Curtis conjecture. Every balanced presentation of the trivial group is $Q^{**}$-equivalent to

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…

Let $X$ be a compact $2$-complex with negative irreducible curvature, the combinatorial negative-curvature property described in the source. Wilton's conjecture. The fundamental gr…

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

Exotic-presentation conjecture. The existence of an exotic presentation for $Q_{4n}$ should hold unconditionally, without assuming that $Q_{4n}$ has the D2 property.

D2 and presentation-form conjecture. If $n \ge 2$, then $Q_{4n}$ has the D2 property and every balanced presentation is homotopy equivalent to a presentation of the form…

A labelled oriented tree (LOT) presentation is a group presentation associated to a labelled oriented tree, and its associated presentation complex is the corresponding $2$-complex…

A $2$-complex is a CW complex of dimension at most $2$, and a $3$-deformation is a deformation between $2$-complexes involving no expansion beyond $3$-cells. Two complexes are simp…

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.

Gaster–Wise conjecture. If $X$ is a compact 2-complex with the generalized Dehn property, then its universal cover $\widetilde{X}$ enjoys a linear isoperimetric inequality: there e…

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…