28 problems

Gromov's conjecture. Every functorial seminorm on singular homology with real coefficients vanishes for simply connected spaces.

Let $M$ be a closed, nonpositively curved, locally symmetric manifold with no local $\mathbb{R}$ factors. The Gromov norm of $M$, namely the simplicial volume, is positive. Gromov'…

For any link $L$, let $J_N(L)$ be the colored Jones polynomial evaluated at $\exp\left(2\pi\sqrt{-1}/N\right)$. Let $\Vert L\Vert$ denote the simplicial volume of the complement of…

For a closed oriented connected $n$-dimensional manifold $M$ admitting a hyperbolic metric, let $g$ be a Riemannian metric on $M$, let $\|M\|_{\Delta}$ denote its Gromov simplicial…

Let $M$ be a closed nonpositively curved $4$-manifold, and let $\|M\|$ denote its simplicial volume and $\chi(M)$ its Euler characteristic. Connell–Ruan–Wang's equivalence conjectu…

Let $M$ be a closed, connected, oriented $n$-dimensional manifold. Its simplicial volume is … A Riemannian metric on $M$ has negative definite Ricci curvature when its Ricci curvat…

Let $M$ be a closed affine manifold, and let $\|M\|$ denote its simplicial volume. Simplicial-volume conjecture. The simplicial volume of $M$ vanishes: … This is a stronger form of…

Let $M$ be a manifold. Suppose $M$ admits a Riemannian metric with nonpositive sectional curvature everywhere and negative definite Ricci curvature at some point. Connell–Wang's co…

Let $M$ be a closed nonpositively curved $4$-manifold, with simplicial volume $||M||$ and Euler characteristic $\chi(M)$. Zero simplicial volume characterization conjecture. Then ……

Let $M$ be a closed, connected, oriented topological $n$-manifold. Suppose that $M$ admits a Riemannian metric whose sectional curvature is nonpositive everywhere and whose Ricci c…

Detcherry–Kalfagianni simplicial-volume conjecture.

Let $M$ be a compact and orientable $3$-manifold with empty or toroidal boundary. For odd integers $r$, set … Let $v_3\approx 1.0149$ be the volume of a regular ideal tetrahedron,…

Simplicial volume conjecture for open 3-manifolds.

Let $M$ be a closed oriented manifold covered by $(\mathbb{H}^2)^n$, and let $\Theta_n$ be the bounded volume form defined in the preceding conjecture. Simplicial-volume equality c…

Nonexistence conjecture. Simply-connected strongly inflexible manifolds do not exist.

Toroidal simplicial-volume conjecture.

Detcherry–Kalfagianni–Yang generalized Turaev–Viro volume conjecture.

Murakami–Murakami simplicial-volume conjecture.

Strengthened Gromov conjecture. Then $M$ has positive simplicial volume:

Let $M$ be a closed hyperbolic manifold, let $g$ be an auxiliary Riemannian metric on $M$, and let $V_{hyp}(R)$ denote the volume of a geodesic ball of radius $R$ in hyperbolic spa…

Let $M$ be a closed manifold supporting an affine structure, meaning a closed manifold admitting an atlas whose transition maps are restrictions of affine isomorphisms. Let $\|M\|$…

Vanishing simplicial volume conjecture. If $M$ is a closed manifold supporting an affine structure, then

Let $f:M\to\mathbb R$ be a Morse function on a closed manifold $M$ of dimension $n+1$, and let $v$ be its gradient-like field satisfying the Morse–Smale transversality condition. F…

For each $g\geq 2$, let $\mathcal{M}_g$ be the set of $3$-manifolds with boundary that admit an ideal triangulation by $g$ tetrahedra and have Euler characteristic $\chi(M)=1-g$, a…

Let $M$ be a finite-volume hyperbolic $n$-manifold with torus cusps, let $V$ be its complete hyperbolic interior, and let $M(T_1,\ldots,T_m)$ denote the manifold obtained by fillin…