54 problems

Let $M$ be a compact orientable 3-manifold with empty or toroidal boundary. For odd integers $r\geq 3$, let $TV_r(M;e^{\pi i/r})$ denote the Turaev–Viro invariant at the specified…

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 $n\in\mathbb{N}$. For an $n$-dimensional, closed, aspherical, orientable manifold $M$, write $\widetilde{M}$ for its universal cover, $b^{(2)}_i(\widetilde{M})$ for its $i$th…

Bonahon's conjecture. The volume satisfies

Let $M$ be a closed, connected, oriented topological $n$-manifold, with simplicial volume $||M||$ and Euler characteristic $\chi(M)$. Say that $M$ is aspherical when its universal…

Let $K$ be a knot, let $\|K\|$ denote its simplicial volume, let $v_3$ be the volume of the ideal regular tetrahedron in $\mathbb{H}^3$, and let $J_N(K)$ denote the colored Jones p…

Conjecture. Then

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

Gromov's conjecture. For every $p\in\mathbb{N}$,

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…

Let $M$ be a smooth manifold. Let ${\mathcal{C}}_k(M)$ be the vector space of signed measures on $C^0(\Delta^k,M)$ with compact support and finite total variation, with boundary in…

Gromov's simplicial volume vanishing conjecture. If $M$ admits a metric of positive scalar curvature, then

Converse signature conjecture. There is a smooth fibre bundle

Gromov's conjecture. If

In a fixed dimension, let $M$ be an aspherical manifold equipped with a Riemannian metric whose scalar curvature has a lower bound and whose volume has an upper bound. The Gromov c…

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 an aspherical manifold, meaning that its universal cover is contractible. Let $\|M\|$ denote its simplicial volume and let $\chi(M)$ denote its Euler characteristic. Gro…

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 an aspherical manifold, and let $\lVert M\rVert$ denote its simplicial volume. Gromov's conjecture. If … then the Euler characteristic of $M$ is zero. This is the second…

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.