42 problems

Let $G$) be a connected, non-compact, semi-simple Lie group with finite center. Write … for the comparison map from continuous bounded cohomology to continuous cohomology. Dupont's…

Monod's isomorphism conjecture. The continuous bounded cohomology of $G$ with real coefficients is naturally isomorphic to its ordinary continuous cohomology:

Guilloux's conjecture. Outside an analytic neighborhood of $[\pi_n\circ i]$, the Borel function is bounded away from its maximum value…

Let ${\cal M}_g$ be the mapping class group. Recall that a group ${\varGamma}$ is uniformly perfect if there is a natural number $N$ such that every element of ${\varGamma}$ is a p…

Let ${\cal M}_g$ be the mapping class group and let $H_b^2({\cal M}_g)$ denote its second bounded cohomology. Bounded-cohomology conjecture. For every genus $g$, the comparison map…

Infinite-dimensionality conjecture. If

Let $g\geq 1$, let $\mathcal{D}_g$ be the Dehn quandle of a closed orientable surface of genus $g$, and let $\mathbb{Z}[\mathcal{D}_g]$ be its integral quandle algebra. An element…

Let $M$ be a closed orientable surface of positive genus, and let $H^2_{(1)}({\rm Homeo}_0(M),{\mathbf R})$ denote the $1$-bounded cohomology of the identity component of the homeo…

Let $G$ be a connected simple Lie group and let $(X,\mu,Y)$ be an integrable transverse $G$-system with $\mu$ $G$-ergodic; let $(\mathcal{G},\nu)$ be its transverse measured groupo…

Let $M$ be a pinched negatively curved manifold of infinite volume, of dimension $n$, and let $[hat{\omega}]in H^{n}_b(M,\mathbb{R})$ denote its bounded fundamental class. Kim–Kim'…

Dupont's conjecture. All real continuous cohomology classes of $G$ are bounded.

Bowden's conjecture. The stable mapping class group $\Gamma_\infty$ is boundedly acyclic.

Let $\Gamma$ be a lattice in a rank-one group. The Serre conjecture. Such lattices should not have the congruence subgroup property. This conjecture is presented as an obstacle to…

Let $M$ be a complete Riemannian manifold of dimension $n$ with a properly discontinuous, cocompact, isometric action by a group $G$, and let … be the isomorphism obtained from the…

Let $M$ be a closed connected Riemannian manifold, and let $[\omega] \in H_\mathrm{dR}^2(M) \cong H^2(M;\mathbb R)$ be represented by a smooth differential $2$-form $\omega$. Let…

Isomorphism conjecture. This comparison map is an isomorphism in every degree $n$.

Vanishing conjecture. The second bounded cohomology vanishes for the groups of piecewise linear homeomorphisms of flows:

Calegari's amenability conjecture. $G$ is amenable.

Frigerio's conjecture. The Euler class of every topologically flat sphere bundle admits a bounded representative.

Let $X$ be a closed Riemannian manifold, and let $\alpha \in H^2(X;\mathbb{R})$. The class $\alpha$ is $\widetilde{d}$-bounded if its pullback to the universal cover…

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…

Maximal-norm conjecture. The norm of $\Theta_n$ is attained at the points satisfying $\theta_i^k=e^{\frac{2ki\pi\sqrt{-1}}{2n+1}}$ for $k=1,\ldots,n$ and $i=0,\ldots,2n$; that is,

Gromov's conjecture. Every closed aspherical $4$-manifold $M$ satisfies

Let $V$ be a closed Riemannian manifold, and let $\alpha\in H^2(V,{\mathbb{R}})$. A class $\alpha$ is $\widetilde{d}$-bounded if it has a bounded primitive after pullback to the un…

Let $F_2$ be the free group of rank $2$, let $\operatorname{PSL}_2\mathbb{C}$ be the group of orientation-preserving isometries of hyperbolic $3$-space, and let…