16 problems

Gromov–Lawson–Thurston conjecture. A necessary condition for the disc bundle to admit a hyperbolic structure is

Let $B=\Sigma_h$ with $h\geq 1$, and consider the corresponding sets and bijections in the holomorphic category. Geometric Shafarevich Conjecture. These sets are finite, and the ho…

Linear-combination conjecture. The class $v_1$ is a linear combination of $\alpha$ and $e_1$:

Let $B$ be the base of a surface bundle, let $p$ be a prime, and let $\kappa_i$ denote its Morita–Mumford classes. Vanishing conjecture. For $s\geq 1$ and $v\geq 0$, one has … The…

Let $M=M(e,g)$ be homeomorphic to an $\mathbb R^2$-bundle over a closed orientable surface of genus $g>1$, with Euler number $e=\langle[F],[F]\rangle$ given by the self-intersectio…

Let $M$ be an arithmetic closed hyperbolic surface bundle, and let the fiber have genus $2$. Fiber-genus-2 extension conjecture. Theorem should hold for this case as well. The conj…

Complex hyperbolic Gromov–Lawson–Thurston conjecture. An oriented disc bundle over a closed Riemann surface admits a complex hyperbolic structure if and only if

Hain's conjecture. Every section of this universal bundle is homotopic to one of the sections $s_1,\ldots,s_n$.

Morita's boundedness conjecture. All MMM-classes are bounded in the sense of Gromov.

Let $M^{d}\to E\to B$ be an oriented bundle with closed $d$-dimensional fiber, let $T_ME$ be its vertical tangent bundle, and let $P$ be a polynomial in the Euler and Pontryagin cl…

Let $M$ be a hyperbolic 3-manifold. A surface bundle over $S^1$ is a mapping torus of an orientation-preserving homeomorphism of an orientable surface, and a finite quotient is a q…

Let $S=S(g,p)$ be an orientable, connected surface of genus $g$ with $p$ punctures, and let $M$ be a hyperbolic surface bundle over $S^1$ with fiber of type $(-\chi,p)$, where…

For each $g>2$, let $\Sigma_g$ be the closed surface of genus $g$, and let $M_g$ be the explicit closed hyperbolic $\Sigma_g$-bundle over the circle obtained by the Dehn fillings d…

Let $\Gamma_\infty$ be the stable mapping class group of oriented surfaces, let $B\Gamma_\infty$ be its classifying space, and let $\Omega^\infty \mathbb{C}P^\infty_{-1}$ denote th…

Let $\pi\colon E\to B$ be an oriented surface bundle with closed fibers of genus $g$, equipped with a fiberwise metric. Let $\mathcal{H}(E)$ be the associated Hodge bundle, with fi…

Let $\psi$ be a diffeomorphism of a surface, let $M(\psi)$ denote its mapping torus, and let $\mathfrak{s}_{k-2}$ be the indicated $\operatorname{Spin}^c$ structure. Let $HF(\psi)$…