20 problems

Let $M$ be a closed hyperbolic $3$-manifold. Thurston's virtual fibering conjecture. The manifold $M$ has a finite cover that fibers over the circle, equivalently, that is the mapp…

Let $M$ be a closed oriented $3$-manifold, and let $S^1\times M$ denote its product with the circle. Suppose that $S^1\times M$ admits a symplectic structure. The symplectic produc…

A fibered three-manifold is a three-manifold admitting a submersion $p:M\to S^1$; equivalently, it is a mapping torus … for a surface $F$ and an orientation-preserving isomorphism…

Lackenby's Heegaard gradient conjecture. If $\chi^h_{\mathcal{L}}(M)=0$ for a family of covers, then $M$ virtually fibres over a circle. The source motivates this by noting that vi…

Let $N$ be a closed $3$--manifold. The symplectic product conjecture. If $S^1 \times N$ is symplectic, then there exists a $\phi \in H^1(N;\mathbb{Z})$ such that $(N,\phi)$ fibers…

Let $N$ be a compact, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold whose boundary is a non-empty collection of incompressible tori and Klein bottles. Assume that every subg…

Let $M$ be a closed, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold with infinite fundamental group. Closed Virtual Bundle Conjecture. If $M$ is hyperbolic, then some finite-…

Twisted Alexander polynomial detection conjecture. If, for every epimorphism $\alpha:\pi_1(N)\to G$ onto a finite group, $\Delta^{\alpha}_{N,\phi}$ is monic and

Symplectic product conjecture. If $S^1\times N$ is symplectic, then there exists $\phi\in H^1(N)$ such that $(N,\phi)$ fibers over $S^1$.

Let $H$ be the handlebody used to construct tunnel number one 3-manifolds, let $\partial H$ be its boundary, and let $\mathcal{MCG}(\partial H)$ be its mapping class group. Choose…

Let $N$ be a $3$-manifold and let $S^{1} \times N$ admit a symplectic structure $\omega$. Write the Künneth component of $[\omega]$ in $H^{1}(N,\mathbb R)$ as the corresponding cla…

Let $N$ be a $3$-manifold, and let $S^{1} \times N$ admit a symplectic structure $\omega$. Taubes's conjecture. Then $N$ admits a fibration over $S^{1}$. This conjecture relates sy…

Let $Y_1$ and $Y_2$ be fibered 3-manifolds with boundary, with fibrations $\pi_1:Y_1\to S^1$ and $\pi_2:Y_2\to S^1$, and glue them along a fibered torus-boundary component to obtai…

Let $M$ be a 3-manifold that fibers over a circle. Such an $M$ cannot be a homology 3-sphere, but it can be a homology $S^2\times S^1$. Let $H(M,G)$ be the paper's invariant, analo…

Let $X$ be a finite-volume hyperbolic $3$-manifold. A finite cover of $X$ is a covering space of finite degree, and the first Betti number of a space is the rank of its first homol…

For a closed 3-manifold $M$, let ${\mathcal X}^{TV}(M)$ denote the set of homeomorphism classes of closed orientable 3-manifolds having the same Turaev–Viro invariants as $M$. Let…

A fibered 3-manifold is a 3-manifold that is a surface bundle over the circle. A 3-manifold is virtually fibered if it has a finite cover that is fibered. Virtually fibered conject…

Let $(N,\phi)$ be a manifold pair, where $N$ is a $3$-manifold and $\phi\in H^1(N;\mathbb Z)$ is the associated cohomology class. For an epimorphism $\alpha:\pi_1(N)\to G$ onto a f…

Let $M$ be a symplectic $4$-manifold equipped with a free $S^1$-action, and let $N=M/S^1$ be its orbit space. Friedl–Vidussi conjecture. The orbit space $N$ is fibered. This conjec…

Symplectic–fibered cone conjecture. The manifold $S^1\times N$ admits a symplectic structure if and only if $N$ fibers over the circle.