14 problems

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

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…

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…

Let $M$ be a closed oriented 3--manifold. Taubes' conjecture. The product $S^1 \times M$ admits a symplectic structure if and only if $M$ fibers over $S^1$. The conjecture relates…

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…

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…

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…

Thurston's virtual fibration conjecture. Every finite-volume hyperbolic $3$-manifold has a finite cover that is fibered.

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…

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…