4 problems

Virtual Betti number conjecture. One has

Let $M$ be a symplectic $4$-manifold with canonical class $K=0$, and let $vb_1(M,\Bbb F_p)$ denote its virtual first Betti number with coefficients in the finite field $\Bbb F_p$.…

Li–Ni's conjecture. If $M$ is a closed oriented reducible $3$-manifold, then

Let $M$ be a hyperbolic $3$-manifold. Property $(FAb)$ means that every finite-index subgroup has finite abelianization. Thurston–Waldhausen's VPBN conjecture. The manifold $M$ has…