3 problems

Let $M_i$ ($i=1,2$) be irreducible 3-manifolds with connected boundary, with $\partial M_1\cong\partial M_2\cong F$. Let $\mathcal{D}_i$ be the set of essential curves in $F$ that…

Let $M$ be a compact orientable 3-manifold with boundary component $F$, let $V\cup_{S}W$ be a Heegaard splitting of $M$ with $F\subset V$, and let $r$ be an essential simple closed…

Let $\mathcal C$ be the curve complex of the surface $\Sigma$, let $\mathcal{EL}$ be its ending lamination space, and let $MCG(\Sigma)$ be the mapping class group of $\Sigma$. Writ…