3 problems

Let $M\neq S^3$ be a closed orientable $3$-manifold, let $[S^1,M]$ denote the set of free homotopy classes of loops in $M$, and let $\mathcal{C}^M_x$ be the set of concordance clas…

Let $M$ be a closed orientable $3$-manifold and let $x\in[S^1,M]$ be a free homotopy class. Celoria–Friedl–Nagel–Orson–Powell conjecture. Aside from certain exceptional cases, ther…

Let $M$ be a closed orientable $3$-manifold with $M\neq S^3$, and let $x\in[S^1,M]$ be a free homotopy class. A dual $2$-sphere for $x$ is a $2$-sphere in $M$ representing the rele…