72 problems

Let $Y$ be a $3$-manifold and let $\xi$ be a contact structure on $Y$. Let ${\rm Tor}(Y,\xi)$ denote the torsion, defined as the supremum of the integers $n\geq 1$ for which there…

Let $L$ be a transverse representative of an alternating smooth link. Write $\psi(L)\in\widetilde{Kh}(L)$ for the transverse-link invariant in reduced Khovanov homology, let…

Let $Y$ be a rational homology sphere with $|H_1(Y;\mathbb Z)|=h$. Suppose that $Y$ bounds a negative-definite four-manifold $X$ whose first homology has no torsion. Write…

Equivariant Seiberg–Witten Floer and Heegaard Floer correspondence. For every such $Y$ and $\mathfrak{s}$, there are isomorphisms

Spectral sequence differential conjecture. The spectral sequence collapses after the $E_3$ stage, and its differential

Let $Y$ be a three-manifold with $b_1(Y)=1$ and a nontorsion $spin^c$ structure ${\mathfrak{s}}$. Let ${\text{SWF}}$ and ${\text{SWF}}_0$ be the associated periodic Seiberg–Witten…

Let $(Y,\tau,\mathbf{p})$ be a multi-based real three-manifold, let $\widetilde{\operatorname{HMR}}_*(Y,\tau,\mathbf{p})$ denote framed real monopole Floer homology, and let…

Let $W_4$ denote the 4-twisted Whitehead link, a two-component link formed from the fourth twist knot and an unknot. Ribbon minimality conjecture. The link $W_4$ is ribbon minimal.…

L-space conjecture. The following statements are equivalent:

Let $C$ be the underlying curve, let $\Sigma$ be a real exact Lagrangian in $T^*C$, and let $q_1,\dots,q_\kappa$ be points of $C$. The maps … and … are respectively defined by holo…

Let $Y$ be a $3$-manifold and $K\subset Y$ a knot, with meridian $\mu$. Let $N$ be the twisted $I$-bundle over the Klein bottle, and let $Y_N^\mu$ be the N-filling of…

Let $(Y,\iota)$ be a real three-manifold with a compatible real $\mathrm{spin^c}$ structure $\mathfrak{s}$. Let $\widetilde{\mathit{HMR}}$, $\widecheck{\mathit{HMR}}$,…

Let $Y$ be a rational homology $3$-sphere. The SU(2)-abelian implies Heegaard Floer L-space conjecture. If $Y$ is $SU(2)$-abelian, then it is a Heegaard Floer L-space. Here, a Heeg…

Ghiggini's conjecture. The contact invariant vanishes:

Let $S$ be a genus $g$ surface embedded in a homology sphere $Y$, and let $H$ be an abelian subgroup of the Torelli group. Boundedness conjecture. The set … is bounded. This conjec…

Let $L$ be a lens space and let $K\subset L$ be a knot. A knot is Floer simple when its Floer-theoretic complexity is minimal, and it is simple when it belongs to the standard clas…

Lin geography restriction conjecture. The Heegaard Floer module $\operatorname{HF}^-(Y)$ satisfies the Lin geography restriction for all rational homology spheres $Y$.

The $L$-space conjecture. The following statements are equivalent:

Let $(M,\Gamma,\xi)$ be a sutured contact $3$-manifold. For $A\in H_1(M;\mathbb{Z})$, let $\mathfrak{s}_\xi$ be the canonical Spin$^c$-structure determined by $\xi$, let…

The $ADE$ Link Conjecture. If $L$ is not an $ADE$ link, then all cyclic branched covers $cSigma_{\psi}(L)$ of $L$ are $NLS$, $LO$, and $CTF$.

Let $\widehat{\sigma}$ be the link associated to a braid $\sigma$, and let $Kh^{\sharp,k}(\widehat{\sigma})$ denote the degree-$k$ part of the higher-dimensional Floer homology, wh…

Let $E_0$ and $E_1$ be the objects appearing in the surgery-formula construction, and let $\operatorname{End}_{w\mathrm{Fil}}(\beta_0^{E_0}\oplus\beta_1^{E_1})$ denote the endomorp…

Let $\mathcal{C}$ be the smooth knot concordance group. A satellite operator $P$ is rank-expanding along $\{nK\}_{n\in\mathbb{Z}}$ if $\{nK\}_{n\in\mathbb{Z}}$ is a rank-one subgro…

Let $\Sigma$ be the surface appearing in the definition of the higher-dimensional Heegaard Floer algebra, let $q_i$ be the chosen points, and let $c$ denote the parameter in…

Let $G$ be a plumbing tree, let $Y(G)$ be its associated plumbed 3-manifold, and let $\boldsymbol{\mathbf{\mathit{HF}}}^-(Y(G))$ denote the completed Heegaard Floer homology. Write…