143 problems

Let $M$ be a rational homology $3$-sphere. Its fundamental group is left-orderable if it admits a left-invariant total order, and $M$ supports a co-orientable taut foliation if it…

Let $Y$ be a hyperbolic integral homology sphere, and let $\lambda_1^*$ denote the first positive eigenvalue of the Laplacian on coexact $1$-forms. Ozsváth–Szabó spectral-gap predi…

Let $K$ be a knot, let $s(K)$ denote Rasmussen's invariant from Lee homology, and let $\tau(K)$ denote Ozsváth–Szabó's invariant from Heegaard Floer homology. Rasmussen's conjectur…

Let $Y_\Gamma$ be a negative-definite plumbed rational homology $3$-sphere with $N$ plumbing vertices, and let $\mathfrak{s}_{\rm can}$ denote its canonical ${\rm Spin}^c$ structur…

Let $L\subset S^3$ be a link, let $\overline{OKh}(L)$ denote its reduced odd Khovanov homology, and let $\Sigma(L)$ denote the branched double-cover of $S^3$ branched along $L$. Oz…

Ozsváth–Szabó's degeneration conjecture. All subsequent differentials in the universal coefficient spectral sequence vanish, so that its $E^\infty$ term is $HC^\infty_*(Y)$.

Ghiggini's conjecture. The contact invariant vanishes:

In Example 1, let $Y$ be a closed connected $3$-manifold with a Morse function having a unique maximum and minimum, and let $\underline{Y}$ and $\underline{f}$ be the associated cy…

Let $Y$ be an irreducible $L$-space, where an $L$-space is a rational homology sphere satisfying $\dim \widehat{\mathit{HF}}(Y)=|H_1(Y)|$. Ozsváth–Szabó's conjecture. The manifold…

Involutive Heegaard Floer–Seiberg–Witten correspondence. For every rational homology sphere $Y$ with a self-conjugate $\operatorname{spin}^c$ structure $\mathfrak{s}$, there is an…

Let a closed graph manifold be decomposed along incompressible tori into Seifert fibered JSJ components. Properties $(1a)$, $(1b)$, and $(2)$ denote, respectively, the existence of…

Let $(Y,\gamma)$ be a sutured manifold, and let $ECH_{\mathrm{sut}}(Y,\gamma)$ and $SFH(Y,\gamma)$ denote its sutured embedded contact homology and sutured Heegaard Floer homology,…

Ozsváth–Szabó's differential conjecture. The only possibly nontrivial differential in this spectral sequence is $d^3$, and this differential is given by a formula in terms of the t…

Planar torsion vanishing conjecture. If $(M,\xi)$ has planar torsion of any order, then its untwisted Ozsváth–Szabó contact invariant vanishes; moreover, the twisted invariant vani…

A contact 3-manifold $(Y,\xi)$ has a contact invariant $c(\xi)\in\widehat{HF}(-Y)$, and Giroux torsion means that the contact structure contains a positive amount of Giroux torsion…

A right-veering open book is an open book whose monodromy is right-veering, and it is minimal when it is not destabilizable. Minimal right-veering open book conjecture. Every minim…

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…

Reduced branched-cover contact-invariant conjecture. The invariant $psi(L)$ is mapped to the Ozsváth–Szabó contact invariant $c(xi_L)$ under this spectral sequence.

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…

CMT–CF comparison conjecture. For all sufficiently large $L$, there is an isomorphism of $R[U]$-modules

Heegaard–Monopole–Taubes comparison conjecture. There are isomorphisms of $R[U]$-modules

Heegaard Floer–Seiberg–Witten Floer equivalence conjecture. There are isomorphisms of $R[U]$-modules

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…

Heegaard Floer homology and Seiberg--Witten monopole Floer homology are two constructions of Floer-theoretic invariants associated with three-manifolds. Isomorphism conjecture. The…

Ozsváth–Szabó–torsion–Casson–Walker identity. Under the stated vanishing assumption, the sign-refined Reidemeister–Turaev torsion and the Casson–Walker invariant satisfy