103 problems

Atiyah–Floer conjecture. Under appropriate hypotheses, there should be an isomorphism

Kronheimer–Mrowka conjecture. Appropriate versions of instanton Floer homology and Heegaard Floer homology are isomorphic.

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 $M$ be a $3$-manifold presented by a Heegaard splitting with Heegaard surface a Riemann surface, and let the associated Lagrangians lie in the moduli space of flat connections…

A selector is a minimax-type invariant assigned to a Hamiltonian, and a symplectic manifold is said here to admit a nice Floer homology when its Floer homology has the properties r…

The ν–barmu conjecture. The $\nu$-invariant equals the $\bar\mu$-invariant for every Seifert fibered homology sphere $Y$:

Let $L\hookrightarrow T^4$ be a 2-dimensional Lagrangian submanifold in the Lagrangian torus fibration $T^4\rightarrow T^2$, endowed with a flat connection, and let…

Fibred Dehn twist exact sequence conjecture. Under conditions which render $a$ and $b$ well-defined, they fit into a long exact sequence

Let $\Sigma$ be a surface and let $L,L'\subset\Sigma$ be disjoint circles. Write $\widehat{V}_L^n$ and $\widehat{V}_{L'}^n$ for the associated Lagrangian correspondences between th…

Let $(Y,\pi)$ be a fibred three-manifold, let $\operatorname{Sym}^n_{S^1}(Y)$ be its relative symmetric product, and let $\Omega\in\Omega^2(\operatorname{Sym}^n_{S^1}(Y))$ be a clo…

Let $Y$ be a closed oriented $3$-manifold equipped with a fibration $f:Y\to S^1$ of fiber genus at least $2$, let $h\in H_1(Y;\mathbb{Z})$, let $c\in H^2(Y;\mathbb{R})$ evaluate po…

Owens' H-thin conjecture. For any H-thin knot $K$,

Floer-theoretic string topology operations conjecture. For every marked chord diagram $\Gamma$, there is an umkehr map

ECH–Floer correspondence conjecture. Up to a grading shift,

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

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

Let $f_-$ be the map defined by the holomorphic triangles in the surgery construction, and let $\delta_1$ be the differential component described in the comparison with Ozsváth–Sza…

Let $\Sigma$ be an integral homology sphere, let $\tau:\Sigma\to\Sigma$ be an orientation-preserving diffeomorphism, and let $X$ be its mapping torus. Let $\lambda_{FO}(X)$ be the…

Let $\Sigma$ be an integral homology sphere, let $\tau:\Sigma\to\Sigma$ be an orientation-preserving diffeomorphism, and let $X$ be its mapping torus. Let $\lambda_{FO}(X)$ be the…

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

General Floer homology and asymptotic growth conjecture.

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 $Y$ be a three-manifold with $b_1(Y)=1$ and a nontorsion $spin^c$ structure, and let ${\text{SWF}}$ and ${\text{SWF}}_0$ denote the associated periodic Seiberg–Witten pro-spect…