21 problems

Diagram-independence conjecture. The spectral sequence in Theorem 1.3 should be independent of the planar diagram $\mathcal{D}$.

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 $(L_+,L_-,L_0)$ be an oriented skein triple, and let $\mathbf{p}$ be a set of basepoints away from the resolutions. Denote by $\widetilde{\operatorname{HMR}}_*(L,\mathbf{p})$ t…

Let $L$ be a link and let $\mathbf{p}$ be a set of basepoints containing exactly one basepoint in each component of $L$. Write $\mathsf{\Sigma}_2(S^3,L)$ for the double branched co…

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 $K$ be a nonzero determinant link. Write $\widetilde{\operatorname{HMR}}_{\bullet}(K)$ for its reduced real monopole Floer homology and let $\operatorname{SWF}_R(K)$ denote the…

Trivial-torsion conjecture. The torsion group $G_{\mathrm{tor}}$ in Lemma 4 is trivial:

Weak symplectic cobordism conjecture. The map

Let $Y$ be an oriented rational homology 3-sphere with an involution or finite-order diffeomorphism $\tau$ as in the paper’s main theorem, and let $\operatorname{Lef}(\tau_*)$ deno…

Let $Y$ be a rational homology sphere with a diffeomorphism $\tau:Y\to Y$ of order $n\geq 2$. Suppose that the quotient $Y/\tau$ is an integral homology sphere $Y'$ and that the br…

Let $Y$ be a $G_2$-manifold, let $\psi$ denote its definite $4$-form, and let $\beta$ be the relevant homology class. Let ${\rm CMA}_\beta^\times(\psi)$ be the refined chain comple…

An $L$-space is a rational homology sphere $Y$ for which the reduced monopole Floer homology $\mathit{HM}(Y)$ vanishes. A taut foliation is a taut foliation on $Y$, and a group is…

Let $L\subset S^3$ be a link, let $\Sigma(L)$ denote its branched double cover, and let $\bar{L}$ be the mirror of $L$. Write $\widetilde{\mathrm{BN}}^3_{*,*}(\bar{L})$ for the rel…

Integral L-space classification conjecture. The connected sums of Poincaré homology spheres, with either orientation, are all the integral homology spheres that are integral $L$-sp…

Derived-groups conjecture. The groups $A'$ and $A”$ are isomorphic, up to a grading shift, to the groups $B'$ and $B”$, respectively.

Let $mathcal{K}0$ and $mathcal{K}1$ be Legendrian knots in $(Y,xi)$, and suppose that $mathcal{K}0prec{Sigma}mathcal{K}1$ is a Lagrangian cobordism of arbitrary genus. Write…

Let $mathcal{K}subset Y$ be a Legendrian knot. Let $S{+}(mathcal{K})$ and $S{-}(mathcal{K})$ denote its positive and negative stabilizations, respectively, and let…

Let $mathcal{K}subset(Y,xi)$ be a Legendrian knot, and let $ellg(mathcal{K})in KHM(-Y,K)otimesmathcal{R}$ denote the invariant constructed using a closure of genus $g$, for…

Let $mathcal{K}$ be a Legendrian representative of a topological knot $K$. The invariant $ell(mathcal{K})$ is defined using sutured monopole Floer homology, while the LOSS invari…

Let $\mathcal{S}^n=T(3,6n+1)$ and $\mathcal{T}^n=T(3,6n-1)$ for $n greater than or equal to 1$. Define the auxiliary polynomials $fn(t,q)=sum{k=0}^{n-1}t^8q^{12}$ and…

Let $T(3,6n+1)$ and $T(3,6n-1)$ be the indicated families of torus knots, and let $E^k$ denote the pages of the link surgery spectral sequence associated to a choice of analytic da…