24 problems

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

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…

Kronheimer–Mrowka's conjecture. These equivariant Seiberg–Witten Floer homologies are well-defined, meaning that they are independent of the particular choices of metrics and pertu…

Let $X_1$ be a cobordism between three-manifolds $Y_1$ and $Y_2$, and let $X_2$ be a cobordism between $Y_2$ and $Y_3$. Denote by $\mathcal{D}_{X_1}$, $\mathcal{D}_{X_2}$, and…

Let $X$ be a closed $3$-manifold with $b_1(X)>0$, let $o$ be a homology orientation, and let $\operatorname{Eul}(X)$ be the set of Euler structures. Turaev’s torsion defines a map…

Let $X$ be a closed $3$-manifold with $b_1(X)>0$, let $o$ be a homology orientation, and let $\operatorname{Spin}^c(X)$ denote the set of spin-c structures on $X$. Let…

Let $X$ be a manifold of simple type. For a choice of $Spin_c$-structure $L$, let $M_L$ be the Seiberg–Witten moduli space and let $d=\dim(M_L)$. Simple-type conjecture. The only n…

Let $Y^+$ be a cylindrical-end $3$-manifold with harmonic Morse function $f^+$, let $Z$ be the set of its index-$1$ and index-$2$ critical points, and let a t-orbit be a weighted c…

Let $X$ be a closed, connected, oriented, smooth four-dimensional manifold with $b^+(X)>1$. A symplectic four-manifold is a four-manifold equipped with a symplectic form. Bogomolov…

Let $X$ be a closed, connected, oriented, smooth four-dimensional manifold with $b^+(X)>1$ and odd $b^+(X)-b_1(X)$. A Seiberg–Witten basic class is a cohomology class associated wi…

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…

Simple type conjecture. Every closed, connected, oriented and smooth four-manifold with $b_2^+\ge 2$ is of simple type.

Universal linear bound conjecture. There is a universal constant $C$ such that

Let $X$ be a smooth spin rational homology $S^1\times S^3$ which is oriented and homology oriented, and let $-X$ denote $X$ with reversed orientation but the same homology orientat…

Let $X$ be a smooth closed oriented $4$-manifold such that … where $\widetilde X$ is the universal abelian cover of $X$. The invariants $\lambda_{\,\rm{SW}}(X)$ and…

Let $S$ be a smooth projective surface satisfying $H_1(S,\mathbb Z)=0$ and $p_g>0$. For any polarization $H$ and first Chern class $c_1$, let…

Let $a\in\{\operatorname{cob},\operatorname{ell}\}$, let $S$ be a smooth projective surface with $b_1(S)=0$ and $p_g(S)>0$, and choose $H,c_1,c_2$ so that there are no rank $2$ str…

Let $M$ be the underlying manifold, and let $(\varepsilon_i,\Psi_i,{\bm\xi}_i,A_i)$ be a sequence of solutions of the blown-up ADHM$_{1,k}$ Seiberg--Witten equation satisfying … wi…

A blow-up set $Z$ for the Seiberg–Witten equations with two spinors on a three-dimensional manifold is a closed subset arising as the blow-up set of a sequence of solutions. An int…

Small perturbation Seiberg–Witten conjecture. Under these assumptions, $M$ satisfies property $(\ast)$.

Naturality conjecture. The diagram of filtered Seiberg–Witten Floer cohomology groups, with horizontal maps given by the chain maps of and vertical maps given by the canonical iden…

Let $X$ be a closed oriented smooth 4-manifold with $b_2^+(X)\geq 2$, let $\mathfrak{s}\in\operatorname{Spin}^c(X)$, and let $d(\mathfrak{s})$ be the expected dimension of the Seib…

Furuta–Ohta–Seiberg–Witten correspondence conjecture.

Let $X$ be a smooth $4$-manifold with the $\mathbb Z[\mathbb Z]$ homology of $S^1\times S^3$. Denote by $\lambda_{\,\operatorname{SW}\,}(X)$ the Seiberg–Witten invariant and by…