33 problems

In the Artin spun case, let knots be called essentially different when they are not isotopic, are not mirrors of one another, and are not isotopic under mirroring of connect-summan…

Let $(M^{4},g)$ be a compact anti-self-dual Einstein manifold with negative scalar curvature. Anti-self-dual Einstein vanishing conjecture. All the Seiberg–Witten invariants vanish…

Némethi–Nicolaescu conjecture. For certain singularities,

Let $N$ be a rational homology sphere, and let the augmentation-free part of its Seiberg–Witten invariant be denoted by the corresponding augmentation-free invariant. Let the refin…

Let $N$ be a rational homology sphere. Seiberg–Witten–Frøy conjecture. For every rational homology sphere $N$ we have [the asserted formula is not present in the supplied source sp…

Let $X$ be a closed, oriented, smooth four-manifold with $b_2^+(X)>0$. Let ${\mathfrak{s}}={\mathfrak{s}}_0\otimes L_1$ be a $\operatorname{Spin}^c$ structure defined by…

Let $R=R_8$ be the manifold obtained in Case 1 by rationally blowing down the configuration $C_3$ in $P_{10}$. The construction gives that $R$ is homeomorphic to $P_8$, and its Sei…

For every $5\leq m\leq 8$, let $R_m$ be a manifold homeomorphic to $P_m$, constructed so that it has vanishing Seiberg–Witten invariants and contains a nullhomologous torus with th…

Let $V$ be a surface with $p_g(V)>0$, let $K$ be an effective canonical divisor, and fix $m\in H^2(V,\mathbb{Z})$. Let … be the zero-cycle on…

Let $(V,g)$ be a surface with a Kähler metric $g$, let $m\in H^2(V,\mathbb{Z})$, and let $\beta$ be a real closed $2$-form of type $(1,1)$. Let $\tau$ be a $Spin^c(4)$-structure re…

Let $V$ be a smooth projective surface. Denote by $P^\bullet_V$ its Poincaré invariants and by $SW^\bullet_V$ its full Seiberg–Witten invariants, with the canonical orientation dat…

Poincaré–Seiberg–Witten correspondence conjecture. If $p_g(V)=0$, then

Poincaré–Seiberg–Witten conjecture. This invariant coincides with the full Seiberg–Witten invariant computed with respect to the canonical orientation data.

Let $M$ be a $G_{2}$-manifold, and let $C$ and $C^{\prime}$ be nearby coassociative submanifolds of $M$. Consider instantons in $M$ with small volume and boundary lying on…

Let $M^{4}=\overline{{\mathbb C}{\mathcal H}_{2}/\Gamma}$ be a reverse-oriented compact complex-hyperbolic $4$-manifold. Reverse-oriented complex-hyperbolic vanishing conjecture. A…

Vanishing conjecture.

Let $\ell$, $F$, $L_1$, $\mathfrak{s}_0$, $SW(\mathfrak{s}_0\otimes L_1)$, and $Q_X$ denote the level, bundle and line-bundle data, Seiberg–Witten invariant, and intersection form…

Let $X$ be a four-manifold, let $E$ be a rank-two bundle, and let $z\in\mathbb A(X)$. Let $n_a$, $d_a$, $\overline{\operatorname{Red}}(W,E)$, $\overline V(z)$,…

Knot and link replacement faithfulness conjecture. $X_K$ is diffeomorphic to $X_{K'}$ if and only if $K$ is isotopic to $K'$.

Let $S$ be the surface and $M$ the moduli space defined above, let $s\in\mathbb Z$, and let $\alpha\in K(S)$ have $\operatorname{rk}(\alpha)=s$. The universal functions…

Let $Z_1$ denote the quotient of $S^2\times S^2$ by the fixed-point-free involution that applies the antipodal map to both $S^2$ factors. For the manifolds and values of $l$ descri…

Let $X$ be a compact, oriented, smooth spin $4$-manifold with $b_+(X)>1$, let $\mathfrak{s}$ be a spin structure on $X$, and let $d(X,\mathfrak{s})$ be the dimension of the Seiberg…

Let $X$ be a simply connected, closed, oriented, smooth $4$-manifold with $b_2^+(X)>1$ and $b_2^-(X)>1$. Orientation-vanishing conjecture. The Seiberg–Witten invariant vanishes for…

Let $X$ be a closed, connected, oriented, smooth $4$-manifold with $b_2^+(X)>1$. Simple type conjecture. $X$ has Seiberg–Witten simple type. This conjecture asserts a fundamental c…

Seiberg–Witten invariant conjecture. The equivariant geometric genera are topological and are given by the Seiberg–Witten invariant of the link.