Unsolved Problems

Is there a closed oriented smooth 4-manifold X for which every finite subgroup Gof the mapping class group $\pi_{0}(\operatorname{Diff}^{+}(X))can$ be...

Is there a closed orientable smooth 4-manifold X for which the identity component $Diff_{0}(X)$ of the diffeomorphism group is not uniformly perfect?...

Is it the case that for every closed, smoothable topological 4manifold X, there exists a locally linear finite group action on X, such that for every ...

Is there an exotic action of $\mathbb{Z}/n$ on $S^{4}$ with 0-dimensional fixed point set? 1-dimensional? 2-dimensional?...

Let $\tau: S^{4} \to S^{4}$ be a free (hence orientation-reversing) involution. Is there an embedded $S^{2} \subset S^{4}$ that is invariant under $\t...

Classify smooth, effective circle actions on simply connected 4-manifolds with boundary. (a) Classify simply connected4-manifolds with boundary that ...

Do the Chern numbers $c^{2}_{1}$ and $c_{2}$ of every closed, symplectic 4–manifold X that is not a ruled surface satisfy the following? (a) $c^{2}_{...

Present a topological construction of symplectic fake projective planes. Does there exist a symplectic fake projective plane that is not a complex bal...

Is every symplectic Calabi-Yau surface diffeomorphic to either the K3 surface, the Enriques surface or $a T^{2}-bundle$ over $T^{2}$?...

Is every symplectic form on the standard K3 surface symplectomorphic to a Kähler form?...

Are homotopy equivalent Horikawa surfaces in different deformation classes diffeomorphic as 4–manifolds? Are they symplectomorphic?...

(a) Is there a closed hyperbolic oriented 4-manifold that admits a symplectic structure? (b) Do the Seiberg–Witten invariants vanish on every closed ...

Does there exist a pair of symplectic 4–manifolds $(X_{1}, \omega_{1})and (X_{2}, \omega_{2})$, where there is a diffeomorphism $f: X_{1} \to X_{2}$ s...

Let $\lambda:=c^{2}_{1}/c_{2}$ be the Chern slope of a closed, almost complex 4–manifold X. What is the supremum of $\lambda as X$ ranges over the fol...

Does every closed symplectic 4-manifold admit inequivalent Lefschetz pencils with the same fiber genus g, for sufficiently large g? How about infinite...

Let X be a closed symplectic 4-manifold. Let $T \subset X$ be a symplectic submanifold that is diffeomorphic to a 2-dimensional torus such that $[T]^{...

Given a closed, connected, symplectic 4-manifold (X, $\omega)$ and $c \in H_{2}(X,\mathbb{Z})$ represented by an embedded, connected, oriented, smooth...

Is every smooth symplectic surface in $(\mathbb{CP}^{2}, \omega_{FS})$ symplectically isotopic to a complex curve? Equivalently, is there a unique sym...

Is every symplectic rational cuspidal curve in $(\mathbb{CP}^{2}, \omega_{FS})$ equisingularly symplectically isotopic to a complex curve? More genera...

(a) What polynomials can occur as the Alexander polynomials of complex plane algebraic curves? (b) More generally, what are the conditions that must ...

Does there exist a transverse link $L \subset (S^{3}, \xi_{std})$ bounding a pair of complex curves in $B^{4} \subset \mathbb{C}^{2}$ that are isotopi...

Does there exist a planar contact 3-manifold that has infinitely many distinct Stein fillings?...

Is the exact symplectomorphism type of $T*X^{4}$ sensitive to the smooth structure on a 4-manifold X, or does it depend only on the simple-homotopy or...

Problems about contact hypersurfaces: (a) Let (Y, $\xi)$ be a contact manifold and $(\mathbb{R} \times Y, \omega)$ its symplectization. Let f:Y $\to ...

Let $W_{+}$ and $W_{-}$ be two 4-dimensional Liouville domains with a contactomorphism $\Phi :\partial W_{-} \cong \partial W_{+}$. This determines an...

If $\Sigma \subset$ (X, $\omega)$ is a symplectic surface in a closed symplectic 4-manifold with $[\Sigma] = P D(k[\omega]), k \in \mathbb{Z}$, does (...

Does there exist a 2-handlebody W that admits an exact symplectic structure with convex contact boundary that does not admit a Weinstein structure fil...

Is trisection genus additive? In other words, must it be the case that $g(X\#X1)$ =g(X) +g(X1)....

Is every trisection of the 4-sphere with positive genus a stabilization of the genus zero trisection?...

Which closed, oriented, smooth 4–manifolds admit genus–3 trisections? Which ones admit genus–3 simplified trisections? How about genus–4?...

For a given Heegaard splitting of a closed orientable3–manifold, classify self-indexing Morse functions that give the given Heegaard splitting, up to ...

(a) Find two diffeomorphic but non-isotopic trisections of the same4–manifold. (b) Find two non-diffeomorphic balanced trisections of the same genus ...

Is there an algorithm to compute ‘distance’ in the cut complex of a trisection? Is the L-invariant computable?...

Let X be a closed, orientable, smooth 4-manifold, with $g(X)$ the trisection genus of X. Does $g(X) =\chi(X) -2+3rk(\pi_{1}(X))$?...

Does every simply connected, closed, smooth 4-manifold admit a handle decomposition without any 1-handles? Without 1-handles and 3-handles?...

Is every topological 4-manifold homeomorphic to a CW complex?...

Which closed, smooth 4–manifolds admit achiral Lefschetz pencils? Does every simply connected 4–manifold have one?...

Which closed, smooth4–manifolds admit open book decompositions? In particular, does every closed, simply connected 4–manifold with signature zero admi...

Is there a universal branching surface $S \subset S^{4}$ such that every closed, orientable 4-manifold W admits a branched covering $W \to S^{4}$ with...

Is every closed leaf of a two dimensional co-orientable smooth taut foliation of an oriented 4-manifold smoothly genus-minimizing in its homology clas...

Does there exist a hyperbolic integer homology four-sphere? What about an arithmetic one? Homology four-spheres have Euler characteristic 2, so it mak...

Is there a noncompact, finite volume, orientable hyperbolic four-manifold without a spin structure?...

(a) If M is a closed, orientable hyperbolic 4-manifold then it always has signature 0, because its Pontryagin class vanishes [Che55]. This implies tha...

Given an aspherical closed (or compact and bounded by flat 3-manifolds) 4-manifold M and a self-diffeomorphism f of M, find necessary and sufficient c...

What is the structure of 4-manifolds that admit a Riemannian metric of positive scalar curvature? There are variations of this problem for different c...

Given a closed, 4-dimensional PSC manifold M, is there a (possibly disconnected) 4-dimensional orbifold $M^{1}$ with isolated singularities such that ...

Does longitudinal knot surgery using a knot K, along a fiber in a K3 surface always yield a reducible 4–manifold? A completely decomposable 4–manifold...

Does every Lipschitz 4-manifold admit a smooth structure? Is this smooth structure unique if so? Some more specific, related questions are as follows....

Does every cellular set in the plane have the fixed point property?...

(Doubly-Small Morphisms of Manifolds). - Suppose that $h: \R^{n} \to \R^{n}$ is a homeomorphism (or diffeomorphism) which satisfies two smallness hyp...