Unsolved Problems

Can the skein lasagna module detect exotic smooth structures on closed 4-manifolds?...

(a) Compute $\pi_{0}(\operatorname{Diff}^{+}(S^{4}))$. Do we have $\pi_{0}(\operatorname{Diff}^{+}(S^{4})) =$ \{1\}? (b) In particular, does some imp...

Does every closed smooth 4-manifold admit an exotic diffeomorphism? How about the following special cases? (a) Is there a definite smooth closed 4-ma...

Does there exist a diffeomorphism of a closed3-manifoldf: $M \to M$ such that f is topologically but not smoothly pseudo-isotopic to the identity?...

Do there exist $k \geq$ 0 and a smooth closed 4-manifold X such that the map $\pi_{k}(\operatorname{Diff}(X)) \to \pi_{k}(\operatorname{Homeo}(X))$ in...

(a) Do there exist $k \geq$ 0 and a smooth closed orientable 4-manifold X such that $\pi_{k}(\operatorname{Diff}(X))$ is finitely generated? (b) Do t...

Let X be a closed orientable topological 4-manifold with finite $\pi_{1}(X)$. (a) Is $\pi_{k}(\operatorname{Homeo}(X))$ finitely generated for every ...

Does the Morlet correspondence $BDiff_{\partial}(D^{n}) \cong \Omega^{n}_{0}(Top(n)/O(n))$ (10) hold for n=4?...

Does there exist a closed, smooth 4-manifold X and a diffeomorphism f: $X \to X$ such that f is smoothly pseudo-isotopic to the identity, but f is not...

Let X be a connected smooth 4-manifold with nonempty boundary, with finite $\pi_{1}(X)$, and let $k \geq$ 0. Let $Diff_{\partial}(X)$ denote the group...

Let X be a closed, oriented, simply connected, smooth 4manifold and fix $k >$ 0. Is there $N \geq$ 0 such that, for every $n \geq N$, the natural map ...

For which $k \geq$ 0 and closed smooth 4-manifold X does the equality $\ker(i_{*}: \pi_{k}(Diff_{\partial}(X^{\circ})) \to \pi_{k}(Homeo_{\partial}(X^...

Let X be a simply connected closed smooth 4-manifold. Does $BDiff(X)$ satisfy homological stability over $\mathbb{Q}$?...

Given $k >$ 0, is there a closed, simply connected, smooth 4manifold X and a nonzero homotopy class $\alpha \in \pi_{k}(Diff_{\partial}(X^{\circ}))$ s...

Is there $n >$ 2 and a smooth closed simply connected 4manifold X for which there is an element of ordernin the $subgroupker(\pi_{0}\operatorname{Diff...

Is there a smooth, closed, simply connected 4-manifold X for which the group $\ker(\pi_{0}(\operatorname{Diff}(X)) \to \pi_{0}(\operatorname{Homeo}(X)...

Let $\phi$ be a self-diffeomorphism of a closed, simply-connected, smooth 4-manifold X. Suppose that for every smooth surface $\Sigma$ in X, the surfa...

For which 4-manifolds does there exist a smooth structure such that there exists a non-smoothable homeomorphism with respect to that smooth structure?...

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 ...