Unsolved Problems

Let $\Sigma$ be a surface embedded in $S^{4}$. Can $\Sigma$ be unknotted by a sequence of torus surgeries in its complement, such that the ambient man...

(a) Give an algebraic classification of groups that arise as the fundamental group of the complement of a smooth or locally flat 2-sphere in $S^{4}$. ...

Are homotopy types of 2-knot complements determined by their homotopy 2-types?...

(Kinoshita conjecture). Does every projective plane in $S^{4}$ decompose as a connected sum of a knotted 2-sphere and an unknotted projective plane?...

Let $\Delta \subset B^{4}$ be a ribbon disk. Is $B^{4} \setminus \Delta$ aspherical, i.e. is $\pi_{i}(B^{4} \setminus \Delta)$ =0 for all i>1?...

Let K be a closed surface smoothly embedded in a 4-manifold $M^{4}$. Describe the subgroup of the mapping class group $Mod(K)$ of diffeomorphisms of K...

Is every link of 2-spheres in $S^{4}$ slice?...

Let $\Sigma$ be a compact surface and let X be a connected 4manifold. Let $f_{0}, f_{1}: \Sigma \to X$ be $\pi_{1}-negligible$ embeddings that agree o...

(a) If X is a smooth, closed, simply connected 4-manifold $withb_{2}(X) \geq$ 2, are all knots slice in X? (b) In particular, are all knots slice in ...

Let K be a knot on the boundary of $X \setminus B^{\circ 4}$, where X is a negative definite, smooth 4-manifold. Suppose K bounds a smoothly embedded,...

(a) Let X be a closed, simply connected 4-manifold. Let $g \geq$ 0 and $d \geq$ 0 be integers. Fix $x \in H_{2}(X;\mathbb{Z})$. Does there exist a (sm...

Let X be a closed simply connected smooth 4-manifold and let $\Sigma$ be a closed, orientable surface. Fix a smooth embedding of $f: \Sigma \hookright...

Are all groups good?...

(Round handle problem). Is there a link $L \subset S^{3}$ with vanishing pairwise linking numbers that is not round handle slice?...

Let $\Delta$ be a contractible, compact 4-manifold. Is the space of homeomorphisms of $\Delta$ that fix the boundary pointwise, with the compact-open ...

Give an effective necessary and sufficient condition for an open 4-manifold to be homeomorphic to the interior of a compact 4-manifold....

Classify closed topological 4-manifolds (orientable and not) with finite fundamental group, up to homeomorphism. The following infinite families of fu...

Let X be a closed, smooth 4-manifold with fundamental group isomorphic to $\mathbb{Z}$. Is the $\mathbb{Z}[\mathbb{Z}]-valued$ intersection form on $\...

Let M and N be closed, orientable topological 4-manifolds with $\pi_{1}(M) \cong \pi_{1}(N)a$ good group. Suppose that M and N are simple homotopy equ...

Does there exist an algorithm that takes as input a closed, triangulated 4-manifold, and outputs in finite time whether or not that 4-manifold is home...

The quadratic 2 type of a 4-manifold M is the data $(\pi_{1}(M), \pi_{2}(M), \lambda_{M}, k_{M})$ of the fundamental group $\pi_{1}(M)$, the second ho...

Let M and N be closed, orientable, connected 4-manifolds with isomorphic quadratic 2-types. If $\pi_{1}(M) \cong \pi_{1}(N)are$ finite, are M and N ho...

(4D s-cobordism conjecture). Let $(W^{4};M_{0}^{3}, M_{1}^{3})$ be a smooth 4-dimensionals-cobordism between closed 3-manifolds. Is W diffeomorphic to...

Let X and Y be closed, oriented, smooth 4-manifolds with the same Euler characteristic and signature. Is there a torus link L in X with trivial normal...

(a) Which Seifert fibered homology spheres $\Sigma(a_{1}$, . . . , $a_{n})bound$ acyclic manifolds? Are there any examples with four or more singular ...

Are lens spaces topologically homology cobordant if and only if they are homeomorphic?...

(a) Let X be an open, spin, smooth4-manifold. Does X have a proper smooth embedding in $\mathbb{R}^{6}$? (b) By choosing a proper exhaustion function...

What do different 4-manifold gauge theories see?...

Let X be a smooth, closed, connected, oriented 4-manifold with $b^{+}_{2}(X)$ >1. (a) Does X have Donaldson simple type? (b) Does X have Seiberg–Wit...

How many independent basic classes can a simply connected smooth 4-manifold X have, as measured $bybr(X)$, the rank of the span of the basic classes? ...

Find an irreducible, closed, smooth 4-manifold with nontrivial Bauer–Furuta invariant but with trivial Seiberg–Witten invariant....

Suppose X is a smooth4-manifold with the homology of $S^{1} \times S^{3}$ whose infinite cyclic cover $\widetilde{X}$ has $H^{1}(\widetilde{X})=0$. Fu...

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?...