Unsolved Problems

(a) Do there exist closed, oriented, smooth, irreducible 4-manifolds with $b^{+}_{2} >$ 1 and $c^{2}_{1}:=2\chi+3\sigma<0$? (b) Is there an irreducib...

Is there an irreducible, closed, simply connected, oriented 4– manifold with $b^{+}_{2}$ and $b^{-}_{2}$ both even?...

(a) Does there exist a pair of smooth, closed 4-manifolds that are homotopy equivalent but not simple homotopy equivalent? (b) Does there exist a pai...

What are the possible Euler characteristics of closed, aspherical 4-manifolds? More specifically, we ask the following. (a) Is it always the case tha...

Is $*\mathbb{RP}^{4}\#*\mathbb{RP}^{4}$ smoothable? Is *En\#*En smoothable?...

Is every topological closed 4–manifold M the union of submanifolds $Y \cup Z$, where Y is smoothable, Z is acyclic, and $Y \cap Z$ is their common bou...

Let $\pi$ be a good group, and let X be a smooth 4-manifold with $\pi_{1}(X) = \pi$. Does $L^{s}_{5}(\mathbb{Z}[\pi])$ act on the smooth structure set...

(Schoenflies problem). If $\Sigma$ is a smoothly embedded $S^{3}$ in $S^{4}$, then its closed complements are smooth 4-balls....

Let K be a framed knot in $S^{3} = \partial B^{4}$. Let U be a meridian of K. Does there exist a smoothly embedded disk D in $B^{4} \cup _{\nu K} h^{2...

Under what conditions does a closed, orientable 3-manifold M smoothly embed in $S^{4}$? Is this question algorithmically decidable?...

If Y is a homology three-sphere, does the punctured manifold $Y_{0}$ =Y $\setminus \operatorname{Int}(B^{3})$ smoothly embed in $S^{4}$?...

Find exotic 3-balls in $S^{4}$, considered up to isotopy rel. boundary. That is, find a pair of 3-balls $B_{1}, B_{2}$ smoothly embedded in $S^{4}$ wi...

Every closed, orientable 3-manifold embeds smoothly in some connected sum of copies of $S^{2} \times S^{2}$. Given a closed 3-manifold M, let $s(M) \g...

Let $\Sigma$ be a locally flat surface in $S^{4}$ with $\pi_{1}(S^{4} \setminus \Sigma)$ cyclic. (a) Prove that $\Sigma$ is topologically unknotted. ...

Does there exist a pair of closed, oriented surfaces in $S^{4}$ that are topologically but not smoothly isotopic? If such an exotic pair exists, does ...

Does every knot in $S^{3}$ bound an exotic pair of orientable surfaces in $B^{4}$?...

Does there exist a locally flat embedding $f: \Sigma \to S^{4}$ for some closed surface $\Sigma$ such that f is not topologically ambiently isotopic t...

Let $S_{1}, S_{2}$ be two topologically isotopic, smoothly embedded closed surfaces in a closed, oriented, smooth 4-manifold X. When do $S_{1}$ and $S...

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...