Unsolved Problems

How do Floer homological invariants behave under maps of nonzero degree? For instance, let $Y$ and $Z$ be closed, oriented 3-manifolds, and suppose th...

Give a method for computing the $\eta$ invariant for the Dirac operator, $\eta_{\mathrm{Dirac}}(Y, s)$, associated to a spin structure s on a hyperbol...

Let A be a flat connection on the trivial $\operatorname{SU}(2)$ bundle on a closed three-manifold M. The Chern--Simons invariant $\operatorname{CS}(M...

Let $S_{2,\infty}(Y)$ denote the Kauffman bracket skein module of a closed, oriented 3-manifold $Y$; this is a module over $R=\mathbb{Z}[A,A^{-1}]$. F...

Suppose that $Y$ is a closed, oriented 3-manifold, and let $S_{2,\infty}(Y)$ denote the Kauffman bracket skein module over $R=\mathbb{Z}[A,A^{-1}]$ as...

Categorify the Witten--Reshetikhin--Turaev invariants of 3-manifolds....

(a) Give a mathematical definition of the $\widehat{Z}$ invariants for all 3-manifolds. (b) Categorify the $\widehat{Z}$ invariants....

(a) What is the isomorphism type of $\Theta^3_{\mathbb{Z}}$? (b) Does there exist a torsion element $[Y]$ in $\Theta^3_{\mathbb{Z}}$? (c) Does there...

Is $\Theta^3_{\mathbb{Z}}$ generated by the classes of knot surgeries $[S^3_{1/n}(K)]$, where $n$ ranges over all integers and $K$ ranges over all kno...

Is there a nontrivial element in the kernel of the natural map $$ \Theta^3_{\mathbb{Z}}\longrightarrow \Theta^3_{\mathbb{Z}/2\mathbb{Z}}; $$ that is...

(a) Does the kernel of the map $\Theta^3_{\mathbb{Z}}\to\Theta^3_{\mathbb{Q}}$ contain a subgroup that is isomorphic to $\mathbb{Z}^{\infty}$? (b) If...

(a) Calculate $\Theta^{\mathrm{TOP}}_{\mathbb{Z}/p}$. (b) Calculate $\Theta^{\mathrm{TOP}}_{\mathbb{Q}}$. (c) Is the linking form homomorphism $[\op...

Let $Y$ be a rational homology sphere and $f:Y\to Y$ be a self-diffeomorphism of $Y$. Suppose $W$ is a 4-manifold with boundary $Y$ such that $f$ exte...

Let $Y$ be a rational homology 3-sphere equipped with an action of a cyclic group $\mathbb{Z}/p\mathbb{Z}$. Suppose $W$ is a 4-manifold with boundary ...

What is the structure of the equivariant homology cobordism groups?...

Does there exist a hyperbolic rational homology 3-sphere that is the totally geodesic boundary of a compact, orientable hyperbolic 4-manifold?...

(a) Is there a non-semisimple 3-TQFT whose mapping class group representation is faithful or has an element in its kernel? (b) Define a 4-manifold in...

(4-dimensional Poincaré conjecture). Is there a unique smooth structure on the 4-sphere?...

Does every smooth, closed 4-manifold admit an exotic smooth structure? Infinitely many?...

Are there exotic smooth structures on the following closed, simply-connected 4–manifolds? (a) $\#_{k}\mathbb{CP}^{2}$ for any $k \geq$ 1. (b) $\#_{m...

Is there an exotic smooth structure on some product 4-manifold $S^{1} \times Y^{3}$ or $\Sigma_{g} \times \Sigma_{h}$? Do they all admit exotic smooth...

Does every connected, open 4-manifold admit uncountably many smooth structures?...

Does every closed, orientable 3-manifold bound an absolutely exotic pair of smooth, orientable 4-manifolds?...

(a) If $M_{1},M_{2}are$ two homeomorphic closed, oriented 4-manifolds, is $M_{1}\#S^{2} \times S^{2}$ diffeomorphic to $M_{2}\#S^{2} \times S^{2}$? (...

Let X be a closed, simply connected, smooth 4-manifold, and T a smoothly embedded torus in X with $\pi_{1}(X$ −T) =1 and $[T]^{2}$ =0. Let $X_{K}$ be ...

Is every Gluck twist in $S^{4}$ standard?...

(a) Is every homotopy $B^{4}$ with boundary $S^{3}$ obtained by performing a Gluck twist on some knotted 2-sphere in $B^{4}$? (b) Suppose a homotopy ...

Let M be a smooth 4-manifold and letf: $S^{2} \to M$ be a smooth embedding with trivial normal bundle. Then let $M_{f}$ denote the result of Gluck twi...

For X a closed simply connected smooth 4-manifold, let $g_{X}: H_{2}(X) \to \mathbb{N}$ denote the genus function, which assigns to every homology cla...

(a) Does every large $\mathbb{R}^{4}-homeomorph$ lie $in\mathcal{R}_{K}$ for some Kthat is not smoothly slice? (b) Does there exist an infinite seque...

Is there a universal cork? More precisely, does there exist some cork (C, f) such that given any pair W and $W^{1}$ of closed, simply connected 4-mani...

(11/8 Conjecture). Does every smooth, spin, closed 4-manifold X satisfy $b_{2}(X) \geq 11|\sigma(X)|$, where $\sigma(X)$ is the signature of the inter...

(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...