Unsolved Problems

Is there an algorithm to decide, given an open book, whether the corresponding contact 3-manifold is tight or fillable?...

(a) Are there contact 3-manifolds with support genus greater than one? (b) Are there contact 3-manifolds with arbitrarily large support genus?...

Let $\lambda$ be a contact form on a closed 3-manifold that is not a lens space. Must the associated Reeb flow have infinitely many simple periodic or...

(a) Does every Reeb flow on $S^3$, associated to a contact form giving the standard contact structure, have an elliptic periodic orbit? (b) What abou...

(L-space Conjecture). For prime rational homology 3-spheres Y, are the following equivalent? (a) $\pi_1(Y)$ is left-orderable. (b) Y is not an L-spa...

Are any of the three conditions in the L-space Conjecture equivalent, for all prime rational homology 3-spheres Y, to the condition that Y admits a co...

(The Floer Poincaré Conjecture). If Y is an integral homology sphere that is an L-space, show that Y is $S^3$ or the connected sum of some copies of t...

Suppose Y is a rational homology 3-sphere such that every homomorphism $\pi_1(Y)$ $\to$ $\operatorname{SU}(2)$ has abelian image. Does it follow that ...

(a) Does every closed 3-manifold M besides the 3-sphere admit a nontrivial representation $\pi_1(M)$ $\to$ $\operatorname{SU}(2)$? (b) For which M wi...

Are all strong L-spaces branched double covers of alternating links in $S^3$?...

(a) Is there a closed 3-manifold $M$ whose Heegaard Floer homology $\widehat{HF}(M;\mathbb{Z})$ has torsion? (b) Is there a rational homology 3-spher...

(a) For $K$ a nontrivial knot in $S^3$, does $HFK^-(K)$ always admit an $\mathbb{F}_2$-summand, as an $\mathbb{F}_2[U]$-module? (b) For $Y$ a rationa...

(a) For $Y^3$ a rational homology sphere, is the Seiberg--Witten Floer spectrum $SWF(Y)$ always a wedge of spheres? (b) Is every monopole Floer homol...

Construct an $S^1$ - or $\operatorname{Pin}(2)$-equivariant lattice homotopy type that computes the Seiberg--Witten Floer homotopy type....

Prove that there is an isomorphism relating Heegaard Floer homology and monopole Floer homology that commutes with the cobordism maps in the two setti...

Prove an isomorphism relating instanton Floer homology and Heegaard Floer homology....

Find an algorithm to compute instanton Floer homology of closed 3-manifolds and the Donaldson invariants of closed 4-manifolds....

Is the dimension of Heegaard Floer homology invariant under genus 2 mutation?...

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...