Unsolved Problems

(a) Considering all closed, orientable, $\pi_1$-injective surfaces (possibly non-embedded) in all closed hyperbolic 3-manifolds, what is the infimum o...

Does every closed hyperbolic 3-manifold admit an immersed $\pi_1$-injective surface with only double points? More precisely, if M is a closed, connect...

Can a hyperbolic knot complement in the 3-sphere contain a closed, embedded totally geodesic surface?...

Let $M$ be a closed hyperbolic 3-manifold with positive first Betti number. (a) Which elements of $H^{2}(M;\mathbb{R})$ are realized as the Euler cla...

Does every finite-volume hyperbolic 3-manifold contain infinitely many simple closed geodesics?...

Let $M_1$ and $M_2$ be finite-volume hyperbolic n--manifolds. If the length spectra of $M_1$ and $M_2$ coincide, must the two manifolds be commensurab...

Is there a closed hyperbolic 3-manifold that is foliated with minimal leaves?...

(a) Does every closed hyperbolic 3-manifold have a nowhere zero vector field whose lift to the universal cover has proper flow lines? (b) Can one ens...

Let $M$ be a closed hyperbolic 3-manifold with a faithful homomorphism $\rho:\pi_1(M)\to \operatorname{Homeo}^{+}(\mathbb{R})$. Prove that $M$ support...

In this problem, all 3-manifolds are orientable, while all flows are considered up to orbit equivalence and are assumed to be transitive. (a) Are the...

Let $G=\pi_1(M)$ be the fundamental group of a finite-volume hyperbolic 3-manifold $M$. What is the regularity of the smoothest (virtual) action of $G...

What is the Margulis constant in dimension 3? Is it realized uniquely by the Weeks manifold W, where $\mu(W)$ = 0.77442...?...

(a) (Cannon Conjecture) If G is a finitely presented, Gromov hyperbolic group with space at infinity equal to the 2-sphere, must G be a cocompact Klei...

(Bending Conjecture). (a) Is a quasi-Fuchsian group determined by the hyperbolic metric on the boundary of its convex core? (b) Is a quasi-Fuchsian g...

Let $M$ be a finite-volume hyperbolic 3-manifold, and let $M^1$ be a minimal-index finite cover of $M$ such that $\pi_1(M^1)$ embeds in a right-angled...

(a) What is the computational complexity of the homeomorphism problem for compact, orientable 3-manifolds? (b) Is there a polynomial-time algorithm t...

How many Pachner moves are needed to pass between two triangulations of a compact 3-manifold?...

(a) Given a closed hyperbolic 3-manifold $M$, can one find an explicit bound on the degree of a finite cover $\widetilde M$ having $b_1(\widetilde M)>...

(a) What is the computational complexity of determining whether a compact 3-manifold admits a hyperbolic structure? (b) If a compact 3-manifold does ...

Suppose M is a closed 3-manifold. (a) Can one decide if the fundamental group of M is left-orderable? (b) What is the complexity of a certificate of...

Is there an algorithm to determine whether two closed, embedded surfaces in $\mathbb{R}^3$ are isotopic?...

Let M and N be closed orientable 3-manifolds. Prove that if there is a degree-1 map $f:M\to N$ then $g(M)\geq g(N)$, where $g(M)$ is the Heegaard genu...

Do any two genus-g Heegaard splittings of a closed, orientable 3-manifold M become equivalent after at most g stabilizations?...

Given a compact manifold M, let $r(M)$ denote the rank of its fundamental group and $g(M)$ denote its Heegaard genus. (a) Does every closed orientabl...

(Simple loop conjecture) Let f : F $\to$ M be a 2-sided immersion of a surface into a 3-manifold such that $f_*$ : $\pi_1(F)$ $\to$ $\pi_1(M)$ is not ...

(a) Is every finitely generated 3-manifold group linear (over some field with characteristic zero)? (b) If so, can one bound the dimension of a faith...

Is every PD$_3$-group the fundamental group of a closed, aspherical 3-manifold?...

(a) Is every finitely generated perfect group the normal closure of a single element? (b) Is there an integral homology sphere whose fundamental grou...

Does every closed, orientable, hyperbolic 3-manifold admit a tight contact structure?...

Is it true that for every knot $K\subset S^3$, there is an integer $n_K$ such that $S^3_r(K)$ admits a tight contact structure for all $r\geq n_K$?...

Does every tight contact 3-manifold have finite Giroux torsion?...

Understand how various properties of contact structures behave under different kinds of symplectic cobordism. For instance: (a) Is tightness preserve...

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?...