Unsolved Problems

Is there an algorithm to compute ‘distance’ in the cut complex of a trisection? Is the L-invariant computable?...

Let X be a closed, orientable, smooth 4-manifold, with $g(X)$ the trisection genus of X. Does $g(X) =\chi(X) -2+3rk(\pi_{1}(X))$?...

Does every simply connected, closed, smooth 4-manifold admit a handle decomposition without any 1-handles? Without 1-handles and 3-handles?...

Is every topological 4-manifold homeomorphic to a CW complex?...

Which closed, smooth 4–manifolds admit achiral Lefschetz pencils? Does every simply connected 4–manifold have one?...

Which closed, smooth4–manifolds admit open book decompositions? In particular, does every closed, simply connected 4–manifold with signature zero admi...

Is there a universal branching surface $S \subset S^{4}$ such that every closed, orientable 4-manifold W admits a branched covering $W \to S^{4}$ with...

Is every closed leaf of a two dimensional co-orientable smooth taut foliation of an oriented 4-manifold smoothly genus-minimizing in its homology clas...

Does there exist a hyperbolic integer homology four-sphere? What about an arithmetic one? Homology four-spheres have Euler characteristic 2, so it mak...

Is there a noncompact, finite volume, orientable hyperbolic four-manifold without a spin structure?...

(a) If M is a closed, orientable hyperbolic 4-manifold then it always has signature 0, because its Pontryagin class vanishes [Che55]. This implies tha...

Given an aspherical closed (or compact and bounded by flat 3-manifolds) 4-manifold M and a self-diffeomorphism f of M, find necessary and sufficient c...

What is the structure of 4-manifolds that admit a Riemannian metric of positive scalar curvature? There are variations of this problem for different c...

Given a closed, 4-dimensional PSC manifold M, is there a (possibly disconnected) 4-dimensional orbifold $M^{1}$ with isolated singularities such that ...

Does longitudinal knot surgery using a knot K, along a fiber in a K3 surface always yield a reducible 4–manifold? A completely decomposable 4–manifold...

Does every Lipschitz 4-manifold admit a smooth structure? Is this smooth structure unique if so? Some more specific, related questions are as follows....

Does every cellular set in the plane have the fixed point property?...

(Doubly-Small Morphisms of Manifolds). - Suppose that $h: \R^{n} \to \R^{n}$ is a homeomorphism (or diffeomorphism) which satisfies two smallness hyp...

(Hilbert--Smith Conjecture). - The Hilbert--Smith Conjecture [Smi41] asserts that a locally compact subgroup of the homeomorphism group of a connecte...

Is the homeomorphism group of a manifold an absolute neighborhood retract (ANR)?...

Is the connected sum of (locally flat) pairs $$ (M^{n+k}_{1},N^{n}_{1})\#(M^{n+k}_{2},N^{n}_{2}) $$ well-defined in the topological category?...

- Does every closed, PL, orientable $n$-manifold admit an $n$-fold branched covering map over $S^{n}$? Assuming that the answer to this problem is "y...

(Montgomery--Yang problem). Does there exist a pseudo-free, smooth, $S^{1}$ action on $S^{5}$ with more than three multiple orbits?...

Is there a closed aspherical 5-manifold that is not triangulable?...

Let $M_{1}$ and $M_{2}$ be smooth manifolds of dimension $n$. Suppose $M_{1}$ admits an $S$-map into $\R^{p}$. If $M_{2}$ is homeomorphic to $M_{1}$, ...

The Andrews--Curtis Conjecture [AC65] for the trivial group: a presentation of the trivial group can be changed to the trivial presentation by Andrews...

(Whitehead's Asphericity Question). Is every subcomplex of an aspherical 2-complex aspherical?...

(Zeeman Conjecture). If $K$ is a finite contractible 2-complex, then $K\times I$ collapses to a point [Zee64, Conjecture (1)]....

Let $M$ be a finite-volume hyperbolic $n$-manifold. - Does it always have a finite cover with $b_{1}>0$? - Does it always have a finite cover with f...

Does there exist a 1-cusped finite-volume hyperbolic $n$-manifold for any $n\geq 5$?...

Suppose $M$ is a manifold with a complete Riemannian metric with nonnegative Ricci curvature. Is the fundamental group of $M$ finitely generated?...

Let $R$ be $\Z$ or a field. Let $A$ and $B$ be differential graded algebras so that either: \begin{itemize} - As a graded algebra, $A$ (respectively ...

Let $(W,\omega,V_{i},\phi_{i})$ be two Weinstein structures on a fixed symplectic manifold $(W,\omega)$ (or equivalently consider two Weinstein handle...

- In higher dimensions, find the `non-analytic' cohomology module $\HH^{*}_{?}$ analogous to the analytic lattice cohomology $\HH^{*}_{\mathrm{an}}$, ...

Question $S(S(f)) = S(f)$ for every endo-reloid $f$?...

Conjecture Does every Jordan curve have 4 points on it which form the vertices of a square?...

Question Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?...

Problem Let $M$ be a $3$-dimensional smooth submanifold of $S^4$, $M$ diffeomorphic to $S^3$. By the Jordan-Brouwer separation theorem, $M$ separates ...

Conjecture If a $4$-manifold has the homotopy type of the $4$-sphere $S^4$, is it diffeomorphic to $S^4$?...

Conjecture Given a knot in $S^3$ which is slice, is it a ribbon knot?...

Problem Given a link $L$ in $S^3$, let the symmetry group of $L$ be denoted $Sym(L) = \pi_0 Diff(S^3,L)$ ie: isotopy classes of diffeomorphisms of $S^...

Problem Is there a complete and computable set of invariants that can determine which (rational) homology $3$-spheres bound (rational) homology $4$-ba...

Problem Does there exist a subset of $\mathbb R^3$ such that its fundamental group has an element of finite order?...

Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smooth...

Problem $Diff(S^4)$ has the homotopy-type of a product space $Diff(S^4) \simeq \mathbb O_5 \times Diff(D^4)$ where $Diff(D^4)$ is the group of diffeom...

Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalen...

Problem Does there exist a smooth/PL embedding of $S^2$ in $S^4$ such that the fundamental group of the complement has an unsolvable word problem?...

Problem Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observa...

Question $( \mathsf{RLD})_{\mathrm{out}} (f \cap^{\mathsf{FCD}} ( \mathcal{A} \times^{\mathsf{FCD}} \mathcal{B})) = (( \mathsf{RLD})_{\mathrm{out}} f)...

Conjecture Let $C$ be a Cantor set embedded in $\mathbb{R}^n$. Is there a self-homeomorphism $f$ of $\mathbb{R}^n$ for every $\epsilon$ greater than $...