Unsolved Problems

(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 $\HH^{*}_{\mathrm{an}}$, ...