101 problems

Hilbert–Smith conjecture. Every faithful action of a locally compact group on a connected manifold has a Lie group as its acting group.

Configuration-space homotopy invariance conjecture. The homotopy type of $\mathrm{Conf}_n(X)$ depends only on the homotopy type of $X$.

Let $M$ be a closed manifold and let $S^n$ be the $n$-sphere. The Lusternik–Schnirelmann category $\operatorname{cat}(X)$ is the least integer $k$ such that $X$ admits an open cove…

Let $M$ and $N$ be closed orientable manifolds, and let $f:M\to N$ be a map of degree one. The Lusternik–Schnirelmann category $\operatorname{cat}(X)$ is the least integer $n$ such…

Generalized Poincaré conjecture. Any homotopy $n$-sphere is homeomorphic to $S^n$.

Lambrechts–Stanley model conjecture. The Lambrechts–Stanley model is a model for the configuration spaces of every simply-connected closed oriented manifold.

Let $n,Re\mathbb{N}$ and $d=n+R$, and let $\mathcal{M}\subseteq\mathbb{R}^d$ be a bounded, immersed, $n$-dimensional smooth submanifold with boundary. For $Q\in\mathbb{N}$ and…

Let $M$ be an analytic nondegenerate manifold, and consider the set of points on $M$ that are very well approximable in the dual Diophantine sense. Sprindzhuk's conjecture. The set…

Triangulation Conjecture. Any topological manifold admits a triangulation.

Let $\mathcal{M}\subset\mathbb{R}^n$ be a $d$-dimensional non-degenerate submanifold, let $\mu_d$ be the normalised $d$-dimensional Lebesgue measure induced on $\mathcal{M}$, and l…

Let $Delta_n^k$ be the union of the $k$-dimensional faces of the $n$-simplex, let $M$ be a closed $(k-1)$-connected PL $2k$-manifold, and let $chi(M)$ denote its Euler characterist…

Let $M$ be a tame, complete, finite-volume $n$-manifold of bounded nonpositive curvature. A manifold has geometric rank one when it satisfies the rank-one condition used in the sou…

Let $cmathcal{M}$ be a compact hyperbolic manifold of dimension $n eq 3$. Belolipetsky–Emery conjecture. There exists a noncompact hyperbolic $n$-manifold $cmathcal{M}1$ whose…

Let $M$ and $N$ be manifolds, let $f:M\rightarrow N$ be a continuous map, and let ${\text{Map}}(M,N;f)$ denote the space of maps homotopic to $f$. Fix a basepoint $*$ in $M$, and d…

Let $\Gamma$ be a regular $(n+1)$-valent graph with a “good” $(\mathbb Z_2)^{n+1}$-coloring $\alpha$. A generalized $(n+1)$-skeletal expansion is an expansion denoted…

Let $(\Gamma,\alpha)$ be a finite connected regular $(n+1)$-valent graph whose edges are colored by $(\mathbb Z_2)^{n+1}$, with $\alpha$ a “good” coloring. A moment graph is the gr…

Exactness conjecture. The lower bound

Immersion-dimension conjecture. Every $n$-dimensional manifold can be immersed in

Let $M^n$ be a closed connected smooth $n$-manifold whose Stiefel–Whitney classes all vanish. A Riemannian metric is conformally flat when its Weyl tensor vanishes. Conformal-flat…

Löffler–Raußen conjecture. Non-trivial $\mathbb{Z}_p$-actions should exist on any simply-connected manifold.

Bounded-geometry hyperEuclidean conjecture. Every uniformly contractible manifold of bounded geometry is hyperEuclidean; in particular, the universal cover of an aspherical closed…

The paper considers quasi-null functions arising in Theorems 10.4 and 10.7(1), which are functions defined on the relevant representation spaces and are determined only up to quasi…

A closed symplectically aspherical manifold is a closed manifold whose symplectic form vanishes on every spherical homology class. The symplectically aspherical positive scalar cur…

A tame quasiconformal manifold is a topological manifold equipped with a quasiconformal structure whose chart transition functions are tame quasiconformal mappings. Tame quasiconfo…

Fix an error weight $t$, set $n=n_{\rm min}(t)$, and consider the family of real minimal qubit permutation-invariant quantum error-correcting codes with parameters $(n,t)$. Real mi…