213 problems

Let $T$ be the once-punctured torus, let $\mathcal{C}(T)$ denote the set of closed curves on $T$, and let $\mathcal{O}$ be any mapping class group orbit in $\mathcal{C}(T)$. For ev…

Does there exist an open $4$-manifold $M$ that admits a finite-dimensional compact ANR $\mathcal{Z}$-compactification but is not pseudo-collarable?

For every complete Riemannian manifold $(M,g)$ with nonnegative Ricci curvature, $\operatorname{Ric}_g\ge 0$, the fundamental group $\pi_1(M)$ is finitely generated.

For every integer $n\geq 2$, there exists a constant $C_n>0$ such that, for every closed oriented $n$-manifold $M$, every Riemannian metric $g$ on $M$, and every $\lambda\geq 0$, t…

Let $X$ be a smooth closed manifold. Write $\mathrm{Diff}(X)$ for its diffeomorphism group, $B\mathrm{Diff}(X)$ for its classifying space, and $\pi_k$ and $H_k(-;\mathbb{Z})$ for h…

Spin torus-action conjecture. The manifold $M$ is homeomorphic to one of $S^3$, $S^4$, $S^5$, or

A compact complete affine manifold is a compact manifold equipped with a complete affine structure. Auslander's conjecture. The fundamental group of a compact complete affine manif…

Let $M$ be a manifold of dimension $m$, and let $T^m$ denote the $m$-dimensional torus. Generalized Geroch conjecture. There is no complete metric of positive scalar curvature on ……

Let $X$ be an aspherical 2-complex, and let $Y$ be a subcomplex of $X$. Whitehead's conjecture. The subcomplex $Y$ is aspherical. This is a famous conjecture about asphericity inhe…

Let $G$ be a Poincaré duality group of dimension $3$. Wall's conjecture. $G$ is the fundamental group of a closed aspherical $3$-manifold. This conjecture asks whether the algebrai…

Weighted Singer Conjecture. The weighted $L^2$-homology vanishes above the middle dimension:

Let $M$ be a hyperbolic $3$-manifold with finitely generated fundamental group. The interior of a compact manifold with boundary is a topological model for a tame hyperbolic $3$-ma…

Let $M$ be a closed aspherical $(2n+1)$-manifold. For a cohomology class $\phi\in H^1(M;\mathbb{R})$, let $b_i^{(2)}(\widetilde M;\phi)$ denote the twisted $L^2$-Betti numbers of t…

Let $W_0$ and $W_1$ be closed, connected, oriented $3$-manifolds. A rational homology ribbon cobordism from $W_0$ to $W_1$ is a ribbon cobordism $W$ from $W_0$ to $W_1$ such that t…

Cusp covering conjecture. For each cusp $C$ of $M$, there exists a sublattice $\Lambda_C$ of $\pi_1(C)$ such that, for every choice of a sublattice $\Lambda'_C\subseteq\Lambda_C$ f…

Let $LIP(\mathbb{S}^n)$ denote the orientation-preserving homeomorphisms of the $n$-sphere that are bi-Lipschitz. For a bi-Lipschitz map, its isometric distortion is the least…

Strong finiteness conjecture. There exists a finite collection $\Sigma_1,\dots,\Sigma_k$ of essential subsurfaces $\Sigma_i\subsetpartial M$ such that, for each $i$, the dimension of…

Let a paper Möbius band be a smooth isometric embedding of the flat Möbius band $M_{\lambda}$ in $\mathbb{R}^3$, and consider a sequence of smooth embedded paper Möbius bands whose…

Let $M_{\lambda}=([0,1]\times[0,\lambda])/\sim$ be the flat Möbius band, where $(x,0)\sim(1-x,\lambda)$, and let a paper Möbius band be a smooth isometric embedding…

Modified Bing–Borsuk conjecture. Every locally compact homogeneous $ANR$-space of dimension $n\geq 3$ is a generalized $n$-manifold.

Let $M^4$ be a closed, oriented, connected $4$-manifold. A simple branched covering is a branched covering whose local monodromies around the branch set are transpositions; let…

Živaljević's conjecture. Every simply connected smooth $d$-manifold admits some LC triangulations.

Let $M^n$ be an oriented closed manifold, where $n\ge 2$. A manifold is hyperbolic if it admits a Riemannian metric of constant negative curvature. Kotschick–Löh's conjecture. Ever…

Let $Y^3_\infty$ be a $3$-dimensional compact, simply connected, non-negatively curved Alexandrov space without boundary, and suppose that $Y^3_\infty$ is a topological manifold. S…

Let $X$ be an $n$-dimensional absolute cone, meaning a compact finite-dimensional metric space that is a cone with respect to every point. de Groot's absolute cone conjecture. The…