50 problems

Let $X$ be an aspherical closed manifold, let $\widetilde X$ be its universal cover, and let $\pi_1(X)$ act on $\widetilde X$ by deck transformations. A closed fundamental domain i…

Let $M$ be an aspherical manifold, let $\widetilde M$ be its universal cover, and let $\pi_1(M)$ act on $\widetilde M$ by deck transformations. Contractible-translate conjecture. T…

Let $G$ be a hyperbolic group whose boundary $\partial G$ is homeomorphic to $S^{n-1}$. An aspherical closed manifold is a closed manifold whose universal cover is contractible. Gr…

Partial-link Singer conjecture. If $X$ has spherical links in codimensions at most $2l+1$, where $2l+1\le n$, then

Euler characteristic conjecture.

Let $\Gamma$ be the fundamental group of a closed aspherical manifold. A finitely presented infinite group $\Gamma$ is w.g.s.c. when some compact polyhedron with fundamental group…

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 $M$ be a closed real $2n$-manifold. It is aspherical when its universal covering is contractible. Chern–Hopf–Thurston conjecture. If $M$ is aspherical, then it satisfies … This…

Let $X$ be a closed oriented aspherical 4-manifold. Write $\chi(X)$ for its Euler characteristic and $\sigma(X)$ for its signature. Geography conjecture. One should have … This is…

Let $M$ be a closed aspherical manifold, meaning that its higher homotopy groups satisfy $\pi_k(M)=0$ for all $k\ge2$. A positive scalar curvature metric is a Riemannian metric who…

Let $M$ be an aspherical closed manifold. Consider a descending chain of subgroups … such that each $G_i$ is normal in $G=\pi_1(M)$, the index $[G:G_i]$ is finite, and…

An aspherical manifold is a manifold with contractible universal cover. The Schoen–Yau–Gromov–Lawson conjecture. There is no Riemannian metric of positive scalar curvature on a clo…

Let $M$ be a closed aspherical manifold with RFRS fundamental group $G=\pi_1(M)$, and let $\beta_i^{(2)}(G;\mathbb K)$ denote the $i$th $L^2$-Betti number with coefficients in a fi…

Let $X$ be a finitely dominated aspherical Poincaré complex. The manifold-structure conjecture. The complex $X$ is homotopy equivalent to a closed topological manifold. This is one…

Let $X$ be a closed, oriented, aspherical $4$-manifold, and let $\tau(X)$ denote its signature. Gromov–Lück inequality. One has … This inequality is a refined form of the Hopf prob…

Let $M^n$ be a closed aspherical $n$-manifold with residually finite fundamental group, and set $G=\pi_1(M^n)$. Let $G_k\mathrel{\lhd}G$ be any normal chain of finite-index subgrou…

Singer–Hopf conjecture.

Full Relative Aspherical Conjecture. If $n\ne 4$, $Y$ is aspherical relative to $X$, and $X$ admits no metric of positive scalar curvature, then $Y$ admits no metric of positive sc…

Let $M$ be a closed aspherical $n$-manifold. Singer's conjecture. The $ell^2$-Betti numbers satisfy … for all $p\neq n/2$. The Singer conjecture is a major unresolved problem conce…

Let $M$ be a closed aspherical manifold and let $\widetilde{M}$ be its universal cover. A manifold is inward tame if it has arbitrarily small neighborhoods of infinity that are fin…

Let $M$ and $N$ be closed aspherical manifolds with isomorphic fundamental groups, and let $\widetilde{M}$ and $\widetilde{N}$ denote their universal covers. Weak Borel conjecture.…

Let $M^n$ be a closed, aspherical manifold, meaning that $\pi_k(M)=0$ for every $k\geq 2$. Gromov--Lawson conjecture. The manifold $M$ admits no Riemannian metric with positive sca…

Let $M^n$ be a compact $n$-manifold with non-empty boundary, and let its double be the closed manifold obtained by gluing two copies of $M^n$ along their common boundary. Let…

Let $M^n$ be a closed $n$-dimensional $K(\pi,1)$ manifold, meaning an aspherical manifold whose universal cover is contractible. A Riemannian metric has positive scalar curvature i…

Let $M$ be an aspherical closed manifold with hyperbolic fundamental group $\pi$. Let $j\colon G\to \operatorname{Out}(\pi)$ be an embedding of a finite group $G$ into the outer au…