100 problems

Borel conjecture. Every homotopy equivalence

A closed aspherical manifold is a closed manifold whose universal cover is contractible. Gromov–Lawson conjecture. A closed aspherical manifold cannot support a Riemannian metric o…

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…

Singer–Hopf conjecture.

A group is a $PD_n$ group if its fundamental-group cohomology satisfies Poincaré duality over $\mathbb Z$ in dimension $n$. Wall Conjecture. Conversely, every $PD_n$ group is the f…

Let $N^{2k}$ be a closed aspherical $2k$-manifold. Euler characteristic conjecture. One should have … This conjecture predicts the sign of the Euler characteristic of every closed…

Let $n\in\mathbb{N}$. For an $n$-dimensional, closed, aspherical, orientable manifold $M$, write $\widetilde{M}$ for its universal cover, $b^{(2)}_i(\widetilde{M})$ for its $i$th…

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 locally symmetric space associated to a group $G$ without discrete series, equivalently of the form $M=K \backslash G/\Gamma$ with Euler characteristic zero. A finite…

Let $M$ be a closed aspherical manifold of dimension $2m$, where $m$ is an integer, and let $\chi(M)$ denote its Euler characteristic. Hopf–Thurston conjecture. Every such manifold…

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…

Gromov's rational homology vanishing conjecture. The induced map on fundamental classes satisfies

Let $n$ be a positive integer. A closed manifold is aspherical if its universal cover is contractible. A Riemannian metric has positive scalar curvature when its scalar curvature i…

Let $M$ be a closed, connected, oriented topological $n$-manifold, with simplicial volume $||M||$ and Euler characteristic $\chi(M)$. Say that $M$ is aspherical when its universal…

Hopf–Chern–Thurston conjecture. One has

Let $M^n$ be a closed aspherical $n$-manifold with residually finite fundamental group. Let $\Gamma_k\vartriangleleft\pi_1(M^n)$ be any normal chain with $\bigcap_k\Gamma_k=1$. Lüc…

Gromov's zero-in-the-spectrum conjecture. There is $p \geq 0$ such that zero belongs to the spectrum of $(\Delta_p)_{\operatorname{min}}$.

Let $M$ be a closed oriented aspherical manifold. Its tangent bundle is denoted by $TM$, and a flat structure on $TM$ means that it is induced by a representation of the fundamenta…

A smooth $4$-manifold is symplectic for both choices of orientation if it admits a symplectic structure for each of its two orientations. A $4$-manifold is ruled if it is a ruled m…

Gromov's conjecture. For every $p\in\mathbb{N}$,

Let $M=K(\pi,1)$ be an $n$-dimensional aspherical manifold, and let $h:N\to M$ be a simple homotopy equivalence of manifolds. Topological Borel rigidity conjecture. The map $h$ is…

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…

Generalized Singer conjecture. All the $L^2$-Betti numbers of $\widetilde M$ vanish, possibly except in the middle dimension.