56 problems

Let $M^n$ be a connected closed manifold whose fundamental group $\pi_1(M)$ is finite of odd order. A metric of positive scalar curvature on the universal cover of $M$ need not des…

Conformal concordance conjecture. If $C_0$ and $C_1$ are conformally concordant, then they are isotopic in ${\cal C}^+(M)$.

Let $M$ be an orientable $n$-manifold equipped with a complete Riemannian metric of positive scalar curvature. For a complete Riemannian metric on $M$, say that it has at most $C$-…

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 $M$ 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 $\Sigma$ be a closed manifold of dimension at least $3$, with $\dim\Sigma\ne4$. A positive scalar curvature metric is a Riemannian metric whose scalar curvature is everywhere p…

Let $M^3$ be a three-manifold that has an exhaustion by solid tori and admits a complete metric with scalar curvature $R>0$ and bounded geometry. The solid-torus exhaustion problem…

Let $M^3$ be the interior of a genus-$\gamma$ handlebody, equipped with a complete metric whose scalar curvature satisfies $R>0$. The handlebody positive-scalar-curvature conjectur…

Let $(M^3,g)$ be a contractible $3$-manifold equipped with a complete metric whose scalar curvature satisfies $R>0$. The contractible three-manifold positive-scalar-curvature conje…

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 $N\to B$ be a fiber bundle with base $B$ a connected closed manifold, and with all fibers diffeomorphic to a fixed connected closed manifold $Z$. Suppose there is no fiberwise…

Let $(M,\sigma)$ be a closed connected totally non-spin spin$^c$ manifold with $\dim M=n$, fundamental group $\pi$, and classifying map … Let $L$ be the associated spin$^c$ line bu…

Let $(S^3,g)$ be a Riemannian 3-sphere with scalar curvature $R_g\geq \Lambda_0>0$. A tree-foliation $\{\Sigma_t\}_{t\in T}$ is a foliation of $(S^3,g)$ by surfaces whose parameter…

Gromov's rational essentialness conjecture. A closed $\mathbb{Q}$-essential $n$-manifold does not admit any Riemannian metric with positive scalar curvature.

Let $(M,L)$ be a closed spin$^c$ non-spin manifold with $\dim M\geq 5$ and with $\mathcal R^{{\mathrm{gen}+}}(M,L)\neq\emptyset$. Fix a basepoint…

Schoen–Yau–Schick degree-one dominated $S^2$-stability conjecture. Every Schoen–Yau–Schick manifold $X^n$ with $n\leq 5$ has the degree-one version of dominated $S^2$-stability.

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

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 $M^n$ 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…

Gromov's noncompact injectivity radius conjecture. Then

Let $(M^n,g)$ be a complete Riemannian manifold. Write $\operatorname{Inj}(M)$ for its injectivity radius and $\operatorname{Rad}_{\mathbb S^n}(M)$ for its spherical radius. The in…

Let $M^n$ be a closed, connected $n$-manifold, let $S^1$ be the circle, and let $\sigma$ denote the Yamabe invariant. Rosenberg's $S^1$-stability conjecture. The Yamabe invariants…

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 $\mathrm{SingMet}$ and $\mathrm{ConcSet}$ be the cubical models of, respectively, the space of positive scalar curvature metrics and the space of positive scalar curvature conc…