14 problems

Let $M$ be a compact Riemannian manifold with positive isotropic curvature. A group is virtually free if it contains a free subgroup of finite index. The virtual freeness conjectur…

Virtual-freeness conjecture. The fundamental group of a manifold with PIC is virtually free.

Topping's conjecture. The manifold $M$ should be diffeomorphic to $\mathbb R^n$.

Generalised PIC1 Hamilton conjecture. Then $(M,g(0))$ is flat or compact.

PIC1 volume-ratio conjecture. These conditions should imply

Let $M^n$ be a closed manifold admitting a metric with positive isotropic curvature. The Gromov–Schoen conjecture. The fundamental group $\pi_1(M)$ is virtually free, and a finite…

Higher-dimensional PIC singularity classification conjecture. A similar singularity phenomenon holds in all dimensions $n\geq 4$ when the initial data is PIC.

Let $M$ be a Riemannian manifold whose curvature tensor is strictly PIC, meaning that it has strictly positive isotropic curvature for every orthonormal four-frame. Consider the Ri…

Let $M$ be a compact Riemannian manifold with positive isotropic curvature. Fraser–Gromov conjecture. The fundamental group $\fundamentalgroup(M)$ should contain a free subgroup of…

Schoen's conjecture. The much stronger differential version of the finite-cover classification theorem should hold.

Fundamental-group conjecture. The fundamental group of a compact manifold with positive isotropic curvature should be residually free; equivalently, so far as the fundamental group…

Let $M$ be a closed Riemannian $n$-manifold with positive isotropic curvature. The virtual-freeness conjecture for positive isotropic curvature. The fundamental group $\pi_1(M)$ is…

Schoen's conjecture. A finite cover of $M$ should be diffeomorphic to $\mathbb{S}^n$, $\mathbb{S}^{n-1}\times\mathbb{S}^1$, or a connected sum of such manifolds. In particular, the…

Gromov–Fraser conjecture. $\pi_1(M)$ is virtually free, i.e., it is a finite extension of a free group.