Generalized Geroch conjecture; spectral Geroch conjecture

The generalized conjecture says that connected sums of an $n$-torus with an arbitrary $n$-manifold admit no complete metric of positive scalar curvature; the spectral version seeks an analogous obstruction. Wang–Zhang established the spin case, while later work removed the spin assumption in dimensions through $7$.

Known results

• Wang–Zhang (2022): a closed area-enlargeable summand obstructs complete positive scalar curvature when the other summand is spin, in every dimension.
• Chodosh–Li: the torus case was proved without a spin assumption for $3\le n\le 7$.
• Related work proves the obstruction for closed aspherical summands in dimensions $3$, $4$, and $5$.

August 25, 2026 preprint report

The new preprint claims to extend the Wang–Zhang obstruction from closed to possibly noncompact area-enlargeable summands and to establish a $\gamma$-spectral obstruction for $3\le n\le 7$ when $0\le \gamma<\frac{2n}{n-1}$. This is substantial claimed progress, not an independently verified resolution of the full spectral conjecture.

Current status (as of August 2026): The classical generalized obstruction is settled in the spin case in all dimensions and without spin for $3\le n\le 7$; the noncompact extension and stated spectral obstruction are claimed by a new preprint, while the full spectral conjecture remains unverified.