A note on rational homology vanishing theorem for hypersurfaces in aspherical manifolds

Abstract

In this note, we generalize Gromov's reduction \cite{Gro20} from the aspherical conjecture to the generalized filling radius conjecture to the smooth $\mathbb{Q}$-homology vanishing conjecture for hypersurface. In particular, we can show that any continuous map from a closed $4$-manifold admitting positive scalar curvature to an aspherical $5$-manifold induces zero map in $H_4(\cdot ,\mathbb{Q})$. As a corollary, we obtain the following splitting theorem: if a complete aspherical $5$-manifold has nonnegative scalar curvature and two ends, then it splits into the Riemannian product of a closed flat manifold and the real line.

final version, to appear in PAMS; modification was made in section 2, where homology filling was replaced by homotopy filling due to technical reasons