Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds
Abstract
We construct smooth collapsing sequences with sectional curvature at least −1 whose limits are topological manifolds with nonzero Kirby--Siebenmann class. For every n≥6, the source and limit are simply connected, have dimensions n+3 and n, and the metrics have a common positive Ricci lower bound. The limits are homeomorphic to the product of the Chern four-manifold with Sn−4. We also obtain spin examples and five-dimensional lens-space homotopy types, the latter from finite quotients of one fixed $2$-connected eight-manifold. Finally, we realize aspherical manifolds with no closed PL representatives of their homotopy types as collapsing limits with scalar curvatures tending uniformly to +∞, and establish an obstruction to their approximation with almost nonnegative Ricci curvature.
Related
- Einstein four-manifolds of positive sectional curvature and ADM mass
- Fundamental groups of compact Kähler manifolds with semi-positive holomorphic sectional curvature
- Hessian manifolds of positive constant Hessian sectional curvature
- Minimal Two-Spheres and Manifolds with Positive Isotropic Curvature
- Positive scalar curvature on products of noncompact manifolds
BibTeX
@misc{ge2026collapse,
title = {{Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds}},
author = {Jian Ge and Chao Qian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05117},
eprint = {2610.05117},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05117},
}
RIS
TY - UNPB
TI - Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds
AU - Jian Ge
AU - Chao Qian
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05117
UR - https://arxiv.org/abs/2610.05117
AB - We construct smooth collapsing sequences with sectional curvature at least −1 whose limits are topological manifolds with nonzero Kirby--Siebenmann class. For every n≥6, the source and limit are simply connected, have dimensions n+3 and n, and the metrics have a common positive Ricci lower bound. The limits are homeomorphic to the product of the Chern four-manifold with S^n−4. We also obtain spin examples and five-dimensional lens-space homotopy types, the latter from finite quotients of one fixed $2$-connected eight-manifold. Finally, we realize aspherical manifolds with no closed PL representatives of their homotopy types as collapsing limits with scalar curvatures tending uniformly to +∞, and establish an obstruction to their approximation with almost nonnegative Ricci curvature.
ER -