Geography of pinched four-manifolds

TL;DR

This paper establishes new topological restrictions on 4-manifolds with pinched sectional curvature, showing that certain simply connected cases are homeomorphic to standard 4-manifolds like S^4 or CP^2.

Contribution

It provides novel restrictions on the Euler characteristic and signature of 4-manifolds with pinched curvature, advancing understanding of their topology.

Findings

Simply connected 4-manifolds with sectional curvature between δ and 1 are homeomorphic to S^4 or CP^2.

New curvature bounds lead to topological classification results.

Restrictions apply to manifolds with positive or negative pinched curvature.

Abstract

We prove several new restrictions on the Euler characteristic and signature of oriented 4-manifolds with (positively or negatively) pinched sectional curvature. In particular, we show that simply connected 4-manifolds with $\delta\leq \sec\leq 1$, where $\delta=\frac{1}{1+3\sqrt3}\cong 0.161$, are homeomorphic to $S^4$ or $\mathbb C P^2$.