Mapping class groups of negatively curved four-manifolds

Abstract

For every closed oriented smooth four-manifold $M$ admitting a metric of strictly negative sectional curvature, we prove that $\dim_{\mathbb{Q}}H_1(T(M);\mathbb{Q})=\dim_{\mathbb{Q}}H_1(T_{\mathrm{TOP}}(M);\mathbb{Q})=\infty$ where $T(M)$ and $T_{\mathrm{TOP}}(M)$ denote the Torelli groups. This extends a theorem of Budney--Gabai for real hyperbolic manifolds.