Fukaya-Yamaguchi Conjecture in Dimension Four

TL;DR

The paper proves Fukaya–Yamaguchi's conjecture in dimensions $n\le 4$ by showing that any complete manifold with nonnegative sectional curvature has a fundamental group that is uniformly virtually abelian. The authors reduce to the compact-universal-cover case via the Cheeger–Gromoll soul theorem, then treat the compact case using Mundet i Riera’s finite-group action results and Gromov’s Betti-number bounds to obtain a finite-index abelian subgroup. For the noncompact universal cover, they apply the Toponogov splitting theorem to force a decomposition $\tilde{M}^n=\mathbb{R}^k\times N$ with $N$ compact, and use Bieberbach theory alongside a torus/finite-subgroup analysis to realize a bounded-index abelian subgroup of $\pi_1(M)$. Consequently, there exists a constant $C(n)$ with $[\pi_1(M):A]\le C(n)$ for some abelian $A\le \pi_1(M)$, confirming the conjecture in dimension four. This work highlights the fruitful interaction between low-dimensional geometric structure and fundamental-group behavior under nonnegative curvature, with potential implications for higher dimensions.

Abstract

Fukaya and Yamaguchi conjectured that if $M^n$ is a manifold with nonnegative sectional curvature, then the fundamental group is uniformly virtually abelian. In this short note we observe that the conjecture holds in dimensions up to four.

Theorems & Definitions (6)

• Theorem 1.1
• Lemma 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['l:compact']}
• proof : Proof of Theorem \ref{['t:FY_4d']}
• Remark 2.3