Instability of the fundamental group for non-collapsed Ricci-limits

TL;DR

The paper proves a negative answer to Pan's questions by constructing two sequences $(M_i)$ and $(N_i)$ with ${\rm Ric}_{M_i}\ge 0$, ${\rm diam}(M_i)\le 1$, ${\rm vol}(M_i)\ge v>0$ and $d_{GH}(M_i,N_i)\to 0$, yet ${\pi_1(M_i)}\cong {\mathbb Z}/2{\mathbb Z}$ and ${\pi_1}(N_i)=0$. The construction relies on glueing from the Eguchi–Hanson space, a free ${\mathbb Z}/2{\mathbb Z}$-action, and a desingularization of ${\mathbb C}^2/{\mu_4}$ to obtain two families with ${\rm Ric}\ge 0$. Both sequences converge in the Gromov–Hausdorff sense to the spherical suspension of $cS^3/{\mu_4}$ (for a small $c>0$), showing that the limit does not determine the fundamental group. Thus the fundamental-group data cannot be recovered from Ricci-limit information in the non-collapsed setting.

Abstract

We construct two sequences of closed $4$-dimensional manifolds with non-negative Ricci curvature, diameter bounded from above by $1$, and volume bounded from below by $v>0$, with different fundamental groups but with the same Gromov-Hausdorff limit. This provides a negative answer to the question posed in [J. Pan. Ricci Curvature and Fundamental Groups of Effective Regular Sets. Journal of Mathematical Study, 58(1):3--21, 2025].

Theorems & Definitions (10)

• Theorem
• Theorem 1
• Theorem 2
• Lemma 3
• Theorem 4
• Theorem 5
• Proposition 6
• Lemma 7
• Lemma 8
• proof