Gespeichert in:
| 1. Verfasser: | |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2404.07478 |
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Inhaltsangabe:
- For a Gromov-Hausdorff convergent sequence of closed manifolds $M_i^n\overset{GH}\longrightarrow X$ with $\mathrm{Ric}\ge-(n-1)$, $\mathrm{diam}(M_i)\le D$, and $\mathrm{vol}(M_i)\ge v>0$, we study the relation between $π_1(M_i)$ and $X$. It was known before that there is a surjective homomorphism $ϕ_i:π_1(M_i)\to π_1(X)$ by the work of Pan-Wei. In this paper, we construct a surjective homomorphism from the interior of the effective regular set in $X$ back to $M_i$, that is, $ψ_i:π_1(\mathcal{R}_{ε,δ}^\circ)\to π_1(M_i)$. These surjective homomorphisms $ϕ_i$ and $ψ_i$ are natural in the sense that their composition $ϕ_i \circ ψ_i$ is exactly the homomorphism induced by the inclusion map $\mathcal{R}_{ε,δ}^\circ \hookrightarrow X$.