(untitled) (Journal of Mathematical Study, 2025)
For a Gromov-Hausdorff convergent sequence of closed manifolds M^n_iGH X with Ric-(n-1), diam(M_i) ≤ D, and vol(M_i) ≥ 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) → 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(R^◦_ϵ,δ )→π_1(M_i). These surjective homomorphisms ϕ_i and ψ_i are natural in the sense that their composition ϕ_i◦ψ_i is exactly the homomorphism induced by the inclusion map R^◦_ ϵ,δX.