Ricci curvature and fundamental groups of effective regular sets

For a Gromov-Hausdorff convergent sequence of closed manifolds MᵢⁿGH X with Ric≥-(n-1), diam(Mᵢ)≤ D, and vol(Mᵢ)≥ v>0, we study the relation between π₁(Mᵢ) and X. It was known before that there is a surjective homomorphism φᵢ:π₁(Mᵢ)→ π₁(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 Mᵢ, that is, ψᵢ:π₁(Rε,δ^∘)→ π₁(Mᵢ). These surjective homomorphisms φᵢ and ψᵢ are natural in the sense that their composition φᵢ ∘ ψᵢ is exactly the homomorphism induced by the inclusion map Rε,δ^∘ X.