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 ₁ (Mᵢ) and X. It was known before that there is a surjective homomorphism ᵢ: ₁ (Mᵢ) 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 Mᵢ, that is, ᵢ: ₁ (R, ^) ₁ (Mᵢ). These surjective homomorphisms ᵢ and ᵢ are natural in the sense that their composition ᵢ ᵢ is exactly the homomorphism induced by the inclusion map R, ^ X.