Lemma 18. Let Z=h^ -1 (c_*)_ red be the rational nodal fiber of Lemma 17. Some connected component Z_0 of Z satisfies |(π_1(Z_0)⟶π_1(X))| =∞.

Let F_s⊂U be a punctured smooth fiber, with s∈B' , and choose access paths when comparing its group with π_1(U) . Its image under ρ is dense in Q by Proposition 15. In particular its image in P_f is all of P_f , so its inverse image F'_s⊂U' is connected. Under the covering inclusion, its group is the preimage of N in π_1(F_s) . Thus the image of π_1(F'_s) in Q is ρ(π_1(F_s))∩N. It is dense in N , because N is open and the fiber image is dense in Q . The group N is infinite: a finite open subgroup would make its finite-index overgroup Q finite. Consequently the image of π_1(F'_s) in Q is infinite. By , its image in π_1(X) is infinite as well. We transfer this conclusion to Z using an explicit deformation-retract neighborhood. Regard X as a compact real analytic manifold and Z as a closed semianalytic subset. This pair admits a compatible finite triangulation [Section 3, Theorem 2, p.~464] Lojasiewicz1964 . Write it as a finite simplicial pair ( K, A) , and barycentrically subdivide so that A is a full subcomplex: any simplex whose vertices belong to A belongs to A . For a point of | K| , let s_ A be the sum of its barycentric coordinates at vertices of A . On the open neighborhood V= s_ A >0 retain those coordinates, divide them by s_ A , and set all other coordinates to zero. Fullness ensures that the result belongs to | A| . The straight-line homotopy inside each simplex gives a strong deformation retraction V→Z . In particular the inclusion of V into X is homotopic to a map through Z . Properness of h supplies the needed containment of nearby fibers. The set h(X V) is closed in C' and does not contain c_* . Hence every whole fiber over a sufficiently small neighborhood of c_* lies in V . Choose s∈B' in this neighborhood. Its connected punctured lifted fiber F'_s lies in one connected component of V ; the deformation retraction takes that component into a connected component Z_0 of Z . Therefore its inclusion-induced homomorphism to π_1(X) factors, up to a change-of-basepoint conjugation, through π_1(Z_0) . Since the former has infinite image, so does the latter.