Simple connectivity of the positive surface complement

Determine whether the complement of the genus-five, self-intersection-two symplectic surface in the simply connected manifold X_{2,3} is simply connected; equivalently, determine whether the group of the complement is trivial rather than merely perfect and normally generated by the meridian.

Background

The paper studies the complement C = X_{2,3} \setminus \operatorname{int}\nu F, where X_{2,3} is a simply connected minimal symplectic four-manifold and F is a genus-five symplectic surface obtained by resolving the positive intersection of square-zero surfaces of genera two and three. The authors prove that H_1(C;\mathbb Z)=0, so \pi_1(C) is perfect, and that \pi_1(C) is normally generated by the meridian of F.

These properties do not establish that \pi_1(C) is trivial, because nontrivial perfect groups normally generated by one element can exist. The authors explain that, without a geometric argument killing the meridian after gluing, an explicit relative presentation would be needed; however, the question of whether C itself is simply connected remains unresolved in the paper. The subsequent symplectic-sum fundamental-group calculation avoids requiring a full computation of \pi_1(C).