Converse to simple connectivity of the Lidman–Piccirillo double

Determine whether simple connectivity of the admissible double Z' = V'\cup_\sigma V' implies simple connectivity of the corresponding admissible Lidman–Piccirillo piece V', where V' is obtained from the genus-2 surface bundle R by the specified pair of Luttinger surgeries and \sigma is the boundary involution.

Background

The paper proves that \pi_1(V')=1 implies \pi_1(Z')=1 by van Kampen, and uses the stronger hypothesis \pi_1(V')=1 to control the fundamental group of the reglued manifold yielding the exotic CP^2#^2 result. The converse would ask whether triviality of the double’s fundamental group already forces triviality of the fundamental group of one half.

The issue arises because \pi_1(Z') is described as an amalgamated product of two copies of \pi_1(V') over the boundary group \pi_1(S^3_0(Q)), and the edge maps into the vertex groups need not be injective. Consequently, cancellations in the amalgam could potentially make the double simply connected even when \pi_1(V') is nontrivial. The paper explicitly leaves this implication unresolved, while noting that its computational approach establishes \pi_1(V')=1 directly for the specified surgery piece.