The non-simply connected Price twist for the 4-sphere

A cutting and pasting operation on a $P^2$-knot $S$ in a $4$-manifold is called the Price twist. The Price twist for the $4$-sphere $S^4$ yields at most three $4$-manifolds up to diffeomorphism, namely, the $4$-sphere $S^4$, the other homotopy $4$-sphere $\Sigma_{S}(S^4)$ and a non-simply connected $4$-manifold $\tau_{S}(S^4)$. In this paper, we study some properties and diffeomorphism types of $\tau_{S}(S^4)$ for $P^2$-knots $S$ of Kinoshita type.