Locally flat $2$-knots in $S^ 2\times S^ 2$ with the same fundamental group

Abstract:

We consider a locally flat $2$-sphere in ${S^2} \times {S^2}$ representing a primitive homology class $\xi$, which is referred to as a $2$-knot in ${S^2} \times {S^2}$ representing $\xi$. Then for any given primitive class $\xi$, there exists a $2$-knot in ${S^2} \times {S^2}$ representing $\xi$ with simply-connected complement. In this paper, we consider the classification of $2$-knots in ${S^2} \times {S^2}$ whose complements have a fixed fundamental group. We show that if the complement of a $2$-knot $S$ in ${S^2} \times {S^2}$ is simply connected, then the ambient isotopy type of $S$ is determined. In the case of nontrivial ${\pi _1}$, however, we show that the ambient isotopy type of a $2$-knot in ${S^2} \times {S^2}$ with nontrivial ${\pi _1}$ is not always determined by ${\pi _1}$.