The peripheral subground and the second homology of the group of a knotted torus in \(S^ 4\) (Q1804695)

Let \(F\) be an embedded torus in \(S^4\), and let \(\pi F=\pi_1(S^4-F)\). Let \(\tau F\) be the the image of \(\pi_1(F)\cong Z^2\) in \(\pi F\). If \(A\) and \(B\) are abelian groups write \(A\leq B\) if \(A\) is a quotient of \(B\). Then \(\tau F\leq Z^2\) and \textit{R. A. Litherland} [Q. J. Math., Oxf. II. Ser. 32, 425-434 (1981; Zbl 0506.57013)] showed that \(H_2(\pi F;Z)\leq\tau F\), and that any quotient of \(Z^2\) is realizable as \(H_2(\pi F;Z)\) for some torus \(F\) with \(\tau F\cong Z^2\). This paper considers the question of realizing pairs \(A\), \(B\) such that \(A\leq B\leq Z^2\), and shows that this is possible provided that either \(B\) is torsion free, or \(B\) has rank one and \(A\) is finite, or \(B\) is cyclic. (The remaining cases are undecided).