Torsion linking forms on surface-knots and exact 4-manifolds (Q2725547)

A surface-knot is a connected oriented surface \(F\) which is smoothly (or locally flat PL) embedded in the 4-sphere \(S^4\). The genus of the surface-knot is the genus of \(F\). In this paper, the author considers surface knots \(F\) with positive genus. Two such surface-knots \(F_1\) and \(F_2\) are said to belong to the same knot type if there is a diffeomorphism (respectively a PL homeomorphism) \(f\) of \(S^4\) sending \(F_1\) to \(F_2\) and such that \(f\) and its restriction fo \(F_1\) are orientation-preserving. A basic problem that is considered here is to determine when two surface-knots of the same genus belong to the same knot-type. One invariant of course is the surface-knot group \(\pi_1(S^4-F)\) and its peripheral subgroup, that is, the image of the natural homomorphism \(\pi_1(\partial E_F,b_0)\to\pi_1(E_F,b_0)\), where \(E_F\) is the compact exterior of \(F\) and \(b_0\) a basepoint in \(\partial E_F\). NEWLINENEWLINENEWLINEIn this paper, the author studies the torsion linking \(\ell_F\) of the surface knot \(F\), a \(Q/Z\)-valued non-singular bilinear form on a finite abelian group, which is derived from the infinite cylic covering homology of \(E_F\) by applying a duality which the author established in a previous paper. The torsion linking is a surface-knot invariant which is different from the surface-knot group and its peripheral subgroup. The author observes that there exists a pair of genus one surface-knots whose surface-knot groups are isomorphic by a meridian-preserving and peripheral-subgroup preserving isomorphism, but the torsion linking of one of them is the zero form whereas the torsion linking of the other one is a non-zero form. The author identifies then the torsion linking of a surface-knot with the torsion linking of an associated closed 4-manifold with infinite cyclic first homology. Some of the proofs are postponed to another paper.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00034].