Kawauchi defined a group structure on the set of homology S¹ × S²’s under an equivalence relation called H̃-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. We apply knot concordance invariants derived from knot Floer homology to study the kernel of the zero-surgery homomorphism. As a consequence, we show that the kernel contains a ℤ∞-subgroup in the smooth category. Moreover, this group can be defined in the topological category. There is a surjective homomorphism from the group defined in the smooth category to that defined in the topological category. We prove that if a homology S¹ × S² has the trivial Alexander polynomial, then it is contained in the kernel of the homomorphism by using Freedman and Quinn's result about ℤ-homology 3-spheres.
Read
