Equivariant intersection forms, knots in 𝑆⁴, and rotations in 2-spheres

Abstract

We consider the problem of distinguishing the homotopy types of certain pairs of nonsimply-connected four-manifolds, which have identical three-skeleta and intersection pairings, by the equivariant isometry classes of the intersection pairings on their universal covers. As applications of our calculations, we: (i) construct distinct homology four-spheres with the same three-skeleta, (ii) generalize a theorem of Gordon to show that any nontrivial fibered knot in S 4 {S^4} with odd order monodromy is not determined by its complement, and (iii) give a more constructive proof of a theorem of Hendriks concerning rotations in two-spheres embedded in threemanifolds.