Symmetric surgery and boundary link maps (Q1298113)

The author shows that any link map \(\coprod_i S^N\to S^{N+2}\) which bounds disjoint maps of Seifert surfaces is homotopically trivial. This means that there is a motion to the unlink in which the spheres stay disjoint (but are allowed to self-intersect). The proof generalizes to all dimensions a procedure known as symmetric surgery or contraction/pushoff from 4-manifold theory. This procedure addresses the freedom of choosing one of the two possible spheres when doing surgery on a hyperbolic summand in the intersection form of an even dimensional manifold. The author also gives the first example of a boundary link which is not homotopically trivial and proves that this can only happen for non-parallelizable Seifert surfaces and in dimension \(\geq 7\).