Codimension 2 embeddings, algebraic surgery and Seifert forms

Abstract

We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings $M^m\subset N^{m+2}$, using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincar\'e pairs, which is then applied to describe the behaviour on the chain level of Seifert surfaces of embeddings $M^{2n-1} \subset S^{2n+1}$ under isotopy and cobordism. The second main result (update: which is false) is that the $S$-equivalence class of a Seifert form is an isotopy invariant of the embedding, generalizing the Murasugi--Levine result for knots and links. The third main result is a generalized Murasugi--Kawauchi inequality giving an upper bound on the difference of the Levine--Tristram signatures of cobordant embeddings.