Codimension 2 embeddings, algebraic surgery and Seifert forms
Abstract
We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings , 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 under isotopy and cobordism. The second main result (update: which is false) is that the -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.
Cite
@article{arxiv.1211.5964,
title = {Codimension 2 embeddings, algebraic surgery and Seifert forms},
author = {Maciej Borodzik and András Némethi and Andrew Ranicki},
journal= {arXiv preprint arXiv:1211.5964},
year = {2018}
}
Comments
The paper contains an error. Main Theorem 2 is false. We are currently working on fixing the result (it will be more subtle), but it might take time. To the best of our knowledge Main Theorem 1 is correct and Main Theorem 3 should be restated in an appropriate way to work