Seifert hypersurfaces of 2-knots and Chern–Simons functional

For a given smooth 2-knot in S^4 , we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible SU(2)-representations of its knot group. For example, we see that any smooth 2-knot having the Poincaré homology 3-sphere as a Seifert hypersurface has at le...

Full description

Saved in:
Bibliographic Details
Published in:Quantum topology 2022-03, Vol.13 (2), p.335-405
Main Author: Taniguchi, Masaki
Format: Article
Language:English
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:For a given smooth 2-knot in S^4 , we relate the existence of a smooth Seifert hypersurface of a certain class to the existence of irreducible SU(2)-representations of its knot group. For example, we see that any smooth 2-knot having the Poincaré homology 3-sphere as a Seifert hypersurface has at least four irreducible SU(2)-representations of its knot group. This result is false in the topological category. The proof uses a quantitative formulation of instanton Floer homology. Using similar techniques, we also obtain similar results about codimension-1 embeddings of homology 3-spheres into closed definite 4-manifolds and a fixed point type theorem for instanton Floer homology.
ISSN:1663-487X
1664-073X
DOI:10.4171/qt/165