Seifert hypersurfaces of 2-knots and Chern-Simons functional

Abstract

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.

arXiv comment46 pages, v3, to appear in Quantum Topology

References in corpus (4)

• A splitting theorem for the Seiberg-Witten invariant of a homology $S^1\times S^3$
• A connected sum formula for the SU(3) Casson invariant
• A proof via the Seiberg-Witten moduli space of Donaldson's theorem on smooth 4-manifolds with definite intersection forms
• Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds

Cited by in corpus (1)

• Positive scalar curvature and homology cobordism invariants