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.

Positive scalar curvature and homology cobordism invariants

“…Equation ( 41) is deduced from ( 42), (43). Combining Lemma 3.14 with ( 38) and ( 41), we obtain the desired equality (35).…”

“…This theorem can be seen as a Seiberg–Witten analogue of the result proven by Yang–Mills gauge theory [43, Theorem 1.9]. Using the Heegaard Floer correction term, Levine–Ruberman [20] gave an obstruction of codimension‐1 smooth embeddings into homology $$S^1\times S^3$$'s.…”

Invariants of $2$-knots

“…where p : X c → X is the Z-cover determined by c ∈ H 1 (X), and A a is a connection on P X c = X c × SU (2) that, informally speaking, agrees with p * a on the left of X c and vanishes on the right of X c ; see [128,129] for details. Note that while X c is noncompact, the integral CS X,c (a) is finite because it can be expressed in terms of CS Y for a closed 3-dimensional submanifold Y ⊂ X that is Poincaré dual to c [128, Lemma 4.2].…”

“…The composition H 1 (X) c − → Z → Z j determines a Z j -cover p j : X j,c → X, and therefore a functional cs j X,c := cs X j,c ,p * j (c) , where p * j (c) ∈ H 1 (X j,c ). Taking X to be M K , the result of surgery on a smooth 2-knot K ⊂ S 4 , leads to the following definition, which is due to Taniguchi [129]. Definition 5.5.…”

“…Taniguchi proves that for every j ∈ Z, the set im(cs j M K ,c ) is finite and im(cs j M K ,c ) = {1} if K is ribbon [129]. These invariants also obstruct invertibility [ Remark 5.6.…”