Instantons for 4-manifolds with periodic ends and an obstruction to embeddings of 3-manifolds

Abstract

We construct an obstruction to the existence of embeddings of a homology 3-sphere into a homology S 3 × S 1 under some cohomological condition. The obstruction are defined as an element in the filtered version of the instanton Floer cohomology due to [6]. We make use of the Z-fold covering space of homology S 3 × S 1 and the instantons on it.We will use the following assumption on Y in our main theorem(Theorem 2.4).Assumption 2.2. All SU (2) flat connections on Y are non-degenerate, i.e. the first cohomology group of the next twisted de Rham complex:Example 2.3. All flat connections on the Brieskorn homology 3-sphere Σ(p, q, r) are non-degenerate.([5])Under Assumption 2.2, l Y is finite ([14]). In this paper without the use of Assumption 2.2, we will introduce the following invariants:• Q i X ∈ R ≥0 ∪ {∞} for i ∈ N and X in Definition 4.6 (When X is homotopy equivalent to S 3 × S 1 , Q i X = ∞ for all i ∈ N). Our main theorem is: Theorem 2.4. Under Assumption 2.2, if there exists an embedding f of Y into X with f * [Y ] = 1 ∈ H 3 (X, Z) then [θ r ] vanishes for any r ∈ [0, min{Q 2lY +3 X , 1}]∩ (R \ cs Y ( R(Y )) ∪ {∞})

Seifert hypersurfaces of 2-knots and Chern-Simons functional

“…author also appreciates Aliakbar Daemi for discussing an alternative proof of Theorem 7. The functionals {cs j X,c } j∈Z>0 are originally defined in [36]. Let X be a closed connected oriented 4manifold.…”

“…In this paper, we treat a slightly large class P * (Y ) of perturbations rather than P(Y ) in [36]. This class is used in [34] to calculate the instanton homologies for Seifert homology 3-spheres.…”

“…We also define non-degenerate and regular perturbations for elements in P * (Y ) by the same way as in the case of P(Y ). (See [36].) For an oriented homology 3-sphere Y and a fixed metric g Y on Y , there exist a positive integer d, f ∈ F d and a Bair subset…”

“…We define P * (Y, g, r, s) := P * ε(Y,g,r,s) (Y, g). One can define a class of non-degenerate perturbations in P * (Y ) as in the same definition in [36]. For a non-degenerate perturbation π ∈ P * (Y ), we have a map ind : R(Y…”

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We introduce a real-valued functional csK on the SU (2)-representation space of the knot group for any oriented 2-knot K. In order to define the functionals, we use a generalization for homology S 1 × S 3 's of the Chern-Simons functionals introduced in [36]. We calculate our functionals for ribbon 2-knots and the twisted spun 2-knots of torus knots, Montesinos knots and 2-bridge knots.

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