A Bauer–Furuta-type refinement of Kronheimer and Mrowka’s invariant for 4–manifolds with contact boundary

Abstract

We give Bennequin-Plamenevskaya-Shumakovitch type lower bounds for the concordance invariant s # introduced by Kronheimer and Mrowka. The proof is a consequence of computations for torus knots and the cobordism inequality of s # due to Gong, combined with well-known arguments used for slice-torus invariants. Contents 1. Introduction 1 Acknowledgement 2 2. Background 3 3. Proof of the main theorem 4 References 6

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

“…This comes from the fact that the map (3) is a homomorphism. In an upcoming paper [31], under suitable conditions on Y , we also prove the existence of a Z ∞ -summand in the abelianization of the kernel of π 0 (Diff(W, ∂)) → π 0 (Homeo(W, ∂)).…”

“…the cobordism obtained by removing small open disk from X and X respectively and let N : S 3 → S 3 be the cobordism obtained by removing two small open disks from −CP 2 . By the pairing formula which is proves in [31] and the composition law Poposition 26.1.2 of [40], we have…”

“…We establish the blow-up formula for Kronheimer-Mrowka's invariant for 4manifold with boundary, using the pairing formula and the formal (3 + 1)-TQFT property of the monopole Floer homology. As a partial result, [30,Theorem 1.3] combined with a computation of the Bauer-Furuta invariant for −CP 2 gives the blow up formula for Kronheimer-Mrowka's invariant for 4-manifold with boundary under the condition b 3 = 0. In this section, we remove the condition b 3 = 0 by following the discussion given in Section 39.3 of [40].…”

A note on generalized Thurston--Bennequin inequalities

“…Thus we give a proof of Theorem 1.4 below. The invariant introduced in [9] is used in the proof, so let us explain it. This invariant Ψ(X, ξ, s) is the Bauer-Furuta type homotopical version of Kronheimer-Mrowka's invariant for a 4-manifold with contact boundary.…”

“…Indeed, we can use a connected sum formula as follows: Ψ(W, s W , ξ ω ) = BF (X, s) ∧ Ψ(W (tb(K ′ 1 )−1,...,tb(K ′ n )−1) (L ′ ), s ω , ξ ω ), where BF (X, s) is the non-equivariant Bauer-Furuta invariant. Since the 4-manifold with boundary (W (tb(K ′ 1 )−1,...,tb(K ′ n )−1) (L ′ ), ω) is a Stein filling, from [9], we have Ψ(W (tb(K ′ 1 )−1,...,tb(K ′ n )−1) (L ′ ), s ω , ξ ω ) = Id . Thus we conclude that Ψ(W, s W , ξ ω ) is the same as BF (X, s), which is non-trivial.…”