On 3-manifolds that are boundaries of exotic 4-manifolds

Abstract

In this note we show that any closed, oriented 3-manifold is the boundary of a simply connected 4-manifold that admits infinitely many distinct smooth structures. We also show that any fillable contact 3-manifold is the boundary of a simply connected 4-manifolds that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any contact manifold.

Diffeomorphisms of 4-manifolds with boundary and exotic embeddings

“…gives rise to a diffeomorphism between these moduli spaces corresponding to s R f and s R g , and this implies (18). To prove (19), as in the proof of Proposition 3.19, let γ# : [0, 1] → Ξ cont (Y ) denote the concatenated path of two copies of γ, which represents γ 2 ∈ π 1 (Ξ cont (Y )).…”

“…By construction, Y is smoothly embedded in W and Y bounds a submanifold W 1 with b + 2 (W 1 ) > 1. Now by attaching a simply connected symplectic cap on (Y , ξ ) with b + 2 > 1 (existence of such caps are shown in [18]) we can get our desired simply-connected 4-manifold X. Let s be the Spin c structure on X obtained as the connected sum of the canonical Spin c structure and the Spin c structure t on N considered in Section 4.…”