Round handles, logarithmic transforms and smooth 4-manifolds

Abstract

Round handles are affiliated with smooth 4-manifolds in two major ways: five-dimensional round handles appear extensively as the building blocks in cobordisms between 4-manifolds, whereas four-dimensional round handles are the building blocks of broken Lefschetz fibrations on them. The purpose of this article is to shed more light on these interactions. We prove that if X and X′ are two cobordant closed smooth 4-manifolds with the same euler characteristics, and if one of them is simply connected, then there is a cobordism between them which is composed of round 2-handles only, and therefore one can pass from one to the other via a sequence of generalized logarithmic transforms along tori. As a corollary, we obtain a new proof of a theorem of Iwase, which is a four-dimensional analog of the Lickorish–Wallace theorem for 3-manifolds: Every closed simply connected 4-manifold can be produced by a surgery along a disjoint union of tori contained in a connected sum of copies of ℂℙ2, ℂ ℙ ¯ 2 and S1 × S3. These answer some of the open problems posed by Ronald Stern [Floer homology, gauge theory, and low-dimensional topology, Clay Mathematics Proceedings 5 (American Mathematical Society, Providence, RI, 2006) 225–239], while suggesting more constraints on the cobordisms in consideration. We also use round handles to show that every infinite family of mutually non-diffeomorphic closed smooth oriented simply connected 4-manifolds in the same homeomorphism class constructed up to date consists of members that become diffeomorphic after one stabilization with S2×S2 if members are all non-spin, and with S 2 × S 2 # ℂ ℙ ¯ 2 if they are spin. In particular, we show that simple cobordisms exist between knot surgered manifolds. We then show that generalized logarithmic transforms can be seen as standard logarithmic transforms along fiber components of broken Lefschetz fibrations, and show how changing the smooth structures on a fixed homeomorphism class of a closed smooth 4-manifold can be realized as relevant modifications of a broken Lefschetz fibration on it.