Round handles, logarithmic transforms, and smooth 4-manifolds

Abstract

Round handles are affiliated with smooth 4-manifolds in two major ways: 5-dimensional round handles appear extensively as the building blocks in cobordisms between 4-manifolds, whereas 4-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 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's, which is a 4-dimensional analogue 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 CP^2, -CP^2 and S^1 x S^3. These answer some of the open problems posted by Ron Stern, 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 S^2 x S^2 if members are all non-spin, and with S^2 x S^2 # -CP^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.

17 pages, 1 figure. Submitted version; contains minor revisions and added references

References in corpus (3)

• Will we ever classify simply-connected smooth 4-manifolds?
• An SO(3)-version of 2-torsion instanton invariants
• Existence of broken Lefschetz fibrations

Cited by in corpus (13)

• Stable isotopy in four dimensions
• Isotopy of the Dehn twist on K3#K3 after a single stabilization
• Indefinite Morse 2-functions; broken fibrations and generalizations
• Knotted surfaces in 4-manifolds and stabilizations
• Luttinger surgery and Kodaira dimension
• Lefschetz fibrations on nonorientable 4-manifolds
• A Levine-Tristram invariant for knotted tori
• Surgery on tori in the 4-sphere
• Topologically isotopic and smoothly inequivalent 2-spheres in simply connected 4-manifolds whose complement has a prescribed fundamental group
• On dissolving knot surgery $4$-manifolds under a ${\mathbb{CP}}^2$-connected sum
• Branched covering simply-connected 4-manifolds
• On entropies, $\mathcal{F}$-structures, and scalar curvature of certain involutions
• Generic fibrations around multiple fibers