Round handles, logarithmic transforms and smooth 4-manifolds

Abstract

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 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'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 × S 3 . These answer some of the open problems posted by Ron Stern in [14], 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 × S 2 if members are all non-spin, and with S 2 × 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.

Knotted surfaces in 4-manifolds and stabilizations

“…Exotic 4-manifolds of course provide examples where at least one stabilization is necessary, whereas in [5], we showed that all construction methods employed up to date to generate infinite families of exotic 4-manifolds always yield 4-manifolds which become diffeomorphic after a single stabilization. 1 We will examine the analogous question for surface stabilizations: how many are needed to make exotic knottings smoothly isotopic?…”

“…On the other hand, by restricting themselves to surfaces whose complements have trivial fundamental group [14], the rim-surgered surfaces are indeed all topologically isotopic to the original surface. 5 It is only known that Σ K 1 and Σ K 2 are smoothly distinct in the case that Σ is symplectically embedded, the annulus' core is nontrivial in H 1 (Σ), and K 1 and K 2 have different Alexander polynomials.…”

On entropies, $${\mathcal {F}}$$ F -structures, and scalar curvature of certain involutions

“…Unlike the group actions involved in Theorem 1.6, the actions in [54] need not be free. A key ingredient in the proof of Theorem 1.6 is the recent work of Baykur-Sunukjan, which shows that the inequivalent smooth structures on simply connected manifolds under consideration become diffeomorphic after taking a connected sum with S 2 × S 2 [4].…”

Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds

“…If L were disjoint from the exceptional sphere S in X, we could blow-down S to get a Lagrangian embedding of L in CP 2 , which contradicts the main theorem of [16,18].) [7] show that if one only considers the underlying smooth structures, the two 4-manifolds would be equivalent via smooth surgeries along tori. However, when we in addition take compatible symplectic Lefschetz pencils on them, Theorem B dictates that there is no sequence of Luttinger surgeries taking one to the other.…”

“…[3] This is a Lefschetz pencil version of the -still open-question on the equivalence of integral symplectic 4-manifolds with the same characteristic numbers listed above via Luttinger surgeries [2], which in turn is a symplectic version of Stern's question on the equivalence of homeomorphic smooth 4-manifolds via smooth surgeries along tori [20], settled positively in [7].…”