4-Manifolds and Kirby Calculus

Abstract

This text is intended to be an introduction and reference for the differential topology of 4-manifolds as it is currently understood. It is presented from a topologist's viewpoint, often from the perspective of handlebody theory (Kirby calculus), for which an elementary and comprehensive exposition is given. Additional topics include complex, symplectic and Stein surfaces, applications of gauge theory, Lefschetz pencils and exotic smooth structures. The text is intended for students and researchers in topology and related areas, and is suitable for an advanced graduate course. Familiarity with basic algebraic and differential topology is assumed.

The symplectic topology of some rational homology balls

“…The smooth handlebody description of B p,q consisting of just one 1-handle and one 2-handle seems to be not widely known for q > 1 (see Figure 8.41 in [18] for q = 1 which matches with the above picture) . Here, we provide not only a smooth handlebody description but also a Legendrian realization of the attaching circle of the 2-handle as a Legendrian knot in S 1 × S 2 such that the smooth framing is given by tb − 1, hence this description equips B p,q with a Stein structure (see [18] Chapter 11).…”

“…The bottom figure in Figure 4 depicts the quotient diagram for the action Γ p,q on S p−1 yielding B p,q . (See Section 6.3 [18] for a discussion of finite covers of handlebody diagrams). Here, there is a unique 2-handle that passes through the 1-handle p times and it has framing tb − 1 = −pq − 1.…”

“…Here, we provide not only a smooth handlebody description but also a Legendrian realization of the attaching circle of the 2-handle as a Legendrian knot in S 1 × S 2 such that the smooth framing is given by tb − 1, hence this description equips B p,q with a Stein structure (see [18] Chapter 11). From our description, it also follows that the Stein structure that we obtain this way is the same as the Stein structure induced from S p−1 via the action Γ p,q .…”

A Note on Scalar Curvature Functions of Almost-Kähler Metrics

“…Recently, there has been much interest on symplectic manifolds [5,6,10]. So we would like to study an analogous question on the scalar curvature functions of almost Kähler metrics on symplectic manifolds.…”

Hyperbolic structure on a complement of tori in the 4-sphere

“…Since H 4 N = H 0 N = Z and H 1 N = H 3 N = 0 this means H 2 N = 0. Now due to Freedman's (see [2,3]) classification of simply-connected 4-manifolds, the only simply-connected 4 manifold N with H 2 N = 0 is the 4-sphere.…”