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.

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The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained -they correspond to open handlebodies with all handles of index ≤ 2. An uncountable collection of exotic R 4 's is shown to admit Stein structures. New invariants of contact 3-manifolds are produced, including a complete (and computable) set of invariants for determining the homotopy class of a 2-plane field on a 3-manifold. These invariants are applicable to Seiberg-Witten theory. Several families of oriented 3-manifolds are examined, namely the Seifert fibered spaces and all surgeries on various links in S 3 , and in each case it is seen that "most" members of the family are the oriented boundaries of Stein surfaces.One of the main techniques of this paper is to describe handle decompositions of Stein surfaces explicitly using Kirby calculus. While this method has already been applied in simple cases without 1-handles [E5], the general case is more delicate. In Section 2, we establish a standard form for any handle decomposition obtained from a strictly plurisubharmonic function on a compact Stein surface. We do this via a standard form for Legendrian links in the connected sum #nS 1 × S 2 that allows us to define and compute the rotation number and Thurston-Bennequin invariant of each link component (even those that are nontrivial in H 1 ). We also provide a complete reduction from Legendrian link theory in #nS 1 × S 2 to a theory of diagrams by introducing a complete set of "Reidemeister moves." These diagrams allow us to construct Stein surfaces by drawing pictures. For example, we obtain the above exotic R 4 's in this manner. A Legendrian link diagram in #nS 1 × S 2 also determines a (positively oriented) contact 3-manifold (M, ξ), namely the oriented boundary of the corresponding compact Stein surface. We say that (M, ξ) is obtained by contact surgery on the Legendrian link, and that (M, ξ) is holomorphically fillable. In Section 5, we construct several families of examples. We realize "most" oriented Seifert fibered 3-manifolds by contact surgery, including all with (possibly nonorientable) base = S 2 , and both orientations on many Brieskorn homology spheres (Theorem 5.4 and Corollary 5.5). We show that any Seifert fibered space can be realized in this manner after possibly reversing orientation. For hyperbolic examples, we realize "most" rational surgeries on the Borromean rings (Theorem 5.9).We also introduce new invariants for distinguishing contact structures. We define a complete set of invariants for determining the homotopy class of an oriented 2-plane field on an oriented 3-manifold M . These invariants are readily computable for the boundary of a compact Stein surface presented in standard form. An explicit formula for the 2-dimensional obstruction (which measures the associated spin c -structure) is given by Theorem 4.12. The 3-dimensional obstruction (which, by recent work of Kronheimer and Mrowka [KM], distinguishe...

Witten's 2 + 1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant.

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