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.

Abstract. We modify the construction of knot Floer homology to produce a one-parameter family of homologies tHFK for knots in S 3 . These invariants can be used to give homomorphisms from the smooth concordance group C to Z, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms.

Abstract. We define invariants of null-homologous Legendrian and transverse knots in contact 3-manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non-loose knots in overtwisted 3-spheres. Moreover, we apply the invariants to find transversely non-simple knot types in many overtwisted contact 3-manifolds.

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