4-manifolds: Introduction Surfaces in 4-manifolds Complex surfaces Kirby calculus: Handelbodies and Kirby diagrams Kirby calculus More examples Applications: Branched covers and resolutions Elliptic and Lefschetz fibrations Cobordisms, $h$-cobordisms and exotic ${\mathbb{R}}^{4,}$s Symplectic 4-manifolds Stein surfaces Appendices: Solutions Notation, important figures Bibliography Index.

4-manifolds and Kirby calculus

Holomorphic disks and knot invariants

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

We use the Ozsvath-Szabo theory of Floer homology to define an invariant of knot complements in three-manifolds This invariant takes the form of a filtered chain complex, which we call CF_r It carries information about the Floer homology of large integral surgeries on the knot Using the exact triangle, we derive information about other surgeries on knots, and about the maps on Floer homology induced by certain surgery cobordisms We define a certain class of \em{perfect} knots in S^3 for which CF_r has a particularly simple form For these knots, formal properties of the Ozsvath-Szabo theory enable us to make a complete calculation of the Floer homology This is the author's thesis; many of the results have been independently discovered by Ozsvath and Szabo in mathGT/0209056

Floer homology and knot complements

Topological field theory

In this paper we investigate the relationship between isotopy classes of knots and links in S and the dieomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of dieomorphism types of manifolds homeomorphic to the K3 surface. In order to state our theorems we need to view the Seiberg-Witten invariant of a smooth 4-manifold as a multivariable (Laurent) polynomial. To do this, recall that the Seiberg-Witten invariant of a smooth closed oriented 4-manifold X with b2 X  > 1 is an integer valued function which is de®ned on the set of spinc structures over X , (cf. [W], [KM], [Ko1], [T1]). In case H1X ;Z has no 2-torsion (which will be the situation in this paper) there is a natural identi®cation of the spinc structures of X with the characteristic elements of H2X ;Z. In this case we view the Seiberg-Witten invariant as

Knots, links, and 4-manifolds

In this paper we introduce a surgical procedure, called a rational blowdown, for a smooth 4-manifold X and determine how this procedure affects both the Donaldson and Seiberg-Witten invariants of X.

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Rational blowdowns of smooth 4-manifolds

Knots, Links, and 4-Manifolds

For an oriented integral homology 3-sphere 2, A. Casson has introduced an integer invariant A(2) that is defined by using the space £%(2) of conjugacy classes of irreducible representations of ^ ( 2 ) into SU(2) (see [1]). This invariant A(2) can be computed from a surgery or Heegard description of 2 and satisfies A(2) = JU(2) (mod 2), where /i(2) is the Kervaire-Milnor-Rochlin invariant of 2. This powerful new invariant was used to settle an outstanding problem in 3-manifold topology; namely, showing that if 2 is a homotopy 3-sphere, then ju(2) = O. C. Taubes [17], utilizing gauge-theoretic considerations, has reinterpreted Casson's invariant in terms of flat connections. Refining this approach, A. Floer [13] has recently defined another invariant of 2 , its 'instanton homology', which takes the form of an abelian group /*(2) with a natural Z8-grading that is an enhancement of A(2) in that

Ronald Fintushel

Papers

Knots, links, and 4-manifolds

Rational blowdowns of smooth 4-manifolds

Knots, Links, and 4-Manifolds

Instanton Homology of Seifert Fibred Homology Three Spheres

Immersed spheres in 4-manifolds and the immersed Thom conjecture