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

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation - the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

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Quandle cohomology and state-sum invariants of knotted curves and surfaces

Function complexes in homotopical algebra

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

3-Manifold Groups

A Survey of Knot Theory

A linking is understood as a pair (G, ~b) such that G is a finite abelian group and ~b is a nonsingular, symmetric bilinear pairing G x G~Q/Z. It is convenient to identify a linking (G, ~) with a matrix which represents ~b relative to the generators of a cyclic splitting of G unless confusion might occur. The linking occurs often in the study of topology. For example, given a closed oriented 3-manifold M, we have a unique linking (~HI(M),q~M) defined by the Poincar~ duality, where "rHI(M ) is the torsion part of the integral homology group Hi(M). The purpose of this paper is to determine completely the structure of the abelian semigroup 9l of all linkings (up to isomorphism) under block sum and to observe that any linking is isomorphic to the linking ~b~ of a closed connected oriented 3-manifold M. To do this, we shall present a complete system of invariants of isomorphic linkings, which arises naturally from our purpose. Such a complete system had already been known by Seifert [ 11] in the case of odd-primary groups, and in general by Burger [2, Satz 5] in terms of Minkowski's beautiful theory. Though they are related directly or indirectly to each other (cf. Fox [5]), we do not discuss here any relation between them.

Akio Kawauchi

Papers

A Survey of Knot Theory

Algebraic classification of linking pairings on 3-manifolds

Descriptions on Surfaces in Four-Space I Normal Forms

Proposals for unknotted surfaces in four-spaces

Polynomials of amphicheiral knots