The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theory

Invariants of 3-manifolds via link polynomials and quantum groups

© Publications mathématiques de l’I.H.É.S., 1977, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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Infinitesimal computations in topology

0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four. Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five. There is such a principle. It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary. The main impact, as outlined in §1, is to the classification of 1-connected 4-manifolds and topological end recognition. However, certain applications to nonsimply connected problems such as knot concordance are also obtained. The discovery of this principle was made in three stages. From 1973 to 1975 Andrew Casson developed his theory of "flexible handles". These are certain pairs having the proper homotopy type of the common place open 2-handle H = (D X D, dD X D) but "flexible" in the sense that finding imbeddings is rather easy; in fact imbedding is implied by a homotopy theoretic criterion. It was clear to Casson that: (1) no known invariant—link theoretic

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The topology of four-dimensional manifolds

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

V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century. This report is my response. Arnold's invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay. I have listed 18 problems, chosen with these criteria:

Mathematical problems for the next century

Since Poincare's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.

Foundational Essays on Topological Manifolds, Smoothings, and Triangulations.

A calculus for framed links in S 3

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The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, C)

Handlebodies and framed links.- Intersection forms.- Classification theorems.- Spin structures.- T Lie 3 and .- Immersing 4-manifolds in R 6.- 3-Manifolds a digression.- Bounding 5-manifolds.- p 1(M) = 3?(M), ? 4 so = Z and ? 4 spin = Z.- Wall's diffeomorphisms and H-cobordism.- Rohlin's theorem.- Casson handles.- Freedman's work.- Exotic R 4's.

Robion Kirby

Papers

Foundational Essays on Topological Manifolds, Smoothings, and Triangulations.

A calculus for framed links in S 3

The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, C)

The topology of 4-manifolds

Deformations of Spaces of Imbeddings