The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [66-70]. He has shown that geometry has an important role to play in the theory in addition to the use of purely topological methods. The basic aim of this article is to discuss the various geometries which arise and explain their significance for the theory of 3-manifolds. The idea is that many 3-manifolds admit 'nice' metrics which give one new insight into properties of the manifolds. For the purposes of this article, the nicest metrics are those of constant curvature. An observer in a manifold with a constant curvature metric will see the same picture wherever he stands and in whichever direction he looks. Such manifolds have special topological properties. However, we will also need to consider nice metrics which are not of constant curvature. In this article, I will explain what is meant by a 'nice' metric and describe their classification in dimension three which is due to Thurston. Then I will discuss some of the 3-manifolds which admit these nice metrics and the relationship between their geometric and topological properties. In this introduction all manifolds and metrics will be assumed to be smooth so that the objects of interest are all Riemannian manifolds. It has been known since the nineteenth century that in dimension two there is a very close relationship between geometry and topology. I will start by describing some basic facts about closed surfaces. I will discuss these in more detail in §1. Each

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The geometries of 3-manifolds

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Groupes et algébres de lie

http://compalg.inf.elte.hu/~tony/Kutatas/BinPacking/Johnson-BinPacking-NP-Column-JAlg-2002.pdf

The NP-completeness column: An ongoing guide

In [D], Dehn considered the following method for constructing 3-manifolds: remove a solid torus neighborhood N(K) of some knot X in the 3-sphere S and sew it back differently. In particular, he showed that, taking X to be the trefoil, one could obtain infinitely many non-simply-connected homology spheres o in this way. Let Mx = S —N(K). Then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus oMK that bounds a meridional disk in the re-attached solid torus. We denote the resulting closed oriented 3-manifold by MK(), and say that it is obtained by r-Dehn surgery on X. More generally, one can consider the manifolds ML(*) obtained by r-Dehn surgery on a fc-component link L = Xi U • • • U Kk in S, where r = (r\,..., ;>). It turns out that every closed oriented 3-manifold can be constructed in this way [Wal, Lie]. Thus a good understanding of Dehn surgery might lead to progress on general questions about the structure of 3-manifolds. Starting with the case of knots, it is natural to extend the context a little and consider the manifolds M(r) obtained by attaching a solid torus V to an arbitrary compact, oriented, irreducible (every 2-sphere bounds a 3-ball) 3-manifold M with dM an incompressible torus, where r is the isotopy class (slope) on dM of the boundary of a meridional disk of V. We say that M(r) is the result of r-Dehn filling on M. An observed feature of this construction is that

Dehn surgery on knots

This answers a question apparently first raised by Tietze [T, p. 83]. It was previously known that there were at most two knots with a given complement [CGLS, Corollary 3]. The notion of equivalence of knots can be strengthened by saying that K and K' are isotopic if the above homeomorphism h is isotopic to the identity, or, equivalently, orientation-preserving. The analog of Theorem 1 holds in this setting too: if two knots have complements that are homeomorphic by an orientation-preserving homeomorphism, then they are isotopic. Theorem 1 and its orientation-preserving version are easy consequences of the following theorem concerning Dehn surgery.

/pdf/knots-are-determined-by-their-complements-4wg4xmlhom.pdf

Knots are determined by their complements

Reducing heegaard splittings

The main purpose of this paper is to show that any embedding of K7 in three-dimensional euclidean space contains a knotted cycle. By a similar but simpler argument, it is also shown that any embedding of K6 contains a pair of disjoint cycles which are homologically linked.

Knots and links in spatial graphs

On the signature of a link

/pdf/dehn-surgery-and-satellite-knots-q1hrckiwq8.pdf

C. McA. Gordon

Papers

Knots are determined by their complements

Reducing heegaard splittings

Knots and links in spatial graphs

On the signature of a link

Dehn surgery and satellite knots