Colin Rourke

Introduction to piecewise-linear topology

1. Polyhedra and P.L. Maps.- Basic Notation.- Joins and Cones.- Polyhedra.- Piecewise-Linear Maps.- The Standard Mistake.- P. L. Embeddings.- Manifolds.- Balls and Spheres.- The Poincare Conjecture and the h-Cobordism Theorem..- 2. Complexes.- Simplexes.- Cells.- Cell Complexes.- Subdivisions.- Simplicial Complexes.- Simplicial Maps.- Triangulations.- Subdividing Diagrams of Maps.- Derived Subdivisions.- Abstract Isomorphism of Cell Complexes.- Pseudo-Radial Projection.- External Joins.- Collars.- Appendix to Chapter 2. On Convex Cells.- 3. Regular Neighbourhoods.- Full Subcomplexes.- Derived Neighbourhoods.- Regular Neighbourhoods.- Regular Neighbourhoods in Manifolds.- Isotopy Uniqueness of Regular Neighbourhoods.- Collapsing.- Remarks on Simple Homotopy Type.- Shelling.- Orientation.- Connected Sums.- Schonflies Conjecture.- 4. Pairs of Polyhedra and Isotopies.- Links and Stars.- Collars.- Regular Neighbourhoods.- Simplicial Neighbourhood Theorem for Pairs.- Collapsing and Shelling for Pairs.- Application to Cellular Moves.- Disc Theorem for Pairs.- Isotopy Extension.- 5. General Position and Applications.- General Position.- Embedding and Unknotting.- Piping.- Whitney Lemma and Unlinking Spheres.- Non-Simply-Connected Whitney Lemma.- 6. Handle Theory.- Handles on a Cobordism.- Reordering Handles.- Handles of Adjacent Index.- Complementary Handles.- Adding Handles.- Handle Decompositions.- The CW Complex Associated with a Decomposition.- The Duality Theorems.- Simplifying Handle Decompositions.- Proof of the h-Cobordism Theorem.- The Relative Case.- The Non-Simply-Connected Case.- Constructing h-Cobordisms.- 7. Applications.- Unknotting Balls and Spheres in Codimension ? 3.- A Criterion for Unknotting in Codimension 2.- Weak 5-Dimensional Theorems.- Engulfing.- Embedding Manifolds.- Appendix A. Algebraic Results.- A. 1 Homology.- A. 2 Geometric Interpretation of Homology.- A. 3 Homology Groups of Spheres.- A. 4 Cohomology.- A. 5 Coefficients.- A. 6 Homotopy Groups.- A. 8 The Universal Cover.- Appendix B. Torsion.- B. 1 Geometrical Definition of Torsion.- B. 2 Geometrical Properties of Torsion.- B. 3 Algebraic Definition of Torsion.- B. 4 Torsion and Polyhedra.- B. 5 Torsion and Homotopy Equivalences.- Historical Notes.

RACKS AND LINKS IN CODIMENSION TWO

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3–manifolds, and also for the 3–manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3–manifold and for the 3–manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

Trunks and classifying spaces

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks [8]. A rackX gives rise to a trunkT (X) which has a single vertex and the setX as set of edges. Therack spaceBX ofX is the realisation of the nerveNT (X) ofT(X). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural mapBX ↦BAs(X) whereBAs(X) is the classifying space of the associated group ofX. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown-Higgins classifying space of the associated crossed module [8, Section 2] and [3].The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.

On Kirby's calculus of links

(Receiued in revised form 21 July 1978) 4 FRAMED OR labelled link in S3 is a finite collection L of embedded circles in S’, each one of which is labelled by an integer. In [3] and [7] Lickorish and Wallace showed that any orientable 3-manifold can be obtained by surgery on S3 using such a link. Furthermore in [21 Kirby shows that two such manifolds are homeomorphic if and only if the links are related by a series of combinatorial moves. Thus there is a classification of orientable 3-manifolds in terms of equivalence classes of links. In this paper we present an exposition of Kirby’s theorem in a form which applies to links in a general 3-manifold and we also give a classification of non-orientable 3-manifolds by equivalence classes of links in the non-orientable S* bundle over S’ denoted S’ 5 S2. Our exposition reduces the dependence on ‘Cerf Theory’ which plays a central role in Kirby’s paper[2], and clarifies the connection between the various allowable moves. This clarification allows us to state the main classification theorem in the following simpler form: THEOREM. Orientation preserving homeomorphism classes of compact closed oriented 3-manifolds corresponds bijectively to equivalence class of labelled links in S’ where the equivalence is generated by a single move-the “Kirby move” (see §l

The braid-permutation group

Abstract We consider the subgroup of the automorphism group of the free group generated by the braid group and the permutation group. This is proved to be the same as the subgroup of automorphisms of permutation-conjugacy type and is represented by generalised braids (braids in which some crossings are allowed to be “welded”). As a consequence of this representation there is a finite presentation which shows the close connection with both the classical braid and permutation groups. The group is isomorphic to the automorphism group of the free quandle and closely related to the automorphism group of the free rack. These automorphism groups are connected with invariants of classical knots and links in the 3-sphere.

The rack space

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James bundle. We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from de. ning many new knot and link invariants (including generalised James-Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for pi(2) of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links.

Markov's theorem in 3-manifolds

Abstract In this paper we first give a one-move version of Markovs braid theorem for knot isotopy in S 3 that sharpens the classical theorem. Then we give a relative version of Markovs theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markovs theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary orientable 3-manifold.

An Introduction to Species and the Rack Space

Racks were introduced in [FR]. In this paper we define a natural category like object, called a species.* A particularly important species is associated with a rack. A species has a nerve, analogous to the nerve of a category, and the nerve of the rack species yields a space associated to the rack which classifies link diagrams labelled by the rack. We compute the second homotopy group of this space in the case of a classical rack. This is a free abelian group of rank the number of non-trivial maximal irreducible sublinks of the link.

The singular braid monoid embeds in a group

We prove that the singular braid monoid of [2] and [5] embeds in a group. This group has a geometric interpretation as singular braids with two type of singularities which cancel.

The existence of combinatorial formulae for characteristic classes

Given a characteristic class on a locally ordered combinatorial manifold M there exists a cocycle which represents the class on M and is locally defined, i.e. its value on a E M depends only on the ordered star st(a, M). For rational classes the dependence on order disappears. There is also a locally defined cycle which carries the dual homology class. For some time it has been known that there is a simple combinatorial representation for the homology duals of the Stiefel-Whitney classes of a combinatorial manifold (Whitney (8), cf. Cheeger (1), Halperin and Toledo (2)). It is natural to ask whether there is an analogous result for other characteristic classes. For instance, can one give a simple combinatorial formula for the Pontrjagin classes or for their homology duals? What is being sought is a formula which depends only on the local structure of the combinatorial manifold K (as a simplicial complex). In this paper we prove a theoretical result. We establish that formulae of this type exist for all characteristic classes and for their homology duals. But the method of proof makes it extremely difficult to actually give such a formula explicitly. Our formulae depend, in general, on local ordering of the complex, but for rational classes (such as the rational Pontrjagin classes) this dependence disappears. Miller (4) has shown that the rational characteristic numbers of K are in fact the only numerical invariants of K which admit formulae in terms of the local (unordered) structure of K. Thus, for a general characteristic class, some other datum such as our local ordering is necessary. One corollary to the existence of local formulae is that any manifold which can be triangulated so that the links of q-simplexes admit orientation revers-