Colin Rourke's WWW Homepage

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What is a welded link? corrected version (2018)

Hello, and welcome to my homepage. I'm a topologist, in other words I am interested in fundamental properties of spaces, though recently my interests have spread to include group theory, singularity theory and cosmology. I've been a member of the Mathematics Institute of the University of Warwick since 1968. Before that I was at the Princeton Institute for Advanced Study and Queen Mary College, London. I've also worked at Madison, Wisconsin and, for several years, at the Open University, where I helped rewrite the mathematics course. A good deal of my work has been in collaboration with Brian Sanderson who has been at Warwick since its foundation in 1966 which is when we started collaborating. I've also collaborated a good deal with Roger Fenn and the three of us have an longstanding project to understand knots and links in codimension 2 using racks.

Recently I have concentrated on constructing a new paradigm for the universe.

My students include David Stone, who is now at Brooklyn College, New York; Sandro Buoncristiano who is now at Rome; Jenny Harrison now at Berkeley, California; Hamish Short who is at Marseille, France; Daryl Cooper at Santa Barbara, California; Gena Cesar de Sa and Eduardo Rego both at Oporto, Portugal; Sofia Lambropoulou at Athens, Greece; and Bert Wiest at Rennes, France.

I am a founding editor of Geometry and Topology and Algebraic and Geometric Topology. For GT the other founding editors were John Jones, Rob Kirby and Brian Sanderson. For AGT there was a founding committee which also included Joan Birman and Haynes Miller. These journals are produced entirely by the topological community. Both publish electronically with a paper copy produced at the end of each year. Rob Kirby's letter on mathematics journal pricing is well worth reading in this context. Mathematical Sciences Publishers (of which I am vice-chairman) has now taken over the management of these journals.

Below is a list of publications, papers, preprints and notes available for collection. This excludes papers (and book) on cosmology. For these follow this link.

I've given a brief summary of each item, but you'll find a better summary included in each item at the start. All items are PS files or PDF files or both.

If you are interested, you'll find a complete list of my papers in my CV, PDF version.

Foundational book on PL Toplogy (Ergebnisse version)

Introduction to PL Topology by Colin Rourke and Brian Sanderson

This is a facsimile of the original Springer Ergebnisse Volume (Band 69, 1972). The revised 1982 version is available in a Springer Study Edition directly from Springer (electronic or print-on-demand).

High dimensional topology and algebraic topology

The compression theorem I by Colin Rourke and Brian Sanderson PDF version

This the first of a set of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem, which solves a 20 old problem, can be deduced from Gromov's theorem on directed embeddings. In this paper we give a direct proof that leads to an explicit description of the finishing embedding. The paper is published in Geometry and Topology 5 (2001) 399-429.

The compression theorem II: directed embeddings by Colin Rourke and Brian Sanderson PDF version

This is the second of the three papers. It contains a new short proof of Gromov's theorem on directed embeddings which leads to a second proof of the compression theorem. The new proof (like Gromov's proof) does not lead to the explicit description of the resulting immersions given by our original proof. The difference between the proofs is illustrated in the pictures at the top of this page. The original proof gives a complete description shown in a special case on the left. The new proof produces a very rippled surface with lots of local double set, which is hard to describe explicitly, see the picture on the right. The paper is published in Geometry and Topology 5 (2001) 431-440.

The compression theorem III: applications by Colin Rourke and Brian Sanderson PDF version

This is the third of the set of three and gives applications which include: short new proofs for immersion theory and for the loops-suspension theorem of James et al; a new approach to classifying embeddings of manifolds in codimension one or more; a theoretical solution in the codimension >0 case to the general problem of controlling the singularities of a smooth projection up to C^0-small isotopy.
The proof given in the Part I uses dynamical systems. We define flows which straighten vector fields and which then allow a given embedding or immersion to be `compressed' to an immersion in a lower dimension. The technique gives explicit descriptions of the resulting immersions and can be seen as a way of desingularising certain maps. An example is the transition from the non-immersion of the projective plane in 3-space as a sphere with cross-cap to Boy's surface (see the picture on the left at the top of this page). Another example (to be pursued in a later paper) is a new way of turning the sphere inside-out.

A new approach to immersion theory by Colin Rourke and Brian Sanderson

This paper was comissioned by Selman Akbulut for the Gokova proceedings in honour of Rob Kirby. It is published in the proceedings of the Gokova Geometry-Topology Conference (1998) pages 57-72. It gives a quick introduction to the proof of immersion theory contained in "The compression theorem III".

Equivariant configuration spaces by Colin Rourke and Brian Sanderson

This paper is published in J. London Math. Soc. 62 (2000) 544-552. It applies the compression theorem to tidy up an untidy corner of mathematics left over from the late '70's. Using the same methods as in "The compression theorem III", we prove equivariant versions of the loops-suspension theorem. The result are significantly sharper than were previously known.

Homology stratifications and intersection homology by Colin Rourke and Brian Sanderson PDF version

This paper is publised in The Proceedings of the Kirbyfest, Geometry and Topology Monographs, Volume 2 (1999) pages 455-472. It gives a short new proof of topological invariance of intersection homology in the spirit of the original Goresky-MacPherson proof. The proof uses homology stratifications and homology general position.

The Princeton Notes on the Hauptvermutung by Tony Armstrong, George Cooke and Colin Rourke

This is ancient history dating from the 1960's. The version here is an excerpt from "The Hauptvermutung Book" edited by Andrew Ranicki and published in the K-theory Journal book series by Kluwer (1996).

Papers on group theory using topological methods

Klyachko's methods and the solution of equations over torsion-free groups by Roger Fenn and Colin Rourke

This paper is published in L'Enseignment Mathematique 42 (1996) 49-74. The version here is identical (apart from format) to the published version. The paper gives an exposition of Klyachko's proof of the Kervaire conjecture for torsion-free groups and extends his methods to solve equations of arbitary exponent over torsion-free groups under a mild technical condition on t-shape.

The braid-permutation group by Roger Fenn, Richard Rimanyi and Colin Rourke

This paper is published in Topology 36 (1997) pages 123-135. The version here is "the director's cut" : Topology requested that all material of a semi-expository nature be removed (which resulted in a reduction in length of about 20%). In the opinion of the authors, this makes the paper considerably more difficult to read. The version here is the original uncut version. The paper examines the subgroup of the automorphism group of the free group generated by braid automorphisms and permutations of the generators. A suggestive geometric interpretation is given and used to establish a finite presentation.

On dunce hats and the Kervaire conjecture by Colin Rourke

This paper was contributed to the collection of papers presented to Christopher Zeeman on his 60th birthday. The paper reduces the general Kervaire conjecture to a problem about diagrams based on a (generalised) dunce hat. The dunce hat connection is then used to suggest a family of possible counterexamples.

The singular braid monoid embeds in a group by Roger Fenn, Ebru Keyman and Colin Rourke

This short paper is published in the Journal of Knot Theory and its Ramifications 7 (1998) 881-892. It contains a proof that singular braid monoid of Baez and Birman embeds in a group, which we call the singular braid group. Further the properties of this group will be proved in a later paper.

Characterisation of a class of equations with solutions over torsion-free groups by Roger Fenn and Colin Rourke

This paper is an addendum to the Klyachko paper (in l'Enseignment) above. It is published in "The Epstein Birthday Schrift", I.Rivin, C.Rourke and C.Series (editors), Geometry and Topology Monographs, Volume 1 (1998) 163-171. We examine in detail the "mild technical condition" (amenability) under which we can solve equations over torsion-free groups.

Ordering the braid groups by Roger Fenn, Michael Greene, Dale Rolfsen, Colin Rourke and Bert Wiest PDF version

This paper is published in the Pacific Journal 191 (1999) pages 49-74. We recover Dehornoy's results on the existence of a right-invariant order for the braid group, construct a new canonical form and prove the existence of a quadratic-time algorithm to detect order.

Order automatic mapping class groups by Colin Rourke and Bert Wiest PDF version

This paper is published in the Pacific Journal 194 (2000) 209-277. It is a sequel to "Ordering the braid groups". It improves the algorithm to linear time and extends the results to a considerably larger class of mapping class groups.

The surjectivity problem for one-generator, one-relator extensions of torsion-free groups by Marshall Cohen and Colin Rourke PDF version

This paper uses the Klyachko's methods to solve a special case of the surjectivity problem. It proves that if G is a torsion-free group and G-hat is obtained from G by adding one new generator t and one new relator w, then the natural map G to G-hat is surjective only when w is conjugate to tg for some g in G. The paper is published in Geometry and Topology 5 (2001) 127-142.

Papers on racks

Racks and links in codimension 2 by Roger Fenn and Colin Rourke

This paper is published in the Journal of Knot Theory and its Ramifications Volume 1 (1992) pages 343-406. The version here is identical (apart from headers and footers) to the published version, which was reproduced from the same electronic source. This partly expository paper establishes the basic theory of racks including the main classification theorem (for irreducible links in general 3-manifolds).

Trunks and classifying spaces by Roger Fenn, Colin Rourke and Brian Sanderson

This paper is published in Applied Categorical Structures, Volume 3 (1995) pages 321-356. The version here is identical (apart from format) to the published version. This paper establishes the formal properties of the rack spaces using the formalism of "trunks" which are loosely analogous to categories, but with preferred squares rather than the preferred triangles (commuting triangles) of a category.

James bundles and applications by Roger Fenn, Colin Rourke and Brian Sanderson

This paper establishes the geometric properties of rack spaces as classifying spaces for links. The classifying bundles are bundles of a type canonically associated to any cubical complex. They have strong connections with classical constructions from stable homotopy theory due to James and others and this is why we have called them James bundles. The theory has many further applications outside rack theory, in particular it provides a natural framework in which to study Vassiliev invariants.

For more information on rack spaces see Bert Wiest's home page.

There are two 2-twist-spun trefoils by Colin Rourke and Brian Sanderson PDF version

This paper gives a short proof inspired by Carter et al [arXiv reference math.GT/9906115] that the 2-twist-spun trefoil is not isotopic to its orientation reverse. The proof uses a computer calculation of the third homology group of the three colour rack. It also gives a new proof using the same calculation of the well-known fact that the left and right trefoil knots are not isotopic. The paper relies on a Maple worksheet.

A new classification theorem for links and some calculations using it by Colin Rourke and Brian Sanderson PDF version

This is a rewritten version of the last paper and includes the underlying classification theorem for links in terms of the rack and the canonical class in \pi_2 of the rack space.


Markov's theorem for 3-manifolds by Sofia Lambropoulou and Colin Rourke

This paper is published in Topology and its Applications, 78 (1997) 95-112. The paper proves a 1-move version of the classical Markov theorem and an extension to links in an arbitrary orientable 3-manifold.

Characterisation of the three sphere following Haken by Colin Rourke

This paper was published in the proceedings of the 1993 Gokova topology conference (Turkish Journal of Maths, 18 (1994) 60-69). The version here is identical (apart from format) to the published version. The paper contains a proof of a characterisation of the 3-sphere, stated without proof in a paper of Wolfgang Haken published in 1968.

Algorithms to disprove the Poincare conjecture by Colin Rourke

This paper was a talk given to the 1994 Gokova topology conference. The main result is the existence of an effective algorithm to find a counterexample to the Poincare conjecture (if one exists). This is obtained by combining the Rego-Rourke and Rubenstein-Thompson algorithms. It is published in the 1996 proceedings pages 99-110.

A program to search for homotopy 3-spheres by Michael Greene and Colin Rourke PDF version

This paper is published in the 1998 Gokova proceedings, pages 73-87; it is a preliminary account of an implementation as a C-program of the Rego-Rourke algorithm. The program is currently being developed and tested.

The Kirby Problem List PDF version

This is a copy of the topology problem list compiled and edited by Rob Kirby. The master copy is in pub/Preprints/Rob_Kirby/Problems in the anonymous ftp directory at and can also be obtained from Rob Kirby's WWW homepage. This copy is placed here as a public service for UK topologists because of the difficulty of transferring a file of this size from the US (except at highly unsocial hours). Warning This post-script file prints out at 377 pages .... don't order a print of the whole file carelessly !!!!