# Colin Rourke's WWW Homepage

What is a welded link? corrected version (2018)

A new paradigm for the
universe
Dimitri Khalezov on 911
Godel's
Theorem
Essay on the
Poincaré conjecture

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.
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 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 the files are PS files but some have alternative PDF versions,
marked PDF version.

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

## High dimensional topology and algebraic topology

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.
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.
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.
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".
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.
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.
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

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.

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.

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.
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.
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.
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.
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

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).

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.

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.

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.
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.
## 3-manifolds

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.

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.
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.
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.
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 math.berkeley.edu 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 !!!!