# Papers by Colin Rourke

## 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 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

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 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.
## Three 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.

## 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.
This paper is a preliminary account of an implementation as a
C-program of the Rego-Rourke algorithm. The program is currently
being developed and tested.
## High dimensional topology and algebraic topology

The compression theorem solves a 20 year old problem. Applications
include : short new proofs for immersion theory and for the
loops-suspension theorem of James et al and a new approach to
classifying embeddings of manifolds in codimension one or more. The
proof introduces a novel technique in differential topology : proof by
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. Another example (to be persued 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 gives a quick introduction to
the new proof of immersion theory contained in "The compression
theorem".
This paper 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 paper, we
prove equivariant versions of the loops-suspension theorem.
The result are significantly sharper than were previously known.
This paper is written for the proceedings of the Kirbyfest. 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).
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 !!!!