#### by Rob Kirby

Colin Patrick Rourke was born on 1 January 1943, in Altrincham, a suburb of Manchester. It was there that his father, Francis Louis Albert Rourke, injured in the Dunkirk evacuation, was convalescing. His mother, Gladys Pauline Rourke (née Levy), had moved up from London to be with her husband along with Colin’s older sister, Sue, then a baby.

Francis was a “quantity surveyor”; he worked on building sites making sure that the builders were using the correct amounts of materials. One might think that Colin’s legendary building skills came from his father, but Francis left the family when Colin was an infant.

Gladys’ family were all tailors, second generation Jewish immigrants from Russia, who had shortened the family name to Ley to sound less Jewish. Gladys and her children eventually moved back to London, near Hampstead Heath, and she went to work in High Holborn (near Lincoln’s Inn Fields) at a shop specializing in clothes for the legal profession. That shop was owned by a brother, Stanley Ley, and still exists, though it has since relocated to Fleet Street, closer to the Royal Courts of Justice. Colin grew up close to the Parliament Hill Lido, a large open air swimming pool, and he swam there frequently. He attended Highgate, a public school, with a scholarship. At Highgate it was known that Colin was good at maths, so he was encouraged to try for a Trinity scholarship a year early. His math teacher gave him a long list of past exams and Colin would sit in a corner during math classes doing problems from these exams. When it came time to sit the Trinity geometry exam, he had already seen eight of the ten problems (from his hours in the corner), so he had plenty of time to work on the other two!

Colin once observed that he was not “pushed” as a child, was not a prodigy, and thus was not “scarred for life”. Yet in 1960 at the age of 17 he matriculated at Trinity College in Cambridge. Moreover, because he had done so well on the exams, he skipped Part I of the undergraduate curriculum, did Part II over the course of two years, and then Part III (graduate courses) in his third year. Life improved during Colin’s second year, for Daphne Cale came to Newnham College as a maths undergraduate. They were married in 1963.

Among his teachers were Christopher Zeeman, Terry Wall and David Epstein. (The “great exodus” had not yet happened; at that time Cambridge limited itself to four professorships in mathematics, so Zeeman left to found the Maths Institute at Warwick in 1964, Wall went to Liverpool, Epstein to Warwick, Michael Atiyah to Oxford, and Frank Adams to Manchester.

While a graduate student Colin took a job as assistant lecturer at Queen Mary College (QMC) in London in 1964. He had not yet received his PhD, but needed the money as he and Daphne had their first child, Jane, in 1964. Jane was followed by Jon in 1965, Simon in 1966, and then Beth in 1973.

Colin became a full lecturer after winning the Raleigh Prize. While at QMC he began collaborating with Brian Sanderson, got a PhD in 1965, and received a prize fellowship at Trinity (taken up for only six months).

Colin met Sanderson when Zeeman asked Colin to come along to an Oberwolfach meeting. Colin’s thesis was (in his own words) “something obscure”, but he needed a notion of PL Grassmanians and Zeeman told him to contact Brian. It turned out that Brian did not have an answer, but their collaboration led to block bundles [2] [3], a PL analogue of vector bundles in the smooth case and topological bundles in that category. This work, plus a locally flat embedding without a normal microbundle [1], was completed before Colin left Queen Mary and Zeeman’s offer to come to Warwick.

After three years at QMC, Colin spent a year at the Institute for
Advanced Study (IAS) in Princeton. While there he lectured in
Deane Montgomery’s
seminar on an exposition of Sullivan’s Hauptvermutung, in
a new version communicated to him by Brian Sanderson,
which relied considerably on
Andrew Casson’s
formulation of the proof from his
fellowship dissertation. Colin’s notes on these lectures “The
Hauptvermutung according to Casson and Sullivan” are published in
[7],
which also contains Casson’s fellowship dissertation. During that
year, Colin wrote a paper with
Dennis Sullivan
[4]
“On the Kervaire Obstruction” containing a proof of the product formula needed for the
Hauptvermutung and the homotopy equivalence between and __\( \mathrm{G}/\mathrm{PL} \)__ and
__\( \Omega^4 \mathrm{G}/\mathrm{PL} \)__.

In 1968 he joined Brian Sanderson at the University of Warwick (Sanderson had arrived in 1965) and remained there, and this began a long period of collaboration.

Brian had obtained his PhD at Liverpool in 1963, working with Michaael Atiyah, while collaborating with another student of Atiyah, Rolph Schwarzenberger. After a Masters degree at Liverpool in 1961, Rolph suggested that Brian go to Oxford to work with Ioan James, but he was too busy, so he switched to Atiyah. After a year at Oxford, Brian returned to Liverpool so that his wife Christine could finish her undergraduate German degree while Brian finished his PhD in 1963. Then Atiyah suggested that Brian learn something new when he went to Yale for a year, so he got Zeeman’s famous notes on PL topology. Then came a year at IAS in Princeton where Brian got the idea for block bundles. Finally Brian went to Warwick in 1965.

Once appointed to Warwick, Colin bought a house in Leamington Spa which he practically rebuilt from scratch (he had previously had fixed up a house he’d bought in Woodford Bridge while at Queen Mary). After three years at Warwick, Colin went to the University of Wisconsin Madison in 1971–72 and then to Stanford in the summer of 1972 where he and his family lived with the Sandersons in a sorority house. With a plethora of available rooms, other mathematicians came, too — John Morgan, among them.

When Colin returned to Leamington Spa, Daphne expressed regret about not having a garden, so they began house-hunting and ended up buying a five-acre farm in Barby in 1973. While there he and Daphne raised chickens, two dairy cows (Jerseys), and eventually, after expanding to 20 acres, a few hundred Hebridean sheep. The sheep had thick wiry wool (which was not then marketable) but the lamb and mutton was excellent. Colin and Daphne essentially fed a family of six from the farm, calling it “a way of life”, if not a commercial venture.

After their work on block bundles, Colin and Brian wrote their book on PL topology [5]. It was originally meant to be a book with Zeeman, amplifying Zeeman’s famous notes on PL topology but Zeeman was too busy to collaborate. So their book was built from their own class notes.

Another book followed
(with Colin’s student
Sandro Buoncristiano)
[6]
which developed the notion of *mock bundles*
to give a more geometric approach to generalized homology
theories (and generated a book review
[e1]
by Frank Adams,
famous for his sharp-tongued reviews). The book includes
transversality for CW-complexes, work that Colin is still quite
pleased with
and feels is underappreciated even though it leads to a geometrization of
homotopy theory.

Wolfgang Metzler
gave a nice description of this book in
*Math Reviews*:

The singular

\( n \)-cycles of a topological space\( X \)can be considered as maps\( f : P \to X \), where\( P \)is a closed oriented pl\( n \)-manifold with a codimension 2 singularity. Two such maps\( (P, f ) \),\( (Q, g) \)are homologous if they are bordant. The central idea of this book is to treat any homology theory as a generalized bordism theory. Its cycles are manifolds with singularities depending on the theory. As cocycles one uses mock bundles with the same “manifolds” as fibres. (A\( q \)-mock bundle\( \xi^q \)is a pl projection\( \xi: E \to |K| \), where\( K \)is a ball complex, and for each ball\( \sigma \in K \),\( p^{-1} (\sigma) \)is a compact pl “manifold” of dimension\( q + \dim \sigma \)with boundary\( p^{-1} (\sigma) \).) Two cocycles are cohomologous if they are cobordant, that is if there exists a mock bundle over\( K \times I \)with the given cocycles corresponding to the restrictions on\( K \times \{0\} [K \times \{1\}] \).A lot of (co-)homology theory easily can be described within this framework: the cup product for instance corresponds to Whitney sum of bundles, the cap product between the cycle

\( M \underset{f}{\rightarrow} |K| \)and the cocycle\( E \underset{\xi}{\to} |K| \)is a is a cycle\( g=f^*(\xi) \)given by an induced mock bundle\[ \begin{CD} E^{\ast} @>{{}}>{{}}> E\\ @V{g}VV @VV{\xi}V\\ M @>{{{f} }}>{{}}> K. \end{CD} \]The Poincaré duality map turns out to be simply the identity on representatives.Coefficients are treated in Chapter III and IV by labelling the components of manifolds with group elements; the singularities that have to be permitted are determined by the relations of the coefficient group. Chapter V extends the results to equivariant theories including power operations and characteristic classes.

\( Z_a \)-operations on pl-cobordism are discussed with relations to work of tom Dieck and Quillen. Sheaves are treated in Chapter VI. In Chapter VII the main theorem is presented, that an arbitrary (co-)homology theory can be obtained by the geometric means introduced before. The central step is that every CW spectrum is the Thom spectrum of a suitable bordism theory. Some of the techniques of this last chapter (the notion of transversality and duality for CW complexes together with some basic theorems) promise to be fruitful beyond the scope of this book.The ideas are presented in a stimulating way, although some of the proofs are only indicated. Because the reviewer didn’t check all the details, he does not take responsibility for them. But he wants to emphasize that this book fills a gap and does it in a beautiful manner because

~~(1) it~~connects ideas of various papers and persons (for instance, D. Sullivan) in a unified treatment without burdening them with too many technical details and (2) it proves not only theorems but the fact that geometry may give vivid impulses (even) in modern algebraic topology.

From 1976–1981 Colin was acting professor of pure mathematics at the Open University (on secondment from Warwick) where he masterminded the rewriting of the pure mathematics course.

In 1996, dissatisfied with the rapidly rising fees charged by the major publishers of mathematical research journals, Colin decided to start his own journal, and was ably assisted by Rob Kirby, John Jones and Brian Sanderson. That journal became Geometry & Topology. Under Colin’s leadership, GT became a leading journal in its field while remaining one of the least expensive per page. GT was joined in 1998 by a proceedings and monographs series, Geometry & Topology Monographs, and in 2000 by a sister journal, Algebraic & Geometric Topology. Colin wrote the software and fully managed these publications until around 2005 when he cofounded Mathematical Sciences Publishers (with Rob Kirby) to take over the running. Mathematical Sciences Publishers has now grown to become a formidable force in academic publishing.

During this century, Colin has been working in cosmology, often with Robert MacKay, an applied mathematician at Warwick. They have several papers together and Colin has written a book [8] proposing a model for the universe without either dark matter or the Big Bang.1