#### by Brian Sanderson

Colin’s thesis was about a highly technical PL problem that Hudson and Zeeman ran into whilst attempting to classify torus knots (knots of a product of two spheres in Euclidean space):

If a sphere __\( S^n \)__ is embedded in a ball __\( B^{n+q} \)__ improperly (i.e., it is
allowed to meet the boundary) then is it unknotted in the sense that
it bounds a ball __\( B^{n+1} \)__ in __\( B^{n+q} \)__? Assume for simplicity that __\( q > 2 \)__
(codimension __\( \geq \)__ 3) so that by Zeeman’s theorem __\( S^n \)__ in unknotted in the
containing Euclidean space __\( R^{n+q} \)__ of __\( B^{n+q} \)__.

The unexpected answer that Colin found is that the answer is yes in
the metastable range __\( 2q > n+3 \)__ (and, as a result of our collaboration
this is best possible).

In order to apply this unknotting theorem to obtain a classification of torus knots, Colin needed a PL notion of the Grassmannian in the smooth category and after asking Zeeman at the 1965 Oberwolfach meeting for help with this, Zeeman pointed to me.

I had been at the Institute for Advanced Study in Princeton the previous year, after a year at Yale. At Yale I had learned PL theory by giving a graduate course on Zeeman’s PL notes. I came up with block bundles in order to reduce an interlocking braid of homotopy groups of Levine [e3], Section 2.2, to homotopy theory and Zeeman had come up with the name block bundles on the flight to Oberwolfach. Meeting Colin was fortuitous as I was having problems with the corresponding PL bundle theory. We started a correspondence about this and next met in person at the British Mathematical Colloquium held in Imperial College in March 1966 and had several long conversations about it and effectively sorted out nearly all the problems. The PL techniques that Colin used in his thesis — in particular, the relative regular neighbourhood theorem or RRNT — were ideally suited to the details needed for the theory. One key point was whether to insist on charts for the new bundles and for a while we toyed with having block bundles without charts. We quickly realised that charts can be reinserted block by block using conical extension and the RRNT so the notions of block bundles with or without charts are the same.

One serious problem remained. We expected that the “group” would be
a semisimplicial group with all the usual face maps and degeneracies.
But the natural way to define degeneracies ran into a famous PL
problem known as “the standard mistake”.
The mistake is to assume radial projection is piecewise linear. On the other hand the
extension condition (the Kan condition) was easy to prove by the same
techniques used to reinsert charts, and we started to think about
setting up semisimplicial theory using just face maps and the Kan
condition. We called this theory
*delta-sets*
[5],
[6]
and we used this
for the final presentation of the new theory. One extra bonus of the
new theory was that transversality, which Zeeman had just proved using
the highly technical PL notion of “transimpliciality”, now had a
natural proof using “block transversality”.

The theory gave all the expected analogues of smooth characteristic classes and huge braid diagrams of associated homotopy groups [2], [3], [4].

#### The embedding without a normal microbundle

One major problem that was left unanswered was whether a
new type of bundle theory was really necessary.
Milnor
had invented
“microbundles”
[e1]
in the hope of providing normal bundles in the PL and
topological categories and
Kister
[e2]
had shown that microbundles are
really __\( R^n \)__ bundles in disguise. Perhaps block bundles were also real
bundles in disguise. This was perhaps the hardest problem that we
tackled and many other mathematicians were working on
it. We worked night and day on this for several weeks.
We struck gold using a wonderful paper of
André Haefliger
[e4].
This was pure PL magic. Haefliger found his own notion of normal
bundle for the PL category via stable microbundle pairs. He used this
to classify links of spheres in codimension __\( \geq 3 \)__ which was complete in
several important cases and contained some useful exact sequences.
Concentrating on links of two copies of __\( S^n \)__ in __\( S^{n+q} \)__, Colin noticed
that if you suspend a link (to get a link of two copies of __\( S^{n+1} \)__ in
__\( S^{n+q+2} \)__) then the resulting link is the boundary of a ribbon (a copy
of __\( S^{n+1}\times I) \)__. This proved by adding a few cones, see the diagram on
page 296 of
[1].
A link in codimension 3 has
two “linking numbers” (elements of a certain homotopy group of
spheres) because the complement of one of the spheres has the homotopy
type of a sphere by Zeeman’s theorem and the other sphere then
determines a linking number. We also noticed that
if a ribbon has a normal bundle with fibre a Euclidean space, then
the ribbon can be “turned over” inside the fibres and this proves that
if q is odd and one linking class of the link which bounds the ribbon
is zero then the other one must have order 2. Now using an exact
sequence found by Haefliger and a search through
Toda’s
tables of
homotopy groups of spheres a link can be found of two 18-spheres in
__\( S^{27} \)__ whose suspension has one linking number zero and the other not
of order two. This proves that there is an embedding of __\( S^{19} \)__ in
__\( S^{29} \)__ with no PL normal microbundle. Since the “turning over”
argument works equally well with topological fibres, this embedding
has no topological normal bundle either. By restricting the example
to one end of the ribbon, there is a PL embedding (necessarily locally
flat by Zeeman’s theorem) of __\( S^{19} \)__ in a 28-manifold with no
topological normal bundle.

#### About the Hauptvermutung

Our joint work on block bundles, delta-sets and the embedding without a normal bundles was completed whilst Colin was at Queen Mary College in London from 1964 to 1967. He moved to Warwick in 1968 and spent the intervening year, half at the Princeton Institute, and half on his fellowship at Trinity College Cambridge. Whilst at Princeton he gave a series of lectures in the Montgomery seminar on the homotopy Hauptvermutung (a homeomorphism between PL manifolds is homotopic to a PL homeomorphism) a proof of which had recently been announced by Dennis Sullivan. In a telephone conversation, Sullivan told Colin that he had a new, and much simpler proof which could be easily understood and I sent Colin an outline of this proof, which he used for his lectures. Later it was made clear by Terry Wall that Sullivan’s new proof followed a simplification of the original proof due to Andrew Casson and the title of the lecture notes was changed from “The Hauptvermutung according to Sullivan” to “The Hauptvermutung according to Casson and Sullivan”. Casson extended the notion of block bundle to allow arbitrary manifold blocks (manifold cross a base disc) an used this to formulate Sullivan’s proof in a coherent and elegant way.

In *The Hauptvermutung Book*
[10],
Andrew Ranicki
assembled all
the original papers, including Casson’s fellowship thesis containing
the new proof, Sullivan’s rough notes on the subject and much else.

Colin’s original contribution to the area was an accurate proof
of the product formula as used in the proof. Sullivan knew it was
right but did not have a proof and Casson didn’t either (he relied on
Wall’s more general product formula which was not yet written up).
The proof of the product formula was published in *On the Kervaire
obstruction* by Rourke and Sullivan which also contained details of
the periodicity map for G/PL
[7].

After Colin came to Warwick we produced what is now a classic text on piecewise linear topology [9]. It included the then recent handle theory and was published in 1972 and reprinted in paperback in 1982.

#### Mock bundles and the geometrisation of generalised homology theories

More general than Casson’s manifold block bundles, mock bundles (mock
for manifold-block) provide a natural geometric setting for homology
theories. A cycle for bordism was given by a map of a manifold into a
space but there was no geometric view of a cobordism representative.
This turned out to be simply a block bundle with blocks replaced by manifolds.
At this point Casson had used the more restrictive notion of manifold block bundles
We called the generalised bundles *mock bundles*, the name suggested by
Marshall Cohen who was
visiting Warwick at the time. Colin and I worked on these
collaborating with Colin’s PhD student
Sandro Buoncristiano
Buoncristiano
[8].
The theory gives geometric representatives for all generalised homology theories
(answering a question posed by Sullivan) and allows a notion of
sheaves of coefficients, again for all generalised homology theories.

#### Fenn Rourke Sanderson papers 1993–2007

In the late seventies I published two papers with
Ulrich Koschorke
[e6],
[e5],
and one on Mahowald orientations
[e7].
The results came from considering situations involving
configuration spaces and embeddings of a manifold __\( M \)__ in a
__\( Q\times R^n \)__ which projects to a transverse immersion in __\( Q \)__. The
multiple point manifolds can represent characteristic classes or
generalised Hopf invariants and we are into algebraic topology. Ulrich
and I could prove that the embedding would project after a bordism.
This got Colin and I thinking about projecting after isotopy. We
proved that it could be done in a way that gave a new proof of the
immersion theorem with the advantage of an explicit description of the
resulting immersion. The details are in our
compression theory papers
[12],
[13],
[14].

The collaboration with Colin and Roger Fenn began when Roger came to give a lecture at Warwick. Roger and Colin had been working for some time on knots and racks and they were puzzling over the significance of the rack space — a cubical complex without degeneracies determined by a rack. We call these complexes square sets. I could see that for the trivial one point rack the space has the homotopy type of the double loop space on the 2-sphere and our work together began.

In
[15]
we investigate square sets. Any square set C determines a
family of square sets __\( J^n(C) \)__ by replacing each __\( k \)__-cube in __\( C \)__ by the
disjoint union of the codimension __\( n \)__ subcubes through
its centre.
These subcubes have a natural order and so we get an embedding of the
realisation of __\( J^n(C) \)__ in __\( C\times R \)__. This leads to James Hopf invariants. By
taking realisations we get a map __\( |J^nC| \rightarrow |C| \)__ which is the
projection of a mock bundle with framed manifold blocks. Now we have
an element in stable cohomotopy which turns out to be a Steifel
Whitney class of a vector bundle over __\( |C| \)__ after applying the Hurewicz
homomorphism. Further for any __\( C \)__ there is an associated cobordism
theory. A __\( C \)__-manifold __\( M \)__ has a codimension-1 immersed ~~submanifold ~~
which has transverse self intersections. Each multiple point manifold
is mapped to the centre of a __\( Q \)____\( k \)__-cube in __\( |C| \)__ and has framed normal
bundle in __\( M \)__ pulled back from the __\( k \)__-cube. Further the stable normal
bundle of __\( M \)__ is required to be trivial away from a neighbourhood of __\( Q \)__
and has twisting in the neighbourhoods of the self intersections
pulled back from the associated __\( k \)__-cube. There are corresponding __\( C \)__-mock
bundles and a __\( C \)__-homology/cohomology theory.

In
[16]
we find the rack space defined as the realisation of a
square set where an __\( n \)__-cube given by an __\( n \)__-tuple of rack elements. We
calculate homology and homotopy groups of the rack space for some
examples. The rack space classifies framed codimension-2 links with
the given rack up to cobordism. For classical links the rack together
with the corresponding element in __\( \pi_2 \)__ of the rack space classifies.
The proof is in
[11]
along with calculations using a homomorphism into
a simple rack space to distinguish the standard trefoils and the
2-twist spun trefoils.

#### Personal memories

After collaborating at a distance I really got to know Colin after he came to Warwick in 1968. I had been there from the beginning in 1965. He seemed to just know how to do things. He became known as the go-to for graduate students with broken down cars and he had to call an official halt on that activity as it was getting to be too much. He bought a house in Leamington and installed the central heating, being a dab hand with a blow torch. Later we both moved to the country in 1973, Colin to Barby and I to Wolfhamcote. Of course like Whitehead before him, Colin had cows and sheep and still does. Again he installed his central heating. I paid a plumber to do mine. In ‘99 our boilers needed replacing. I got boilers from a plumber/hang-gliding friend. Colin installed his and helped me install mine.