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 \( Q \)
which has transverse self intersections. Each multiple point manifold
is mapped to the centre of a \( k \)-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.
