Celebratio Mathematica

C. P. Rourke and B. J. Sanderson: “Block bundles,” Bull. Amer. Math. Soc. 72 (1966), pp. 1036–1039. MR 214079 Zbl 0147.42501 article

C. P. Rourke and B. J. Sanderson: “An embedding without a normal microbundle,” Invent. Math. 3 (1967), pp. 293–299. MR 222904 Zbl 0168.44602 article

C. P. Rourke and B. J. Sanderson: “Block bundles, I,” Ann. of Math. (2) 87 (1968), pp. 1–28. MR 226645 Zbl 0215.52204 article

C. P. Rourke and B. J. Sanderson: “Block bundles, II: Transversality,” Ann. of Math. (2) 87 (1968), pp. 256–278. MR 226646 Zbl 0215.52301 article

C. P. Rourke and B. J. Sanderson: “Block bundles, III: Homotopy theory,” Ann. of Math. (2) 87 (1968), pp. 431–483. MR 232404 Zbl 0215.52302 article

C. P. Rourke and B. J. Sanderson: “Decompositions and the relative tubular neighbourhood conjecture,” Topology 9 (1970), pp. 225–229. MR 261611 Zbl 0213.25004 article

C. P. Rourke and B. J. Sanderson: “Some results on topological neighbourhoods,” Bull. Amer. Math. Soc. 76 (1970), pp. 1070–1072. MR 271952 Zbl 0213.50104 article

C. P. Rourke and B. J. Sanderson: “On topological neighbourhoods,” Compositio Math. 22 (1970), pp. 387–424. MR 298671 Zbl 0218.57005 article

This paper is concerned with the ‘normal bundle’ problem for topological manifolds: Suppose \( M^n \) is a proper, locally flat submanifold of \( Q^{n+r} \); then what structure can be put on the neighbourhood of \( M \) in \( Q \)? If \( r \leqq 2 \) the problem has been solved by Kirby, who has shown that there is an essentially unique normal disc bundle, while if \( r\leqq 3 \) then our counterexample showed that the notion of fibre bundle is too strong a concept. The notion of topological block bundle seems inapplicable since \( M \) might possibly be untriangulable and a triangulation of \( M \) would be unnatural structure for the problem. The answer we propose here is the ‘stable microbundle pair’.

C. P. Rourke and B. J. Sanderson: “\( \Delta \)-sets, I: Homotopy theory,” Quart. J. Math. Oxford Ser. (2) 22 (1971), pp. 321–338. MR 300281 Zbl 0226.55019 article

C. P. Rourke and B. J. Sanderson: “\( \Delta \)-sets, II: Block bundles and block fibrations,” Quart. J. Math. Oxford Ser. (2) 22 (1971), pp. 465–485. MR 300282 Zbl 0226.55020 article

C. P. Rourke and B. J. Sanderson: Introduction to piecewise-linear topology. Ergeb. Math. Grenzgeb. 69. Springer, 1972. MR 350744 Zbl 0254.57010 book

The first five chapters of this book form an introductory course in piecewise-linear topology in which no assumptions are made other than basic topological notions. This course would be suitable as a second course in topology with a geometric flavour, to follow a first course in point-set topology, and perhaps to be given as a final year undergraduate course. The whole book gives an account of handle theory in a piecewiselinear setting and could be the basis of a first-year postgraduate lecture or reading course. Some results from algebraic topology are needed for handle theory and these are collected in an appendix. In a second appendix are listed the properties of Whitehead torsion which are used in the \( s \)-cobordism theorem. These appendices should enable a reader with only basic knowledge to complete the book. The book is also intended to form an introduction to modern geometric topology as a research subject, a bibliography of research papers being included. We have omitted acknowledgements and references from the main text and have collected these in a set of “historical notes” to be found after the appendices.

S. Buoncristiano, C. P. Rourke, and B. J. Sanderson: A geometric approach to homology theory. London Mathematical Society Lecture Note Series 18. Cambridge University Press; London Mathematical Society (Cambridge; London), 1976. MR 413113 Zbl 0315.55002 book

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C. P. Rourke and B. J. Sanderson: Introduction to piecewise-linear topology, Revised reprint of the 1972 original edition. Ergeb. Math. Grenzgeb. 69. Springer, 1982. Revised reprint of the 1972 original. MR 665919 Zbl 0477.57003 book

R. Fenn, C. Rourke, and B. Sanderson: “An introduction to species and the rack space,” pp. 33–55 in Topics in knot theory. Edited by M. E. Bozhüyük. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 399. Kluwer (Dordrecht), 1993. MR 1257904 Zbl 0828.57005 incollection

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R. Fenn, C. Rourke, and B. Sanderson: “Trunks and classifying spaces,” Appl. Categ. Struct. 3 : 4 (1995), pp. 321–356. MR 1364012 Zbl 0853.55021 article

Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks. A rack \( X \) gives rise to a trunk \( \mathcal{T} ( X ) \) which has a single vertex and the set \( X \) as set of edges. The rack space \( B X \) of \( X \) is the realisation of the nerve \( N \mathcal{T} ( X ) \) of \( \mathcal{T} ( X ) \). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural map \( B X \to B As ( X ) \) where \( B As ( X ) \) is the classifying space of the associated group of X. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown–Higgins classifying space of the associated crossed module [Fenn and Rourke 1992] and [Brown and Higgins 1991]. The theory can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariant of the knot or link.

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C. Rourke and B. Sanderson: “A new approach to immersion theory,” Turk. J. Math. 23 : 1 (1999), pp. 57–72. MR 1701639 Zbl 0951.57014 article

C. Rourke and B. Sanderson: “Homology stratifications and intersection homology,” pp. 455–472 in Proceedings of the Kirbyfest (Berkeley, CA, 1998). Institute of Mathematics, University of Warwick (Warwick), 1999. MR 1734420 Zbl 0947.55008 incollection

C. Rourke and B. Sanderson: “Equivariant configuration spaces,” J. Lond. Math. Soc., II. Ser. 62 : 2 (2000), pp. 544–552. MR 1783643 Zbl 1024.55009 article

C. Rourke and B. Sanderson: “The compression theorem, I,” Geom. Topol. 5 (2001), pp. 399–429. MR 1833749 Zbl 1002.57057 article

This the first of a set of three papers about the Compression Theorem: if \( M^m \) is embedded in \( Q^q\times\mathbb{R} \) with a normal vector field and if \( q-m\geq 1 \), then the given vector field can be straightened (i.e., made parallel to the given \( \mathbb{R} \) direction) by an isotopy of \( M \) and normal field in \( Q\times R \).

The theorem can be deduced from Gromov’s theorem on directed embeddings and is implicit in the preceeding discussion. Here we give a direct proof that leads to an explicit description of the finishing embedding.

In the second paper in the series we give a proof in the spirit of Gromov’s proof and in the third part we give applications.

C. Rourke and B. Sanderson: “The compression theorem, II: Directed embeddings,” Geom. Topol. 5 (2001), pp. 431–440. MR 1833750 Zbl 1032.57028 article

C. Rourke and B. Sanderson: “The compression theorem, III: Applications,” Algebr. Geom. Topol. 3 (2003), pp. 857–872. MR 2012956 Zbl 1032.57029 article

This is the third of three papers about the Compression Theorem: if \( M^m \) is embedded in \( Q^q\times\mathbb{R} \) with a normal vector field and if \( q-m\geq 1 \), then the given vector field can be straightened (i.e., made parallel to the given \( \mathbb{R} \) direction) by an isotopy of \( M \) and normal field in \( Q\times R \).

The theorem can be deduced from Gromov’s theorem on directed embeddings [Gromov 1986] and the first two parts gave proofs. Here we are concerned with applications.

We give short new (and constructive) 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, which leads to theoretical solutions.

We also consider the general problem of controlling the singularities of a smooth projection up to \( C^0 \)–small isotopy and give a theoretical solution in the codimension \( \neq 1 \) case.

C. Rourke and B. Sanderson: “Geometry & Topology Publications: A community based publishing initiative,” pp. 144–156 in Electronic information and communication in mathematics (ICM 2002 international satellite conference) (Beijing, China, 2002). Springer, 2003. Zbl 1045.00006 incollection

We are primarily research mathematicians. It’s what we do. As research mathematicians we are ideally placed to see the grip that commercial publishers have established on our literature. We’d like to call it the Publisher’s Fork in analogy with Morton’s Fork. On the one hand, whilst carrying out our research, we need — indeed demand — total access to the literature. If we want to consult a paper, we will do whatever we need to do to find it…in the old days, we would look in our filing cabinets, then colleagues cabinets, then search libraries or telephone more distant colleagues. These days we are more likely to start at the arXiv or Google before resorting to hard-copy search. The methods may have altered, but the purpose has not. We need to check or use previous work and nothing will stop us until we find it in some form or other. This is the first prong of the fork. The other is that, once we have produced a piece of new research, we want everyone to see it. We don’t hide it under a bushel; we want to give it away for all to see. We eagerly talk about it to colleagues and students, freely give lectures on it (in all gory detail if asked), etc. And we do the same for everyone else’s research, including anonymous refereeing, editorial advising, etc. This is the second prong of the fork.

R. Fenn, C. Rourke, and B. Sanderson: “James bundles,” Proc. Lond. Math. Soc. (3) 89 : 1 (2004), pp. 217–240. MR 2063665 Zbl 1055.55005 article

We study cubical sets without degeneracies, which we call \( \square \)-sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a \( \square \)-set \( C \) has an infinite family of associated \( \square \)-sets \( J^i (C), \) for \( i = 1, 2,\dots \), which we call James complexes. There are mock bundle projections \[ p_i: |J^i(C)|\to |C| \] (which we call James bundles) defining classes in unstable cohomotopy which generalise the classical James–Hopf invariants of \( \Omega (S^2) \). The algebra of these classes mimics the algebra of the cohomotopy of \( \Omega (S^2) \) and the reduction to cohomology defines a sequence of natural characteristic classes for a \( \square \)-set. An associated map to \( BO \) leads to a generalised cohomology theory with geometric interpretation similar to that for Mahowald orientation.

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Brian Joseph Sanderson	Related

R. Fenn, C. Rourke, and B. Sanderson: “The rack space,” Trans. Amer. Math. Soc. 359 : 2 (2007), pp. 701–740. MR 2255194 Zbl 1123.55006 article

The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space and the classifying bundle is the first James bundle. We investigate the algebraic topology of this classifying space and report on calculations given elsewhere. Apart from defining many new knot and link invariants (including generalised James–Hopf invariants), the classification theorem has some unexpected applications. We give a combinatorial interpretation for \( \pi_2 \) of a complex which can be used for calculations and some new interpretations of the higher homotopy groups of the 3-sphere. We also give a cobordism classification of virtual links.

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