Celebratio Mathematica — Rourke — Complete Bibliography

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

[2] C. P. Rourke: “A note on complementary spines,” Proc. Cambridge Philos. Soc. 63 (1967), pp. 1–3. MR 203710 Zbl 0149.20904 article

[3] C. P. Rourke: “Improper embeddings of P.L. spheres and balls,” Topology 6 (1967), pp. 297–330. MR 215309 Zbl 0168.21501 article

[4] 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

[5] C. P. Rourke: “Covering the track of an isotopy,” Proc. Amer. Math. Soc. 18 (1967), pp. 320–324. MR 224098 Zbl 0149.20903 article

[6] 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

[7] 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

[8] P. Orlik and C. P. Rourke: “Free involutions on homotopy \( (4k+3) \)-spheres,” Bull. Amer. Math. Soc. 74 (1968), pp. 949–953. MR 229245 Zbl 0159.53901 article

[9] 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

[10] W. B. R. Lickorish and C. P. Rourke: “A counter-example to the three balls problem,” Proc. Cambridge Philos. Soc. 66 (1969), pp. 13–16. MR 238328 Zbl 0182.26301 article

[11] C. P. Rourke: “Embedded handle theory, concordance and isotopy,” pp. 431–438 in Topology of Manifolds (Athens, Georgia, 1969). Edited by J. C. Cantrell and J. C. H. Edwards. Markham (Chicago, IL), 1970. MR 279816 Zbl 0281.57023 inproceedings

James Cecil Cantrell	Related
C. H. Edwards	Related

[12] 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

[13] 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

[14] 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’.

[15] C. P. Rourke: “Block structures in geometric and algebraic topology,” pp. 127–132 in Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 2. Gauthier-Villars (Paris), 1971. MR 420642 Zbl 0245.57009 incollection

[16] 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

[17] 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

[18] C. P. Rourke and D. P. Sullivan: “On the Kervaire obstruction,” Ann. of Math. (2) 94 (1971), pp. 397–413. MR 305416 Zbl 0227.57012 article

The purpose of this paper is to give a geometrical definition, based on immersion theory, of this obstruction, the Kervaire obstruction, and to prove a product formula which gives the obstruction for \( f \times g \) in terms of invariants of \( f \) and \( g \). The definition and formula apply when \( M \) is non-simply connected and \( n \) is any even integer but in this case the obstruction is just part of the surgery obstruction. We also interpret the product formula in terms of the classifying space \( G/PL \) for \( PL \) normal maps. It implies the existence of primitive “Kervaire” classes in the \( \mathbb{Z}_2 \)-cohomology of \( G/PL \) which induce the obstruction. We give formulae due to Brumfiel and Wall relating the various classes.

[19] 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.

[20] N. W. Alcock, C. P. Rourke, and G. S. Pawley: “An improvement in the algorithm for absorption correction by the analytic method,” Acta Crystallographica 28 (1972), pp. 440–444. Has an appendix by M. R. Levine, about whom I couldn’t find anything. article

Nathaniel W. Alcock	Related
G. Stuart Pawley	Related

[21] C. P. Rourke: “Representing homology classes,” Bull. London Math. Soc. 5 (1973), pp. 257–260. MR 339134 Zbl 0266.55004 article

This note is concerned with the problem of finding simple geometric representatives for homology classes. Thom has proved that any \( \mathbb{Z}_2 \)-homology class is represented by a manifold, while his counterexample can be used to show that \( \mathbb{Z}_p \)-classes are not representable by \( \mathbb{Z}_p \)-manifolds (in the sense of Sullivan) for \( p \neq2 \).

We will show that any \( \mathbb{Z}_p \)-class is represented by a “\( p \)-polyhedron” which is ageneralised twisted \( \mathbb{Z}_p \)-manifold (see § 2); this answers a conjecture of Sullivan. We conjecture that \( p \)-polyhedra are in fact the simplest general representatives for \( \mathbb{Z}_p \)-homology. The main theorem, from which the representation result follows easily, is that any connected ring theory which represents the cohomology of the standard \( p \)-Lens space represents \( \mathbb{Z}_p \)-(co)homology in general. The theorem is proved by ashort spectral sequence argument using a result of Serre on the cohomology of \( K(\mathbb{Z}_p, n) \). Since the 2-Lens space is real projective space, we recover (a strengthened form of) Thorn’s result (and the whole proof is considerably simpler and has less algebraic input than Thorn’s original one).

[22] 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

Sandro Buoncristiano	Related
Brian Joseph Sanderson	Related

[23] N. Levitt and C. Rourke: “The existence of combinatorial formulae for characteristic classes,” Trans. Am. Math. Soc. 239 (1978), pp. 391–397. MR 494134 Zbl 0382.55010 article

[24] E. César de Sá and C. Rourke: “The homotopy type of homeomorphisms of 3-manifolds,” Bull. Am. Math. Soc., New Ser. 1 (1979), pp. 251–254. MR 513752 Zbl 0442.58007 article

The purpose of this paper is to announce some results on homeomorphism groups of 3-manifolds. These results together with the Smale conjecture (a proof of which has recently been announced by A. Hatcher) essentially reduce the com- putation of the homotopy type of the homeomorphism group of a 3-manifold to an analysis of a certain associated configuration space and the homotopy types of the homeomorphism groups of the prime factors. For a prime 3-manifold, this homotopy type is known (a) if the manifold is \( P^2 \)-irreducible and sufficiently large, (b) for \( S^1 \times S^2 \) and \( S^1 \underset{\sim}{\times} S^2 \).

The results announced here are an extension of the first part of the first author’s Ph.D. thesis and use the methods of this thesis. None of our results depends on the Smale conjecture but we have indicated where the Smale conjecture simplifies the conclusion.

[25] R. Fenn and C. Rourke: “On Kirby’s calculus of links,” Topology 18 : 1 (1979), pp. 1–15. MR 528232 Zbl 0413.57006 article

[26] R. Fenn and C. Rourke: “Nice spines of 3-manifolds,” pp. 31–36 in Topology of low-dimensional manifolds (Proc. Second Sussex Conf.). Lecture Notes in Math. 722. 1979. MR 547451 Zbl 0406.57002 inproceedings

[27] C. P. Rourke: “Presentations and the trivial group,” pp. 134–143 in Topology of low-dimensional manifolds (Proc. Second Sussex Conf.) (Chelwood Gate, 1977). Edited by R. A. Fenn. Lecture Notes in Math. 722. Springer, 1979. MR 547460 Zbl 0408.57001 inproceedings

[28] 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

[29] L. Bostock, S. Chandler, and C. Rourke: Further pure mathematics. Stanley Thornes (Kingston Upon Thames, Surrey), 1982. book

Linda Bostock	Related
Suzanne Chandler	Related

[30] E. Rêgo and C. Rourke: Heegaard diagrams and homotopy. Preprint, University of Warwick, 1984. techreport

[31] C. Rourke: “A new proof that \( \Omega_3 \) is zero,” J. Lond. Math. Soc., II. Ser. 31 (1985), pp. 373–376. MR 809959 Zbl 0585.57012 article

[32] E. Rêgo and C. Rourke: A characterization of homotopy 3-spheres. Preprint, Mathematics Institute, University of Warwick, 1985. techreport

[33] E. Rêgo and C. Rourke: Characterisation of \( S^3 \). Preprint, University of Warwick, 1986. techreport

[34] E. Rêgo and C. Rourke: Announcement of a proof of the Poincaré conjecture. Preprint, University of Warwick, 1986. techreport

[35] C. Rourke: “Convex ruled surfaces,” pp. 255–272 in Analytical and geometric aspects of hyperbolic space. London Mathematical Society Lecture Note Series 111. 1987. MR 903853 Zbl 0615.53061 incollection

[36] E. Rêgo and C. Rourke: “Heegaard diagrams and homotopy 3-spheres,” Topology 27 : 2 (1988), pp. 137–143. MR 948177 Zbl 0646.57006 article

[37] C. Rourke: “On dunce hats and the Kervaire conjecture,” pp. 221–230 in Papers presented to E. C. Zeeman. Mathematics Institute, University of Warwick (Warwick), 1988. Papers presented to Christopher Zeeman on the occasion of his 60th birthday. incollection

The dunce hat \( Z \) is a CW complex with one 0-cell, one 1-cell and one 2-cell attached to the 1-cell by the word \( t^2\bar{t} \) (throughout the paper \( \bar{x} \) will mean \( x^{-1} \)), see, for example, Zeeman [8]. By a dunce hat I shall mean a CW complex with one each of 0, 1 and 2-cells, as before, but with the 2-cell attached to the 1-cell by an arbitrary (unreduced) word \( w ( t, \bar{t}) \), the simplest interesting dunce hat being \( Z \) . In this paper I shall point out a connection between dunce hats and a conjecture in combinatorial group theory, known as the Kervaire conjecture, and then use the dunce hat connection to suggest possible counterexamples to the Kervaire conjecture.

[38] R. Fenn and C. Rourke: “Racks and links in codimension two,” J. Knot Theory Ramifications 1 : 4 (1992), pp. 343–406. MR 1194995 Zbl 0787.57003 article

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3-manifolds, and also for the 3-manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3-manifold and for the 3-manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

[39] 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

Roger Fenn	Related
Brian Joseph Sanderson	Related

[40] R. Fenn, R. Rimányi, and C. Rourke: “Some remarks on the braid-permutation group,” pp. 57–68 in Topics in knot theory. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 399. Kluwer (Dordrecht), 1993. MR 1257905 Zbl 0830.57005 incollection

Roger Fenn	Related
Richard Rimányi	Related

[41] C. Rourke: “Characterisation of the three sphere following Haken,” Turk. J. Math. 18 : 1 (1994), pp. 60–69. MR 1270439 Zbl 0867.57010 article

[42] G. Masbaum and C. Rourke: “Model categories and topological quantum field theories,” Turk. J. Math. 18 : 1 (1994), pp. 70–80. MR 1270440 Zbl 0864.57032 article

[43] 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.

Roger Fenn	Related
Brian Joseph Sanderson	Related

[44] R. Fenn and C. Rourke: “Klyachko’s methods and the solution of equations over torsion-free groups,” Enseign. Math. (2) 42 : 1–2 (1996), pp. 49–74. MR 1395041 Zbl 0861.20029 article

[45] A. A. Ranicki, A. J. Casson, D. P. Sullivan, M. A. Armstrong, C. P. Rourke, and G. E. Cooke: The Hauptvermutung book: A collection of papers of the topology of manifolds. \( K \)-Monographs in Mathematics 1. Kluwer Academic Publishers (Dordrecht), 1996. MR 1434100 Zbl 0853.00012 book

Andrew J. Casson	Related
Dennis P. Sullivan	Related
Mark Anthony Armstrong	Related
George E. Cooke	Related
Andrew A. Ranicki	Related	Volume

[46] M. A. Armstrong, C. P. Rourke, and G. E. Cooke: “The Princeton notes on the Hauptvermutung,” pp. 105–106 in The Hauptvermutung book. \( K \)-Monogr. Math. 1. Kluwer (Dordrecht), 1996. MR 1434104 Zbl 0871.57020 incollection

Mark Anthony Armstrong	Related
George E. Cooke	Related

[47] C. P. Rourke: “The Hauptvermutung according to Casson and Sullivan,” pp. 129–164 in The Hauptvermutung book. \( K \)-Monogr. Math. 1. Kluwer (Dordrecht), 1996. MR 1434106 incollection

[48] C. P. Rourke: “Coda: connection with the results of Kirby and Siebenmann,” pp. 189–190 in The Hauptvermutung book. \( K \)-Monogr. Math. 1. Kluwer (Dordrecht), 1996. MR 1434108 incollection

[49]R. Fenn, R. Rimányi, and C. Rourke: “The braid-permutation group,” Topology 36 : 1 (1997), pp. 123–135. MR 1410467 Zbl 0861.57010 article

We consider the subgroup of the automorphism group of the free group generated by the braid group and the permutation group. This is proved to be the same as the subgroup of automorphisms of permutation-conjugacy type and is represented by generalised braids (braids in which some crossings are allowed to be “welded”). As a consequence of this representation there is a finite presentation which shows the close connection with both the classical braid and permutation groups. The group is isomorphic to the automorphism group of the free quandle and closely related to the automorphism group of the free rack. These automorphism groups are connected with invariants of classical knots and links in the 3-sphere.

Roger Fenn	Related
Richard Rimányi	Related

[50] C. Rourke: “Algorithms to disprove the Poincaré conjecture,” Turk. J. Math. 21 : 1 (1997), pp. 99–110. MR 1456164 Zbl 0906.57006 article

[51] S. Lambropoulou and C. P. Rourke: “Markov’s theorem in 3-manifolds,” Topology Appl. 78 : 1–2 (1997), pp. 95–122. MR 1465027 Zbl 0879.57007 article

[52] R. Fenn, E. Keyman, and C. Rourke: “The singular braid monoid embeds in a group,” J. Knot Theory Ramifications 7 : 7 (1998), pp. 881–892. MR 1654641 Zbl 0971.57011 article

Ebru Keyman	Related
Roger Fenn	Related

[53] The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Edited by I. Rivin, C. Rourke, and C. Series. Geom. Topol. Monogr. 1. Mathematical Sciences Publishers; Institute of Mathematics, University of Warwick (Berkeley; Warwick), 1998. MR 1668319 Zbl 0901.00063 book

Igor Rivin	Related
Caroline Mary Series	Related

[54] R. Fenn and C. Rourke: “Characterisation of a class of equations with solutions over torsion-free groups,” pp. 159–166 in The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Geometry and Topology Monographs 1. Mathematical Sciences Publihsers; Institute of Mathematics, University of Warwick (Berkeley; Warwick), 1998. MR 1668351 Zbl 0913.20018 incollection

We study equations over torsion-free groups in terms of their “\( t \)-shape” (the occurences of the variable \( t \) in the equation). A \( t \)-shape is good if any equation with that shape has a solution. It is an outstanding conjecture that all \( t \)-shapes are good. In a previous article, we proved the conjecture for a large class of \( t \)-shapes called amenable. Clifford and Goldstein characterised a class of good \( t \)-shapes using a transformation on \( t \)-shapes called the Magnus derivative. In this note we introduce an inverse transformation called blowing up. Amenability can be defined using blowing up; moreover the connection with differentiation gives a useful characterisation and implies that the class of amenable \( t \)-shapes is strictly larger than the class considered by Clifford and Goldstein.

[55] 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

[56] M. Greene and C. Rourke: “A program to search for homotopy 3-spheres,” Turk. J. Math. 23 : 1 (1999), pp. 73–87. MR 1701640 Zbl 0944.57011 article

[57] R. Fenn, M. T. Greene, D. Rolfsen, C. Rourke, and B. Wiest: “Ordering the braid groups,” Pac. J. Math. 191 : 1 (1999), pp. 49–74. MR 1725462 Zbl 1009.20042 article

We give an explicit geometric argument that Artin’s braid group Bn is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical form. The left-consistent form of a braid which is positive (respectively negative) in our order has consistently positive (respectively negative) exponent in the smallest braid generator which occurs. It follows that our ordering is identical to that of Dehornoy (1995) constructed by very different means, and we recover Dehornoy’s main theorem that any braid can be put into such a form using either positive or negative exponent in the smallest generator but not both.

Our definition of order is strongly connected with Mosher’s (1995) normal form and this leads to an algorithm to decide whether a given braid is positive, trivial, or negative which is quadratic in the length of the braid word.

Dale P. O. Rolfsen	Related
Roger Fenn	Related
Michael Thomas Greene	Related
Bertold Wiest	Related

[58] 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

[59] C. Rourke and B. Wiest: “Order automatic mapping class groups,” Pac. J. Math. 194 : 1 (2000), pp. 209–227. MR 1756636 Zbl 1016.57015 article

[60] 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

[61] C. Rourke and B. Sanderson: A new classification of links and some calculations using it. Preprint, 2000. ArXiv math/0006062 techreport

[62] M. M. Cohen and C. Rourke: “The surjectivity problem for one-generator, one-relator extensions of torsion-free groups,” Geom. Topol. 5 (2001), pp. 127–142. MR 1825660 Zbl 1014.20015 article

[63] 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.

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

[65] 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.

[66] 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.

[67] C. Rourke: A new paradigm for the universe. Preprint, 2003. ArXiv abs/astro-ph/0311033 techreport

[68] 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.

Roger Fenn	Related
Brian Joseph Sanderson	Related

[69] M. Forester and C. Rouke: “The adjunction problem over torsion-free groups,” Proc. Natl. Acad. Sci. USA 102 : 36 (2005), pp. 12670–12671. MR 2172892 Zbl 1135.57300 article

[70] M. Forester and C. Rourke: “Diagrams and the second homotopy group,” Commun. Anal. Geom. 13 : 4 (2005), pp. 801–820. MR 2191908 Zbl 1096.55008 article

[71] M. Forester and C. Rourke: “A fixed-point theorem and relative asphericity,” Enseign. Math. (2) 51 : 3–4 (2005), pp. 231–237. MR 2214887 Zbl 1112.55003 article

[72] S. Lambropoulou and C. P. Rourke: “Algebraic Markov equivalence for links in three-manifolds,” Compos. Math. 142 : 4 (2006), pp. 1039–1062. MR 2249541 Zbl 1156.57007 article

Let \( B_n \) denote the classical braid group on \( n \) strands and let the mixed braid group \( B_{m,n} \) be the subgroup of \( B_{m+n} \) comprising braids for which the first \( m \) strands form the identity braid. Let \[ B_{m,\infty}=\bigcup_nB_{m,n} .\] We describe explicit algebraic moves on \( B_{m,\infty} \) such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-manifold. The moves depend on a fixed link representing the manifold in \( S^3 \). More precisely, for link complements the moves are the two familiar moves of the classical Markov equivalence together with ‘twisted’ conjugation by certain loops \( a_i \). This means premultiplication by \( a_i^{-1} \) and postmultiplication by a ‘combed’ version of \( a_i \). For closed three-manifolds there is an additional set of ‘combed’ band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using \( L \)-moves (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov theorem that classifies links in \( S^3 \) up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of three-manifolds.

[73] 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.

Roger Fenn	Related
Brian Joseph Sanderson	Related

[74]C. Rourke: “What is a welded link?,” pp. 263–270 in Intelligence of low dimensional topology 2006 (Hiroshima, Japan July 22–26, 2006). Series on Knots and Everything 40. World Scientific (Hackensack, NJ), 2007. MR 2371734 Zbl 1202.57011 incollection

[75] C. Rourke: “La congettura di Poincaré,” pp. 731–763 in La matematica, II: Problemi e teoremi. Edited by C. Bartocci and P. Odifreddi. Giulio Einaudi (Torino), 2008. Zbl 1290.01007 incollection

[76] R. S. MacKay and C. P. Rourke: “Natural observer fields and redshift,” J. Cosmol. 15 (2011), pp. 6079–6099. article

[77] R. S. MacKay and C. Rourke: “Natural flat observer fields in spherically symmetric space-times,” J. Phys. A, Math. Theor. 48 : 22 (2015), pp. 225204, 11. Id/No 225204. MR 3355220 Zbl 1318.83002 article

An observer field in a space-time is a time-like unit vector field. It is natural if the integral curves (field lines) are geodesic and the perpendicular three-plane field is integrable (giving normal space slices). We prove that a natural observer field determines a coherent notion of time: a coordinate that is constant on the perpendicular space slices and whose difference between two space-slices is the proper time along any field line.

A natural observer field is flat if the normal space slices are metrically flat. For static spherically symmetric space-times we find a necessary and sufficient condition for possession of a spherically symmetric natural flat observer field. In this case, which includes the Schwarzschild and the Kottler space-times, there is in fact a dual pair of spherically symmetric natural flat observer fields. One of these observer fields is expanding and the other contracting and it is natural to describe the expanding field as the ‘escape’ field and the dual contracting field as the ‘capture’ field.

Observer fields are useful for understanding redshift and the fields described here are used in a possible explanation of redshift explored in [McKay and Rourke 2015].

[78] R. S. Mackay and C. Rourke: “Are gamma-ray bursts optical illusions?,” Palest. J. Math. 5 (2016), pp. 175–197. MR 3477629 Zbl 1346.83071 article

[79] C. Rourke: “Approximating a topological section by a piecewise-linear section,” Algebr. Geom. Topol. 16 : 3 (2016), pp. 1373–1377. MR 3523042 Zbl 1347.57028 article

[80] C. Rourke, R. Toalá-Enríquez, and R. S. MacKay: Black holes, redshift and quasars. Preprint, 2017. techreport

[81]C. Rourke: The geometry of the universe. Series on Knots and Everything 71. World Scientific Publishing (Hackensack, NJ), 2021. MR 4375354 book

[82] R. S. MacKay and C. P. Rourke: Polarisation of gamma-ray bursts, 2021. preprint. misc