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

Abstract
People
BibTeX
@article {key222904m,
AUTHOR = {Rourke, C. P. and Sanderson, B. J.},
TITLE = {An embedding without a normal microbundle},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {3},
YEAR = {1967},
PAGES = {293--299},
DOI = {10.1007/BF01402954},
NOTE = {MR:222904. Zbl:0168.44602.},
ISSN = {0020-9910},
}
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

Abstract
People
BibTeX

The purpose of this and two subsequent papers is to establish for the piecewise-linear category a tool analogous to the vector bundle in the differential category.

The fundamental problem is this. Given a smooth submanifold \( M^n\subset Q^{n+q} \) , one can define a normal bundle to \( M \) in \( Q \) . It is a \( q \) -vector bundle uniquely determined up to isomorphism. What is the correct PL analogue of this result?

@article {key226645m,
AUTHOR = {Rourke, C. P. and Sanderson, B. J.},
TITLE = {Block bundles, {I}},
JOURNAL = {Ann. of Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {87},
YEAR = {1968},
PAGES = {1--28},
DOI = {10.2307/1970591},
NOTE = {MR:226645. Zbl:0215.52204.},
ISSN = {0003-486X},
}
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

Abstract
People
BibTeX

Block bundles were designed to play the same role in the PL category as vector bundles in the smooth category. This paper is a continuation of “Block bundles, I” [Rourke and Sanderson 1968] where the basic properties (and notation) were established. Here we apply the techniques developed in [Rourke and Sanderson 1968] to the problem of PL transversality, and at the same time establish several properties in continuing analogy with vector bundle theory.

@article {key226646m,
AUTHOR = {Rourke, C. P. and Sanderson, B. J.},
TITLE = {Block bundles, {II}: {T}ransversality},
JOURNAL = {Ann. of Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {87},
YEAR = {1968},
PAGES = {256--278},
DOI = {10.2307/1970598},
NOTE = {MR:226646. Zbl:0215.52301.},
ISSN = {0003-486X},
}
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

Abstract
People
BibTeX

This paper is a continuation of ‘Block bundles: I and II (Transversality)’ [Rourke and Sanderson 1968a], [Rourke and Sanderson 1968b]. In these papers we established the theory of block bundles as a tool in the PL category, drawing strong analogies with the usage of vector bundles in the differential category. Our methods were geometrical wherever possible.

@article {key232404m,
AUTHOR = {Rourke, C. P. and Sanderson, B. J.},
TITLE = {Block bundles, {III}: {H}omotopy theory},
JOURNAL = {Ann. of Math. (2)},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {87},
YEAR = {1968},
PAGES = {431--483},
DOI = {10.2307/1970714},
NOTE = {MR:232404. Zbl:0215.52302.},
ISSN = {0003-486X},
}
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

Abstract
People
BibTeX
@article {key300281m,
AUTHOR = {Rourke, C. P. and Sanderson, B. J.},
TITLE = {\$\Delta\$-sets, {I}: {H}omotopy theory},
JOURNAL = {Quart. J. Math. Oxford Ser. (2)},
FJOURNAL = {The Quarterly Journal of Mathematics.
Oxford. Second Series},
VOLUME = {22},
YEAR = {1971},
PAGES = {321--338},
DOI = {10.1093/qmath/22.3.321},
NOTE = {MR:300281. Zbl:0226.55019.},
ISSN = {0033-5606},
}
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

People
BibTeX
@article {key300282m,
AUTHOR = {Rourke, C. P. and Sanderson, B. J.},
TITLE = {\$\Delta\$-sets, {II}: {B}lock bundles
and block fibrations},
JOURNAL = {Quart. J. Math. Oxford Ser. (2)},
FJOURNAL = {The Quarterly Journal of Mathematics.
Oxford. Second Series},
VOLUME = {22},
YEAR = {1971},
PAGES = {465--485},
DOI = {10.1093/qmath/22.4.465},
NOTE = {MR:300282. Zbl:0226.55020.},
ISSN = {0033-5606},
}
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

Abstract
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BibTeX

The purpose of these notes is to give a geometrical treatment of generalized homology and cohomology theories. The central idea is that of a ‘mock bundle’, which is the geometric cocycle of a general cobordism theory, and the main new result is that any homology theory is a generalized bordism theory. The book will interest mathematicians working in both piecewise linear and algebraic topology especially homology theory as it reaches the frontiers of current research in the topic. The book is also suitable for use as a graduate course in homology theory.

@book {key413113m,
AUTHOR = {Buoncristiano, S. and Rourke, C. P.
and Sanderson, B. J.},
TITLE = {A geometric approach to homology theory},
SERIES = {London Mathematical Society Lecture
Note Series},
NUMBER = {18},
PUBLISHER = {Cambridge University Press; London Mathematical
Society},
ADDRESS = {Cambridge; London},
YEAR = {1976},
PAGES = {iii+149},
DOI = {10.1017/CBO9780511662669},
NOTE = {MR:413113. Zbl:0315.55002.},
ISSN = {0076-0552},
}
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

People
BibTeX
@book {key665919m,
AUTHOR = {Rourke, C. P. and Sanderson, B. J.},
TITLE = {Introduction to piecewise-linear topology},
EDITION = {Revised reprint of the 1972 original},
SERIES = {Ergeb. Math. Grenzgeb.},
NUMBER = {69},
PUBLISHER = {Springer},
YEAR = {1982},
DOI = {10.1007/978-3-642-81735-9},
NOTE = {Revised reprint of the 1972 original.
MR:665919. Zbl:0477.57003.},
ISBN = {978-3-540-11102-3},
}
C. Rourke and B. Sanderson :
“The compression theorem, I ,”
Geom. Topol.
5
(2001 ),
pp. 399–429 .
MR
1833749
Zbl
1002.57057
article

Abstract
People
BibTeX

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.

@article {key1833749m,
AUTHOR = {Rourke, Colin and Sanderson, Brian},
TITLE = {The compression theorem, {I}},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry \& Topology},
VOLUME = {5},
YEAR = {2001},
PAGES = {399--429},
DOI = {10.2140/gt.2001.5.399},
NOTE = {MR:1833749. Zbl:1002.57057.},
ISSN = {1465-3060},
}
C. Rourke and B. Sanderson :
“The compression theorem, II: Directed embeddings ,”
Geom. Topol.
5
(2001 ),
pp. 431–440 .
MR
1833750
Zbl
1032.57028
article

Abstract
People
BibTeX
@article {key1833750m,
AUTHOR = {Rourke, Colin and Sanderson, Brian},
TITLE = {The compression theorem, {II}: Directed
embeddings},
JOURNAL = {Geom. Topol.},
FJOURNAL = {Geometry \& Topology},
VOLUME = {5},
YEAR = {2001},
PAGES = {431--440},
DOI = {10.2140/gt.2001.5.431},
NOTE = {MR:1833750. Zbl:1032.57028.},
ISSN = {1465-3060},
}
C. Rourke and B. Sanderson :
“The compression theorem, III: Applications ,”
Algebr. Geom. Topol.
3
(2003 ),
pp. 857–872 .
MR
2012956
Zbl
1032.57029
article

Abstract
People
BibTeX

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.

@article {key2012956m,
AUTHOR = {Rourke, Colin and Sanderson, Brian},
TITLE = {The compression theorem, {III}: {Applications}},
JOURNAL = {Algebr. Geom. Topol.},
FJOURNAL = {Algebraic \& Geometric Topology},
VOLUME = {3},
YEAR = {2003},
PAGES = {857--872},
DOI = {10.2140/agt.2003.3.857},
NOTE = {MR:2012956. Zbl:1032.57029.},
ISSN = {1472-2747},
}
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

Abstract
People
BibTeX

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.

@article {key2063665m,
AUTHOR = {Fenn, Roger and Rourke, Colin and Sanderson,
Brian},
TITLE = {James bundles},
JOURNAL = {Proc. Lond. Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society. Third Series},
VOLUME = {89},
NUMBER = {1},
YEAR = {2004},
PAGES = {217--240},
DOI = {10.1112/S0024611504014674},
NOTE = {MR:2063665. Zbl:1055.55005.},
ISSN = {0024-6115},
}
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

Abstract
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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.

@article {key2255194m,
AUTHOR = {Fenn, Roger and Rourke, Colin and Sanderson,
Brian},
TITLE = {The rack space},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {359},
NUMBER = {2},
YEAR = {2007},
PAGES = {701--740},
DOI = {10.1090/S0002-9947-06-03912-2},
NOTE = {MR:2255194. Zbl:1123.55006.},
ISSN = {0002-9947},
}