The rack space

Fenn, Roger; Rourke, Colin; Sanderson, Brian

doi:10.1090/S0002-9947-06-03912-2

The rack space
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by Roger Fenn, Colin Rourke and Brian Sanderson PDF

Trans. Amer. Math. Soc. 359 (2007), 701-740 Request permission

Abstract:

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.

References

S. Buoncristiano, C. P. Rourke, and B. J. Sanderson, A geometric approach to homology theory, London Mathematical Society Lecture Note Series, No. 18, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0413113
J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, and Masahico Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947–3989. MR 1990571, DOI 10.1090/S0002-9947-03-03046-0
J. Scott Carter, Seiichi Kamada, and Masahico Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11 (2002), no. 3, 311–322. Knots 2000 Korea, Vol. 1 (Yongpyong). MR 1905687, DOI 10.1142/S0218216502001639
Jean Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France 89 (1961), 227–380 (French). MR 140120
Adrien Douady, Variétés à bord anguleux et voisinages tubulaires, Séminaire Henri Cartan, 1961/62, Exp. 1, Secrétariat mathématique, Paris, 1961/1962, pp. 11 (French). MR 0160221
Roger A. Fenn, Techniques of geometric topology, London Mathematical Society Lecture Note Series, vol. 57, Cambridge University Press, Cambridge, 1983. MR 787801
R. Fenn, M. Jordan, L. H. Kauffman, Biracks and virtual links, to appear in Topology Appl. available from: http://www.maths.sussex.ac.uk/Staff/RAF/Maths/loumerc.ps
Roger Fenn and Colin Rourke, Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992), no. 4, 343–406. MR 1194995, DOI 10.1142/S0218216592000203
Roger Fenn, Colin Rourke, and Brian Sanderson, An introduction to species and the rack space, Topics in knot theory (Erzurum, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 399, Kluwer Acad. Publ., Dordrecht, 1993, pp. 33–55. MR 1257904
Roger Fenn, Colin Rourke, and Brian Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995), no. 4, 321–356. MR 1364012, DOI 10.1007/BF00872903
R. Fenn, C. Rourke, and B. Sanderson, James bundles and applications, preprint (1996). http://www.maths.warwick.ac.uk/~cpr/ftp/james.ps
Roger Fenn, Colin Rourke, and Brian Sanderson, James bundles, Proc. London Math. Soc. (3) 89 (2004), no. 1, 217–240. MR 2063665, DOI 10.1112/S0024611504014674
R. Fenn, C. Rourke, and B. Sanderson, A classification of classical links, in preparation.
J. Flower, Cyclic bordism and rack spaces, Ph.D. Thesis, Warwick, 1995.
M. Greene, Some results in geometric topology and geometry, Ph.D. Thesis, Warwick, 1996, available from: http://www.maths.warwick.ac.uk/~cpr/ftp/mtg.ps.gz
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
P. J. Hilton, On the homotopy groups of the union of spheres, J. London Math. Soc. 30 (1955), 154–172. MR 68218, DOI 10.1112/jlms/s1-30.2.154
Naoko Kamada and Seiichi Kamada, Abstract link diagrams and virtual knots, J. Knot Theory Ramifications 9 (2000), no. 1, 93–106. MR 1749502, DOI 10.1142/S0218216500000049
Louis H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663–690. MR 1721925, DOI 10.1006/eujc.1999.0314
Ulrich Koschorke and Brian Sanderson, Self-intersections and higher Hopf invariants, Topology 17 (1978), no. 3, 283–290. MR 508891, DOI 10.1016/0040-9383(78)90032-0
Greg Kuperberg, What is a virtual link?, Algebr. Geom. Topol. 3 (2003), 587–591. MR 1997331, DOI 10.2140/agt.2003.3.587
R. K. Lashof and S. Smale, Self-intersections of immersed manifolds, J. Math. Mech. 8 (1959), 143–157. MR 0101522, DOI 10.1512/iumj.1959.8.58010
R. A. Litherland and Sam Nelson, The Betti numbers of some finite racks, J. Pure Appl. Algebra 178 (2003), no. 2, 187–202. MR 1952425, DOI 10.1016/S0022-4049(02)00211-6
J. G. Pastor, Bundle complexes and bordism of immersions, Ph.D. Thesis, University of Warwick, 1982.
Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
Colin Rourke and Brian Sanderson, The compression theorem. I, Geom. Topol. 5 (2001), 399–429. MR 1833749, DOI 10.2140/gt.2001.5.399
C. Rourke and B. Sanderson, There are two $2$-twist spun trefoils, arxiv:math.GT/0006062:v1
Brian J. Sanderson, The geometry of Mahowald orientations, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 152–174. MR 561221
B. J. Sanderson, Bordism of links in codimension $2$, J. London Math. Soc. (2) 35 (1987), no. 2, 367–376. MR 881524, DOI 10.1112/jlms/s2-35.2.367
J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409–428. MR 4123, DOI 10.2307/1968907
Bert Wiest, Rack spaces and loop spaces, J. Knot Theory Ramifications 8 (1999), no. 1, 99–114. MR 1673890, DOI 10.1142/S0218216599000080