Algebraic & Geometric Topology

A theorem of Sanderson on link bordisms in dimension 4

J Scott Carter, Seiichi Kamada, Masahico Saito, and Shin Satoh

Full-text: Open access

Abstract

The groups of link bordism can be identified with homotopy groups via the Pontryagin–Thom construction. B J Sanderson computed the bordism group of 3 component surface-links using the Hilton–Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and their framings. In this paper, we geometrically represent every element of the bordism group uniquely by a certain standard form of a surface-link, a generalization of a Hopf link. The standard forms give rise to an inverse of Sanderson’s geometrically defined invariant.

Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 299-310.

Dates
Received: 9 October 2000
Revised: 11 May 2001
Accepted: 17 May 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882594

Digital Object Identifier
doi:10.2140/agt.2001.1.299

Mathematical Reviews number (MathSciNet)
MR1834778

Zentralblatt MATH identifier
0973.57010

Subjects
Primary: 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}

Keywords
surface links link bordism groups triple linking Hopf $2$–links

Citation

Carter, J Scott; Kamada, Seiichi; Saito, Masahico; Satoh, Shin. A theorem of Sanderson on link bordisms in dimension 4. Algebr. Geom. Topol. 1 (2001), no. 1, 299--310. doi:10.2140/agt.2001.1.299. https://projecteuclid.org/euclid.agt/1513882594


Export citation

References

  • Bartels, A. and Teichner, P., All two dimensional links are null homotopic, Geometry and Topology, Vol. 3 (1999), 235–252.
  • Buoncristiano, S., Rourke, C.P., Sanderson, B.J., A geometric approach to homology theory, London Mathematical Society Lecture Note Series, No. 18. Cambridge University Press, Cambridge-New York-Melbourne, 1976.
  • Carter, J.S., Jelsovsky, D., Kamada, S., Langford, L., and Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces, preprint at http://xxx.lanl.gov/abs/math.GT/9903135.
  • Carter, J.S. and Saito, M., Reidemeister moves for surface isotopies and their interpretations as moves to movies, J. Knot Theory Ramifications 2 (1993), 251–284.
  • Carter, J.S. and Saito, M., Knotted surfaces and their diagrams, American Mathematical Society, Mathematical Surveys and Monograph Series, Vol 55, (Providence 1998).
  • Cochran, T.D., On an invariant of link cobordism in dimension four, Topology Appl. 18 (1984), no. 2-3, 97–108.
  • Cochran, T.D. and Orr, K.E., Not all links are concordant to boundary links, Ann. of Math. (2) 138 (1993), no. 3, 519–554.
  • Fenn, R. and Rolfsen, D., Spheres may link homotopically in $4$-space, J. London Math. Soc. (2) 34 (1986), no. 1, 177–184.
  • Kirk, P.A., Link maps in the four sphere, Differential topology (Siegen, 1987), 31–43, Lecture Notes in Math., 1350, Springer, Berlin-New York, 1988.
  • Kirk, P.A. and Koschorke, U., Generalized Seifert surfaces and linking numbers, Topology Appl. 42 (1991), no. 3, 247–262.
  • Koschorke, U., Homotopy, concordance and bordism of link maps, Global analysis in modern mathematics (Orono, ME, 1991; Waltham, MA, 1992), 283–299, Publish or Perish, Houston, TX, 1993.
  • Koschorke, U., A generalization of Milnor's $\mu$-invariants to higher-dimensional link maps, Topology 36, 2 (1997), 301–324.
  • Massey, W.S. and Rolfsen, D., Homotopy classification of higher-dimensional links, Indiana Univ. Math. J. 34 (1985), no. 2, 375–391.
  • Rolfsen, D., Knots and links, Publish or Perish, Inc., 1976.
  • Roseman, D., Reidemeister-type moves for surfaces in four dimensional space, in Banach Center Publications 42 (1998) Knot theory, 347–380.
  • Ruberman, D., Concordance of links in $S\sp 4$, Four-manifold theory (Durham, N.H., 1982), 481–483, Contemp. Math., 35, Amer. Math. Soc., Providence, R.I., 1984.
  • Sanderson, B. J., Bordism of links in codimension $2$, J. London Math. Soc. (2) 35 (1987), no. 2, 367–376.
  • Sanderson, B. J., Triple links in codimension $2$, Topology. Theory and applications, II (Pécs, 1989), 457–471, Colloq. Math. Soc. János Bolyai, 55, North-Holland, Amsterdam, 1993.
  • Sato, N., Cobordisms of semiboundary links. Topology Appl. 18 (1984), no. 2-3, 225–234.