Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T11:46:22.944Z Has data issue: false hasContentIssue false
  • English
  • Français

Duality in homotopy theory

Published online by Cambridge University Press:  26 February 2010

E. H. Spanier  and
J. H. C. Whitehead
E. H. Spanier
Affiliation:
University of Chicago. Magdalen College, Oxford
J. H. C. Whitehead
Affiliation:
University of Chicago. Magdalen College, Oxford
Get access

Extract

Certain results ([7], [8], [10], [11]) suggest that there should be some principle of duality in homotopy theory. Among other things one is led to expect that cohomotopy groups will appear as dual to homotopy groups. But the fact that a cohomotopy group πn(X), unlike πn(X), is only defined if dim X ≤ 2n—2 is a serious obstacle to the formulation of such a principle. However, the set of S-maps (i.e.S-homotopy classes [11]) X → Y is a group for every pair of spaces X, Y. Therefore, this difficulty does not appear in S-theory [11].

Type
Research Article
Information
Mathematika , Volume 2 , Issue 1 , June 1955 , pp. 56 - 80
Copyright
Copyright © University College London 1955

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alexander, J. W., “The combinatorial theory of complexes”, Annals of Math., 31 (1930), 292–320.CrossRefGoogle Scholar
2.Alexandroff, P. and Hopf, H.., Topologie, Berlin (1935).Google Scholar
3.Barratt, M. G., “Track groups (1)”, Proc. London Math. Soc. (3), 5 (1955), 71–106.CrossRefGoogle Scholar
4.Chang, S. C., “Homotopy invariants and continuous mappings”, Proc. Royal Soc., A, 202 (1950), 253–63.Google Scholar
5.Clarke, B., “A note on Alexander's duality” (to appear).Google Scholar
6.Eilenberg, S. and Steenrod, N. E., Foundations of Algebraic Topology, Princeton (1952).CrossRefGoogle Scholar
7.Massey, W. S., “Exact couples in algebraic topology”, Annals of Math., 56 (1952), 353–96; 57 (1953), 248–56.CrossRefGoogle Scholar
8.Nakaoka, M., “Exact sequences Σp(K, L) and their applications”, J. Inst. Polytech., Osaka City Univ., 3 (1953), 83–100.Google Scholar
9.Serre, J.-P., “Homologie singulière des espaces fibrés”, Annals of Math., 54 (1951), 425–505.CrossRefGoogle Scholar
10.Spanier, E. H., “Borsuk's cohomotopy groups”, Annals of Math., 50 (1949), 203–45.CrossRefGoogle Scholar
11.Spanier, E. H. and Whitehead, J. H. C., “The theory of carriers and S-theory”, Annals of Math. (to appear).Google Scholar
12.Whitehead, J. H. C., “On subdivisions of complexes”, Proc. Camb. Phil. Soc. 31 (1935), 69–75.CrossRefGoogle Scholar
13.Whitehead, J. H. C., “On the groups πr(Vn, m) and sphere bundles”, Proc. London Math. Soc. (2), 48 (1944), 243–91.Google Scholar
14.Whitehead, J. H. C., “On simply connected 4-dimensional polyhedra”, Comm. Math. Helvetici, 22 (1949), 48–92.CrossRefGoogle Scholar
15.Whitehead, J. H. C., “Combinatorial homotopy I”, Bull. Amer. Math. Soc., 55 (1949), 213-45.CrossRefGoogle Scholar
16.Whitehead, J. H. C., “On the realizability of homotopy groups”, Annals of Math., 50 (1949), 261–3.CrossRefGoogle Scholar