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Homotopy Classification of Mappings of a 4-Dimensional Complex into a 2-Dimensional Sphere

Published online by Cambridge University Press:  22 January 2016

Nobuo Shimada
Nobuo Shimada*
Affiliation:
Mathematical Institute, Nagoya University
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Steenrod [1] solved the problem of enumerating the homotopy classes of maps of an (n + 1)-complex K into an n-sphere Sn utilizing the cup-i-product, the far-reaching generalization of the Alexander-Čech-Whitney cup product [7] and the Pontrjagin *-product [5].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 5 , February 1953 , pp. 127 - 144
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

[1] Steenrod, N. E., Products of cocycles and extension of mappings, Ann. of Math., 48 (1947), 290–320.Google Scholar
[2] Steenrod, N. E., Cohomology invariants of mappings, Ann. of Math., 50 (1949), 954–988.Google Scholar
[3] Cartan, H., Une théorie axiomalique des carrés de Steenrod, Comp. Rend. Paris, 230 (1950), 425–427.Google Scholar
[4] Pontrjagin, L., A classification of mappings of the 3-dimensional complex into the 2-rii-mensional sphere, Rec. Math. (Mat, Sbornik), N.S. 9 (51) (1941), 331–363.Google Scholar
[5] Pontrjagin, L., Mappings of the 3-dimensional sphere into an n-dimensional complex, Doklady Akad. Nauk SSSR, 34 (1942), 35–37.Google Scholar
[6] Pontrjagin, L., Homotopy classification of the the mapping of an (n + 2)-dimensional sphere on n-dimensional one, Doklady Akad. Nauk SSSR, 70 (1950), 957–959.Google Scholar
[7] Whitney, H., On products in a complex, Ann. of Math., 39 (1938), 397–432.Google Scholar
[8] Whitney, H., Classification of the mappings of a 3-compIex into a simply connected space, Ann. of Math., 50 (1949), 270–284.Google Scholar
[9] Whitney, H., An extension theorem for mappings into simply connected spaces, ibid., 285–296.Google Scholar
[10] Postnikov, M. M., The classification of continuous mappings of a 3-dimensional polyhedron into a simply connected polyhedron of arbitray dimension, Doklady Akad. Nauk SSSR (N.S.) 64 (1949), 461–462.Google Scholar
[11] Postnikov, M. M., Classification of continuous mappings of an (n + 1)-dimensional complex into a connected topological space which is aspherical in dimensions less than Doklady Akad. Nauk SSSR, (N.S.) 71 (1950), 1027–1028.Google Scholar
[12] Whitehead, G. W., A generalization of the Hopf invariant, Ann. of Math., 51 (1950), 192–237.Google Scholar
[13] Whitehead, G. W.. The (w + 2)-nd homotopy group of the n-sphere, Ann, of Math., 52 (1950), 245–247.Google Scholar
[14] Blakers, A. L. and Massay, W. S., Homotopy groups of a triad I, Ann. of Math., 53 (1951), 161–205.Google Scholar
[15] Eilenberg, S., Cohomology and continuous mappings, Ann. of Math., 41 (1940), 231–251.Google Scholar
[16] Whitehead, J. H. C., On simply connected, 4-dimensional polyhedra, Comm. Math. Helv., 22 (1949), 48–90.Google Scholar
[17] Whitehead, J. H. C., A certain exact sequence, Ann. of Math., 52 (1950), 51–110.Google Scholar
[18] Whitehead, J. H. C., On the theory of obstructions, Ann. of Math., 54 (1951), 68–84.Google Scholar
[19] Nakaoka, M., On Whitney’s extension theorem, Jour, of Inst. Polytechnics Osaka City Univ., vol. 2, No. 1. (Series A) (1951), 31–37.Google Scholar
[20] Shimada, N. and Uehara, H., On a homotopy classfication of mappings of an (n + 2)-dimensional complex into an arcwise connected topological space which is aspherical in dimensions less than n(n>2) , Nagoya Math. Jour., 3 (1951), 67–72.Google Scholar
[21] Shimada, N. and Uehara, H., Classification of mappings of an (n + 2)-complex into an (n —1)-connected space with vanishing (n + l)-st homotopy group, Nagoya Math. Jour., 4 (1952), 43–50.Google Scholar