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On a Homotopy Classification of Mappings of an (n+1) Dimensional Complex Into an Arcwise Connected Topological Space Which is Aspherical in Dimensions Less Than n (n >2)

Published online by Cambridge University Press:  22 January 2016

Nobuo Shimada  and
Hiroshi Uehara
Nobuo Shimada
Affiliation:
Mathematical Institute, Nagoya University
Hiroshi Uehara
Affiliation:
Mathematical Institute, Nagoya University
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Pontrjagin classified mappings of a three dimensional sphere into an n dimensional complex, where he made use of a new type of product of cocycles. By the aid of the generalized Pontrjagin’s product of cocycles Steenrod enumerated effectively all the homotopy classes of mappings of an (n+1) dimensional complex into an n sphere. According to the recent issue of the Mathematical Reviews it is reported that M. M. Postnikov extended Steenrod’s case to the case where an arcwise connected topological space which is aspherical in dimensions less than n, takes place of an n sphere. (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 n. Doklady Akad. Nauk SSSR (N.S.) 71., 1027-1028, 1950 (Russian. No. proof is given.)) But here in Japan no details are yet to hand. We intend to give a solution to this problem in case where n>2, and also to give an application concerning the (n + 3)-extension cocycle.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 3 , October 1951 , pp. 67 - 72
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1951

References

[1] Eilenberg, Cohomology, S. and continuous mappings. Ann. of Math., 41 (1940), 231–251.Google Scholar
[2] Steenrod, N. E., Products of cocycles and extension of mappings, Ann. of Math., 48 (1947).Google Scholar