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A Remark on (π, n)-Type CW-Complexes

Published online by Cambridge University Press:  22 January 2016

Kenichi Shiraiwa
Kenichi Shiraiwa*
Affiliation:
Mathematical Institute, Nagoya University
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Let X be a space whose i-th homotopy group πi(X)vanishes for every i ≧ 0 except i = n ≧ 1, and whose n-th homotopy group is isomorphic to a group π. Then it is well known that the polyhedral homotopy type of X is completely determined by π and n. We call such a space a (π, n)-type space. Also it is well known that the minimal complex of the singular complex of a (π, n)-type space is isomorphic to the complex K(π, n)defined by S. Eilenberg and S. MacLane [1]. We know also that for any n ≧ 1 and any group π(abelian if n > 1) there exists a (π, n)-type space (See [6]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 12 , December 1957 , pp. 25 - 30
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

References

[1] Eilenberg, S. and MacLane, S.: Relations between homotopy and homology groups of spaces I, II, Ann. of Math., 46 (1945), pp. 480–509; 51 (1950), pp. 514–533.Google Scholar
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[7] Whitehead, J. H. C.: On the homotopy type of ANR’s, Bull. Amer. Math. Soc, 54 (1948), pp. 1133–1145.Google Scholar