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April, 2006 Homotopy classes of self-maps and induced homomorphisms of homotopy groups
Martin ARKOWITZ, Jeffrey STROM, Hideaki ŌSHIMA
J. Math. Soc. Japan 58(2): 401-418 (April, 2006). DOI: 10.2969/jmsj/1149166782


For a based space X , we consider the group # n ( X ) of all self homotopy classes α of X such that α # = i d : π i ( X ) π i ( X ) , for all i n , where n , and the group Ω ( X ) of all α such that Ω α = i d . Analogously, we study the semigroups 𝒵 # n ( X ) and 𝒵 Ω ( X ) defined by replacing ' i d ' by ' 0 ' above. There is a chain of containments of the -groups and the 𝒵 -semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on X which ensure that the -groups and the 𝒵 -semigroups are equal. When X is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for the -groups and 𝒵 -groups of certain low dimensional Lie groups.


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Martin ARKOWITZ. Jeffrey STROM. Hideaki ŌSHIMA. "Homotopy classes of self-maps and induced homomorphisms of homotopy groups." J. Math. Soc. Japan 58 (2) 401 - 418, April, 2006.


Published: April, 2006
First available in Project Euclid: 1 June 2006

zbMATH: 1112.55007
MathSciNet: MR2228566
Digital Object Identifier: 10.2969/jmsj/1149166782

Primary: 55P10
Secondary: 55P45 , 55P60 , 55Q05

Keywords: group-like spaces , Homotopy equivalences , Homotopy groups , Lie groups

Rights: Copyright © 2006 Mathematical Society of Japan


Vol.58 • No. 2 • April, 2006
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