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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Category and group rings in homotopy theory


Author: William J. Ralph
Journal: Trans. Amer. Math. Soc. 299 (1987), 205-223
MSC: Primary 55Q05; Secondary 55P50
MathSciNet review: 869408
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Abstract: It frequently arises in algebraic topology that a function $ \beta :G \to H$, between two groups, is not a homomorphism. We show that in many standard situations $ \beta $ induces a group homomorphism $ \overline \beta :{\mathbf{Z}}(G)/{\mathcal{A}^d} \to H$, where $ {\mathcal{A}^d}$ is a power of the augumentation ideal in the group ring $ {\mathbf{Z}}(G)$. A typical example is $ \beta :[X,\,Y] \to [{S^2}X,\,{S^2}Y]$ where $ Y$ is some $ H$-group, in which case $ d$ can be taken to be $ 1 + {\text{cat}}\,X$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0869408-2
PII: S 0002-9947(1987)0869408-2
Article copyright: © Copyright 1987 American Mathematical Society