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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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  • [K] Max Karoubi, 𝐾-theory, Springer-Verlag, Berlin-New York, 1978. An introduction; Grundlehren der Mathematischen Wissenschaften, Band 226. MR 0488029
  • [R] William J. Ralph, An extension of singular homology to Banach algebras, Pacific J. Math. 123 (1986), no. 2, 391–405. MR 840850
  • [St] James Stasheff, 𝐻-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR 0270372
  • [Sw] Robert M. Switzer, Algebraic topology—homotopy and homology, Springer-Verlag, New York-Heidelberg, 1975. Die Grundlehren der mathematischen Wissenschaften, Band 212. MR 0385836
  • [W] George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508

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Article copyright: © Copyright 1987 American Mathematical Society

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