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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Category and group rings in homotopy theory
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by William J. Ralph PDF
Trans. Amer. Math. Soc. 299 (1987), 205-223 Request permission

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 $1 + {\text {cat}} X$.
References
  • Max Karoubi, $K$-theory, Grundlehren der Mathematischen Wissenschaften, Band 226, Springer-Verlag, Berlin-New York, 1978. An introduction. MR 0488029
  • William J. Ralph, An extension of singular homology to Banach algebras, Pacific J. Math. 123 (1986), no. 2, 391–405. MR 840850
  • James Stasheff, $H$-spaces from a homotopy point of view, Lecture Notes in Mathematics, Vol. 161, Springer-Verlag, Berlin-New York, 1970. MR 0270372
  • Robert M. Switzer, Algebraic topology—homotopy and homology, Die Grundlehren der mathematischen Wissenschaften, Band 212, Springer-Verlag, New York-Heidelberg, 1975. MR 0385836
  • 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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 299 (1987), 205-223
  • MSC: Primary 55Q05; Secondary 55P50
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0869408-2
  • MathSciNet review: 869408