Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1987-0869408-2
MathSciNet review: 869408
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [K] Max Karoubi, $ K$-theory. An introduction, Springer-Verlag, Berlin-New York, 1978. MR 0488029 (58:7605)
  • [R] William J. Ralph, An extension of singular homology to Banach algebras, Pacific J. Math. (to appear). MR 840850 (87k:55009)
  • [St] James Stasheff, $ H$-spaces from a homotopy point of view, Lecture Notes in Math., Vol. 161, Springer-Verlag, Berlin and New York, 1970. MR 0270372 (42:5261)
  • [Sw] Robert M. Switzer, Algebraic topology-homotopy and homology, Springer-Verlag, New York and Heidelberg, 1975. MR 0385836 (52:6695)
  • [W] George W. Whitehead, Elements of homotopy theory, Springer-Verlag, New York and Heidelberg, 1978. MR 516508 (80b:55001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55Q05, 55P50

Retrieve articles in all journals with MSC: 55Q05, 55P50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0869408-2
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society