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Infinite loop structures on the algebraic K-theory of spaces

Published online by Cambridge University Press:  24 October 2008

Richard J. Steiner
Richard J. Steiner
Affiliation:
University of Glasgow
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Let X be a topological space with base-point. The algebraic K-theory AX of X is a space invented by Waldhausen in (15) for use in geometric topology. It can be defined in two ways, which I shall call geometric and ring-theoretic; Steinberger ((12)) has shown them to be equivalent.

The geometric method ((15), corollary to lemma 2·1) gives AX as the group-completion of the geometric realization of a permutative category. It follows from the machinery of May ((4), 4) or Segal (11) that AX is an infinite loop space in a well-defined way ((6)).

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 1 , July 1981 , pp. 85 - 111
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Adams, J. F.Infinite loop spaces. Annals of Mathematics Studies, no. 90 (Princeton University Press, 1978).CrossRefGoogle Scholar
(2)Boardman, J. M. and Vogt, R. M.Homotopy invariant algebraic stnictures on topological spaces. Lecture Notes in Mathematics, no. 347 (Springer, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
(3)May, J. P.The geometry of iterated loop spaces. Lecture Notes in Mathematics, no. 271 (Springer, Berlin, Heidelberg, New York, 1972).CrossRefGoogle Scholar
(4)May, J. P.E ∞ spaces, group completions, and permutative categories. In New developments in topology, ed. Segal, G.. London Mathematical Society Lecture Note Series, no. 11 (Cambridge University Press, 1974), pp. 61–93.CrossRefGoogle Scholar
(5)May, J. P.A ∞ ring spaces and algebraic K-theory. In Geometric applications of homotopy theory, vol. II, ed. Barratt, M. C. and Mahowald., M. E. Lecture Notes in Mathematics, no. 658 (Springer, Berlin, Heidelberg, New York, 1978), pp. 240–315.CrossRefGoogle Scholar
(6)May, J. P.The spectra associated to permutative categories. Topology 17 (1978), 225–228.CrossRefGoogle Scholar
(7)May, J. P. Infinite loop space theory revisited. In Algebraic topology, Waterloo 1978, ed. Hoffman, P. and Snaith, V.. Lecture Notes in Mathematics, no. 741 (Springer, Berlin, Heidelberg, New York, 1979), pp. 625–642.Google Scholar
(8)May, J. P. with contributions by Quinn, F., Ray, N., and Tornehavo, J.E∞ ring spaces and E∞ ring spectra. Lecture Notes in Mathematics, no. 577. (Springer, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
(9)May, J. P. and Thomason, R.The uniqueness of infinite loop space machines. Topology 17 (1978), 205–224.CrossRefGoogle Scholar
(10)Segal, G.Classifying spaces and spectral sequences. I.H.E.S. Publ. Math. 34 (1968), 105–112.CrossRefGoogle Scholar
(11)Segal, G.Categories and cohomology theories. Topology 13 (1974), 293–312.CrossRefGoogle Scholar
(12)Steinberger, M. On the equivalence of the two definitions of the algebraic K-theory of a topological space. In Algebraic topology, Aarhus 1978, ed. Dupont, J. L. and Madsen, I. H.. Lecture Notes in Mathematics, no. 703 (Springer, Berlin, Heidelberg, New York, 1979), pp. 317–331.Google Scholar
(13)Steiner, R.A canonical operad pair. Math. Proc. Cambridge Philos. Soc. 86 (1979), 443–449.CrossRefGoogle Scholar
(14)Thomason, R. W.Uniqueness of delooping machines. Duke Math. J. 46 (1979). 217–252.CrossRefGoogle Scholar
(15)Waldhausen, F. Algebraic K-theory of topological spaces. I. In Algebraic and geometric topology, ed. Milgram, R. J.. Proceedings of Symposia in Pure Mathematics, no. 32 A.M.S., (Providence, 1978), part 1, pp. 35–60.CrossRefGoogle Scholar
(16)Woolfson, R.Hyper-Γ-spaces and hyperspectra. Quart. J. Math. Oxford (2) 30 (1979), 229–255.CrossRefGoogle Scholar