Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-09T14:21:18.806Z Has data issue: false hasContentIssue false
  • English
  • Français

A new chain complex for the homology of the Steenrod algebra

Published online by Cambridge University Press:  24 October 2008

William M. Singer
William M. Singer
Affiliation:
Fordham University, New York
Get access Rights & Permissions [Opens in a new window]

Extract

Let A be the Steenrod algebra over the field F2. For any graded A-module M we construct here a small chain complex of A-modules EM. This complex computes the homology of the Steenrod algebra with coefficients in M, in the sense that:

The complexes F2 ⊗AEM are comparable in size but slightly larger than those that arise from the Λ-algebra(4). We are motivated to introduce them because they can be used to prove an embedding theorem for the homology of the Steenrod algebra. In fact, we construct a functor K on the category of graded. A-modules, and using the complexes EM, prove that there is a natural, one-one map:

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 2 , September 1981 , pp. 279 - 292
Copyright
Copyright © Cambridge Philosophical Society 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Adams, J. F. Operations of the n'th kind in K-theory and what we don't know about RP∞ In New Developments in Topology, ed. Segal, G. (London Math. Soc. Lecture Notes, no. 11, University Press, Cambridge, 1974).Google Scholar
(2)Adams, J. F.On the structure and applications of the Steenrod algebra. Com. Math. Helv. 32 (1958), 180–214.CrossRefGoogle Scholar
(3)Adams, J. F. Private communication (08, 1980).Google Scholar
(4)Bousfield, A. K., Curtis, E. B., Kan, D. M., Quillen, D. G., Rector, D. L. and Schlesinger, J. W.The mod-p lower central series and the Adams spectral sequence. Topology 5 (1966), 331–342.CrossRefGoogle Scholar
(5)Kahn, D. S. and Priddy, S. B.The transfer and stable homotopy theory. Math. Proc. Cambridge Philos. Soc. 83 (1978), 103–112.CrossRefGoogle Scholar
(6)Kristensen, L.On a Cartan formula for secondary cohomology operations. Math. Scand. 16 (1965), 97–115.CrossRefGoogle Scholar
(7)Lin, W. H., Davis, D. M., Mahowald, M. E. and Adams, J. F.Calculation of Lin's Ext groups. Math. Proc. Cambridge Philos. Soc. 87 (1980), 459–470.CrossRefGoogle Scholar
(8)Lin, W. H. Algebraic Kahn-Priddy Theorem (preprint; to appear).Google Scholar
(9)Miller, H. and Kuhn, N. Private communication (10, 1980).CrossRefGoogle Scholar
(10)Mui, H.Modular invariant theory and the cohomology algebras of the symmetric groups. Jour. Fac. Sci. Univ. of Tokyo 22 (1975), 319–369.Google Scholar
(11)Singer, W. M.On the localization of modules over the Steenrod algebra. Jour. Pure and Appl. Algebra 16 (1980), 75–84.CrossRefGoogle Scholar
(12)Singer, W. M. A new chain complex for the homology of the Steenrod algebra and the algebraic Kahn-Priddy theorem (preprint, 07, 1980).Google Scholar
(13)Steenrod, N. and Epstein, D. B. A.Cohomology Operations (Annals of Mathematics Studies no. 50, Princeton University Press, Princeton, New Jersey, 1962).Google Scholar
(14)Welkerson, C.Classifying spaces, Steenrod operations, and algebraic closure. Topology 16 (1977), 227–237.CrossRefGoogle Scholar