Hostname: page-component-cb9f654ff-w5vf4 Total loading time: 0 Render date: 2025-09-10T06:22:05.940Z Has data issue: false hasContentIssue false
  • English
  • Français

The connective $K$-theory of the Eilenberg–MacLane space $K\!\left({\mathbb{Z}}_p,\textrm{2}\right)$

Part of: Homology and cohomology theories Spectral sequences

Published online by Cambridge University Press:  11 December 2023

Donald M. Davis  and
W. Stephen Wilson
Donald M. Davis*
Affiliation:
Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA
W. Stephen Wilson
Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, MD, 01220, USA
*
*Corresponding author: Donald M. Davis; Email: dmd1@lehigh.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We compute $ku^*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$ and $ku_*\left(K\!\left({\mathbb{Z}}_p,2\right)\right)$, the connective $KU$-cohomology and connective $KU$-homology groups of the mod-$p$ Eilenberg–MacLane space $K\!\left({\mathbb{Z}}_p,2\right)$, using the Adams spectral sequence. We obtain a striking interaction between $h_0$-extensions and exotic extensions. The mod-$p$ connective $KU$-cohomology groups, computed elsewhere, are needed in order to establish higher differentials and exotic extensions in the integral groups.

Keywords

Adams spectral sequenceconnective K-theoryEilenberg-MacLane spaces

MSC classification

Primary: 55T15: Adams spectral sequences
Secondary: 55N20: Generalized (extraordinary) homology and cohomology theories 55N15: $K$-theory

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 1 , January 2024 , pp. 188 - 220
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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.)

Article purchase

Temporarily unavailable

References

Adams, J. F., Lectures on generalized cohomology, Lecture Notes in Mathematics, vol. 99 (Springer-Verlag, 1969), 1–138.Google Scholar
Anderson, D. W. and Hodgkin, L., The $K$ -theory of Eilenberg-MacLane complexes, Topology 7 (1968), 317–329.CrossRefGoogle Scholar
Browder, W., Torsion in $H$ -spaces, Ann. Math. 74 (1961) 24–51.CrossRefGoogle Scholar
Clément, A., Integral cohomology of finite Postnikov towers, thesis, University of Lausanne (2002), available online.Google Scholar
Davis, D. M. and Greenlees, J. P. C., Gorenstein duality and universal coefficient theorems, Jour. Pure Applied. Alg. 227 (2023) paper 107265, 12 p.Google Scholar
Davis, D. M., Ravenel, D. C. and Wilson, W. S., The connective Morava $K$ -theory of the second mod $p$ Eilenberg-MacLane space, arXiv 2206.14035.Google Scholar
Davis, D. M. and Wilson, W. S., The connected $K$ -theory of the Eilenberg-MacLane space $K({\mathbb{Z}}_2,2)$ . Available at: https://www.lehigh.edu/~dmd1/kkpaper3.pdf.Google Scholar
Davis, D. M. and Wilson, W. S., Stiefel-Whitney classes and immersions of orientable and Spin manifolds, Topol. Appl. 307 (2022) paper107780.Google Scholar
Johnson, D. C. and Wilson, W. S., Projective dimension and Brown-Peterson homology, Topology 12(4) (1973), 327–353.CrossRefGoogle Scholar
Liulevicius, A., Notes on homotopy of Thom spectra, Am. J. Math. 86 (1964), 1–16.CrossRefGoogle Scholar
Serre, J. P., Cohomologie modulo 2 des complexes d’Eilenberg MacLane, Comm. Math. Helv. 27 (1953), 198–231.CrossRefGoogle Scholar
Tamanoi, H., The image of the $BP$ Thom map for Eilenberg-MacLane spaces, Trans. Am. Math. Soc. 349 (1997), 1209–1237.CrossRefGoogle Scholar
Wall, C. T. C., A characterization of simple modules over the Steenrod algebra mod 2, Topology 1 (1962), 249–254.CrossRefGoogle Scholar
Wilson, W. S., A new relation on the Stiefel-Whitney classes of Spin manifolds, Ill. J. Math. 17 (1973), 115–127.Google Scholar