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

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