An algebraic approach to \(BP_*(B{\mathbb Z}/p^k)\)
type: Item
instance of: scholarly article
title: An algebraic approach to \(BP_*(B{\mathbb Z}/p^k)\)
zbMATH Open document ID: 1016.55002
DOI: 10.1016/S0166-8641(01)00109-2
author: Jesús González
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Fix a prime \(p\); \(\mathbb{Z}/p^k\) is the cyclic group of order \(p^k\) with classifying space \(B\mathbb{Z}/p^k\). \(BP_*(B\mathbb{Z}/p^k)\) is the Brown-Peterson homology of \(B\mathbb{Z}/p^k\) and \(BP\langle 1\rangle_*(B\mathbb{Z}/p^k)\) is its connective \(K\)-theory (or at least a summand of it). Bahri, Bendersky, Davis, and Gilkey (BBDG) constructed an additive isomorphism between \(BP_*(B\mathbb{Z}/p)\) and \(BP\langle 1\rangle_*(B\mathbb{Z}/p)\otimes \mathbb{Z}_{(p)}[v_2,v_3,\dots]\), although a richer structure was expected. For all positive integers \(k\), the author constructs a homomorphism over the ring \(\mathbb{Z}_{(p)}[v_2,v_3,\dots]\), \[ \Phi_k: BP_*(B\mathbb{Z}/p^k)\to BP\langle 1\rangle_*(B\mathbb{Z}/p^k)\otimes \mathbb{Z}_{(p)}[v_2,v_3,\dots]. \] For all \(k\geq 1\), \(\Phi_k\) is a monomorphism. When \(k=1\), this is an isomorphism -- an isomorphism over \(\mathbb{Z}_{(p)}[v_2,v_3,\dots]\) -- an enhanced version of the BBDG isomorphism. The author's approach is algebraic, using the \(p\)-divisibility properties of the \([p^k]\)-series.
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 55R35
zbMATH DE Number: 1787515
zbMATH Keywords: classifying space
zbMATH Keywords: Brown-Peterson homology
zbMATH Keywords: connective \(K\)-theory
zbMATH Keywords: \([p^k]\)-series
reviewed by: David C. Johnson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787515