An algebraic approach to \(BP_*(B{\mathbb Z}/p^k)\)
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.
scientific article; zbMATH DE number 1787515