Rational BP operations and the Chern character

Abstract

This paper will study certain rational BP operations. We will work in the category of spaces having the homotopy type of CW complexes. Recall that for − 1 ≤ n ≤ ∞ there is defined in this category the homology theory BP(n)*(X). (See (7).) In particular, BP(∞)*(X) = BP*(X) is the usual Brown Peterson homology of X, BP(1)*(X) = l*(X) is the canonical direct summand of connective K-theory with Qp coefficients (Qp are the integers localized at the prime p), BP(0)*(X) = Hℤ*(X) is ordinary homology with Qp coefficients, and BP(−1)*(X) = Hℤ/p*(X) is ordinary homology with ℤp coefficients (ℤp are the integers reduced mod p). For 1 ≤ n ≤ ∞, BP(n)*(X) is a module over BP(n)*(pt) = Qp[v1, …, vn] (deg vs = 2ps − 2). The BP(n) theories are all related via canonical maps p(m, n):BP(m)*(X) → BP(n)*(X) whose performance is described in (7).