Constructing formal groups III: applications to complex cobordism and Brown-Peterson cohomology

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In parts I and II, cf. [S], [6] and also [7], of this series of papers we constructed a universal p-typical one dimensional commutative formal group and a universal one dimensional commutative formal group. The extraordinary cohomology theories BP (Brown-Peterson cohomology) and MU (complex cobordism cohomology) are complex oriented and hence define one dimensional formal groups over BPt Z, Tz, . . . ] (when localized at p) can be identified with the right unit map pR : BP,(ptj+ BP,(BP) of the Hopf algebra BP&BP). ln Sections 4, 5 belo-w, we use the universal isomorphism of [S] to obtain a recursive description of the homomorphism qR. This description is useful in the calculation of various BP cohomology operatio s, cf. Sections 6,7 below. To obtain this recursive description of qR we need an isomorphism formula (Section 5 below) which is also useful in the theory of formal groups itself, cf. part III of f8].