## The eightfold way to BP-operations or $$E_*E$$ and all that.(English)Zbl 0563.55002

Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, 1, 187-226 (1982).
[For the entire collection see Zbl 0538.00016.]
In this paper the author presents a reinterpretation of Adams work on $$E_*(E)$$ (E being a flat ring spectrum) and gives a very careful analysis of left/right module/comodule structures on homology/cohomology. For this he introduces the notion of universal cohomology operations. Let R be a free right $$\pi =E_*$$ module and let $$m_ r$$ of degree d(r) be basis elements of R. He defines $$R\otimes_{\pi}E$$, the spectrum E with coefficients R, as $$\bigvee_ r\Sigma^{d(r)}E$$. Clearly, $$\pi_*(R\otimes_{\pi}E)\approx R$$. Using this notion an operation $$\psi_ L: E\to R\otimes_{\pi}E$$ is a universal additive operation if given any operation $$\theta$$ : $$E\to M\otimes_{\pi}E$$ with M a free right $$\pi$$-module, there exists a unique homomorphism $$g: R\to M$$ of right $$\pi$$-modules such that $$\theta =(g\otimes E)\circ \psi_ L$$. Multiplicative universal operations are defined similarly. It turns out that if $$R\otimes_{\pi}E$$ admits a universal cohomology operation, R is a ”two-faced Hopf algebra” (Theorem 5.2.).
He then shows (Theorem 5.4.) that under certain assumptions on E which are fulfilled for example if $$E=MU$$, BP, $$H{\mathbb{F}}_ p$$, etc. $$\psi_ L: E=E\wedge S\to E\wedge E\approx A\otimes_{\pi}E$$ is both the universal additive and the universal multiplicative operation $$(A=E_*E)$$. Theorem 5.2. provides all the standard structure on A for free except for the internal conjugation antiautomorphism c which is induced by the switch map $$E\wedge E\to E\wedge E$$. The author then applies his theory to ordinary cohomology $$H{\mathbb{F}}_ p$$, to MU and BP. Finally, he extends the theory of unstable operations, with emphasis on BP-theory. As an example, he shows that the real stunted projective space P(26,16) cannot be desuspended 11 times, confirming a result of Wilson.
Reviewer: U.Würgler

### MSC:

 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55S10 Steenrod algebra 55P42 Stable homotopy theory, spectra 55S05 Primary cohomology operations in algebraic topology 55P40 Suspensions

Zbl 0538.00016