##
**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.

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 |