Stable and unstable operations in mod \(p\) cohomology theories. (English) Zbl 1153.55016

Stacey, Andrew; Whitehouse, Sarah

Algebr. Geom. Topol. 8, No. 2, 1059-1091 (2008).

Given two \(\mathbb Z\)-graded cohomology theories \(F^*(-)\) and \(E^*(-)\) it is a natural and important problem to understand the operations (i.e., natural transformations) between them. This can be interpreted in several ways. For example, given a \(k,\ell\) one could consider unstable operations \(F^k(-)\rightarrow E^{k+\ell}(-)\), or given \(\ell\), compatible families of stable operations \(\{F^k(-)\rightarrow E^{k+\ell}(-)\}_{k\in \mathbb Z}\) where ‘compatible’ means commuting with suspension maps. In the 1950s, the case of \(E=F=H\mathbb F_p\) corresponding to mod \(p\) ordinary cohomology for a prime \(p\) was worked out: the first case corresponds to determining \(H^{k+\ell}(K(\mathbb F_p,k);\mathbb F_p)\), while the second case amounts to determining \(A(p)^\ell\). Here \(K(\mathbb F_p,k)\) is the \(k\)-th Eilenberg Mac Lane space for \(\mathbb Z\) and \(A(p)^*\) is mod \(p\) Steenrod algebra. Both of these involve nontrivial calculations. Note that stable operations always restrict to families of additive unstable operations, but not every additive operation \(F^k(-)\rightarrow E^{k+\ell}(-)\) needs to come from an operation \(F^{k+1}(-)\rightarrow E^{k+\ell+1}(-)\) under the desuspension operation.
The general theory of stable operations between multiplicative complex oriented cohomology theories has been well understood for some time, and was written up by Frank Adams in his influential lecture notes [J. F. Adams, Stable homotopy and generalised homology. Chicago Lectures in Mathematics. (Chicago - London): The University of Chicago Press. (1974; Zbl 0309.55016)]. In the 1970s the theory of Hopf rings was developed as an algebraic framework to describe the dual object \(E_*(\underline{F}_*)\) to the full unstable operation algebra \(E^*(\underline{F}_*)\). Here \(\underline{F}_k\) denotes the \(k\)-th space in the \(\Omega\)-spectrum \(F\), so \(\underline{F}_k\simeq\Omega\underline{F}_{k+1}\). Since then a large body of work by Boardman, Wilson and others has refined the algebraic structure used to handle unstable operations and some extremely difficult examples have been successfully analysed.
The present paper considers the question of when the destabilisation map \(E^\ell(F)\rightarrow [\underline{F}_k,\underline{E}_{k+\ell}]_+\) has a left inverse. Here the domain corresponds to stable operations of degree \(\ell\), while the codomain corresponds to stable operations of degree \(\ell\), \(F^k(-)\rightarrow E^{k+\ell}(-)\). Under some reasonable restrictions on \(E,F\) it is shown that such a left inverse always exists and is suitably functorial in the pair \(E,F\). These include the assumptions that \(E,F\) are commutative and complex orientable ring spectra where \(E_*\) has characteristic \(p\) for an odd prime \(p\), the associated formal group law for \(E\) has finite height and the coefficient of the leading term of its \(p\)-series is invertible. A prime example of such an \(E\) is the \(n\)-th Morava \(K\)-theory \(K(n)\). In the case \(E=K(n)\), the relationship with the Bousfield-Kuhn functor is also established.

Reviewer: Andrew Baker (Glasgow)

MSC:

55S25	\(K\)-theory operations and generalized cohomology operations in algebraic topology
55P47	Infinite loop spaces

Keywords:

cohomology theory; cohomology operation; Morava \(K\)-theory

Citations:

Zbl 0309.55016

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