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Continuous Morava \(K\)-theory and the geometry of the \(I_ n\)-adic tower. (English) Zbl 0828.55003
We consider certain completed cohomology theories and the topology of the \(\Omega\)-spectra representing them. Our main interest is in the Johnson- Wilson spectra \(E(n)\) and their \(I_n\)-adic completions \(\widehat {E(n)}\), but much of our work applies also to completions of other periodic theories.
Given a cohomology theory \(E^*(-)\), there are representing spaces \(\{\mathbb{E}_r; r \in \mathbb{Z}\}\) with natural isomorphisms of groups \(E^r (X) \cong [X, \mathbb{E}_r]\). Let \(\widehat \mathbb{E}_*\) represent some completed version, \(\widehat E^*(-)\) say, of the theory \(E^*(-)\). Given a cohomological functor \(L^*(-)\), we set up a notion of continuous \(L\)-cohomology, \(L^*_c(-)\), which forms a suitable tool for studying these completed \(\Omega \)-spectra \(\widehat \mathbb{E}_*\). As an application, we show how our theory in the case of the spectra \(\widehat {E(n)}\) can be used to discuss problems on the Morava \(K\)- theory of extended powers \(K(n)^* (D_p (X))\).
Reviewer: A.Baker (Glasgow)

55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
19L41 Connective \(K\)-theory, cobordism
55P60 Localization and completion in homotopy theory
55N15 Topological \(K\)-theory
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