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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)

##### MSC:
 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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