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The Tate spectrum \(v_ n\)-periodic complex oriented theories. (English) Zbl 0849.55005

Summary: Let \(G\) be a finite group. We prove that the Tate spectrum with respect to \(G\) of the \(n\)th Morava \(K\)-theory spectrum \(K(n)\) is equivariantly contractible. As a corollary we can deduce results about the Tate spectrum with respect to \(G\) of various \(v_n\)-periodic homology theories. For example, if \(K_n\) is the integral Morava \(K\)-theory spectrum, then \(t_G (K_n)\) is rational. This generalizes the result for \(n=1\) that appears as [the first author with J. P. May, Generalized Tate cohomology, Mem. Am. Math. Soc. 543 (1995), Theorem 16.1].
More succinctly but less precisely, if \(E\) is complex oriented and \(v_n\)-periodic then \(t_G (E)\) is complex oriented and \(v_{n-1}\)-periodic. Our results are related to and partially motivated by certain conjectures about the relationship between Mahowald’s root invariant and \(v_n\)-periodic homotopy.

MSC:

55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
19L41 Connective \(K\)-theory, cobordism
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References:

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