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On real Johnson-Wilson theories


The central object of study in this thesis is a family of generalized cohomology theories ER(n), known as real Johnson-Wilson theories. These theories arise as the homotopy fixed points of the classical Johnson-Wilson theories E(n) under the Z/2-action of complex conjugation. The classical Johnson-Wilson theories E(n) are closely related to another family E_n of cohomology theories, the so-called Lubin-Tate or Morava E-theories. A purely obstruction-theoretic argument given by Hopkins and Miller [Rez98] shows that the E_n admit an action of the Morava stabilizer group of automorphisms of the height n Honda formal group law. We relate the real Johnson-Wilson theories ER(n) to homotopy fixed points of the Morava E- theories En under an action by a certain subgroup of the Morava stabilizer group. In doing so, we obtain a calculation of the coefficients of the homotopy fixed points of E_n for this subgroup and as a corollary we see that after completion the ER(n) are commutative S-algebras (i.e. E_infinity-ring theories). We work entirely at the prime 2

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