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Register a new userHurewicz Images of Real Bordism Theory and Real Johnson-Wilson Theories
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We show that the Hopf elements, the Kervaire classes, and the $bar{kappa}$-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the $C_2$-fixed points of the Real Brown-Peterson spectrum. A subset of these families is detected by the $C_2$-fixed points of Real Johnson-Wilson theory $Emathbb{R}(n)$, depending on $n$.
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We prove that the Real Johnson-Wilson theories ER(n) are homotopy associative and commutative ring spectra up to phantom maps. We further show that ER(n) represents an associatively and commutatively multiplicative cohomology theory on the category of (possibly non-compact) spaces.
We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{mathbb{R}}$.
We show a number of Toda brackets in the homotopy of the motivic bordism spectrum $MGL$ and of the Real bordism spectrum $MU_{mathbb R}$. These brackets are red-shifting in the sense that while the terms in the bracket will be of some chromatic height $n$, the bracket itself will be of chromatic height $(n+1)$. Using these, we deduce a family of exotic multiplications in the $pi_{(ast,ast)}MGL$-module structure of the motivic Morava $K$-theories, including non-trivial multiplications by $2$. These in turn imply the analogous family of exotic multiplications in the $pi_{star}MU_mathbb R$-module structure on the Real Morava $K$-theories.
This is the first part in a series of papers establishing an equivariant analogue of Steve Wilsons theory of even spaces, including the fact that the spaces in the loop spectrum for complex cobordism are even.
We offer a complete description of $THH(E(2))$ under the assumption that the Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an $E_infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2) rightarrow THH(E(2))rightarrow overline{THH}(E(2))$ and describe $overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring spectrum. We state general results about the $K(i)$-local behaviour of $THH(E(n))$ for all $n$ and $0 leq i leq n$. In particular, we compute $K(i)_*THH(E(n))$.
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