On fibrations related to real spectra

TL;DR

This paper studies fibrations in real spectra, establishing connections between homotopy fixed points, real Johnson--Wilson spectra, and their complex counterparts, with explicit fibrations and evaluations of related maps.

Contribution

It introduces new fibrations linking real and complex spectra, specifically connecting ER(n) to E(n), and evaluates the associated maps, advancing understanding of real spectra structures.

Findings

Established a fibration connecting ER(n) and E(n) spectra.

Evaluated the map analogous to the forgetful functor from complex to real spectra.

Derived explicit formulas for the fibrations involving real Johnson--Wilson spectra.

Abstract

We consider real spectra, collections of Z/(2)-spaces indexed over Z oplus Z alpha with compatibility conditions. We produce fibrations connecting the homotopy fixed points and the spaces in these spectra. We also evaluate the map which is the analogue of the forgetful functor from complex to reals composed with complexification. Our first fibration is used to connect the real 2^{n+2}(2^n-1)-periodic Johnson--Wilson spectrum ER(n) to the usual 2(2^n-1)-periodic Johnson--Wilson spectrum, E(n). Our main result is the fibration Sigma^{lambda(n)} ER(n) --> ER(n) --> E(n)$, where lambda(n) = 2^{2n+1}-2^{n+2}+1.