题目: Hurewicz Images of Real Bordism Theory and Real Johnson--Wilson Theories
摘要: 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 bordism spectrum and the Real Brown--Peterson spectrum. A subset of these families is detected by the $C_2$-fixed points of Real Johnson--Wilson theories $E\mathbb{R}(n)$, depending on $n$. The proof relies on a computation of the map from the classical Adams spectral sequence of the sphere spectrum to the $C_2$-equivariant Adams spectral sequence of $BP\mathbb{R}$. We also prove that the $C_2$-equivariant May spectral sequence of $BP\mathbb{R}$ is isomorphic to the associated graded $C_2$-slice spectral sequence of $BP\mathbb{R}$. This is joint work with Guchuan Li, Guozhen Wang, and Zhouli Xu.
