2025/01/05 by Carlisle, Jack, Wisdom, Noah, Yan, Guoqi
#55N22 #55P91 #55Q91 #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.05469
Recently, Hausmann defined global group laws and used them to prove that MUG_* is the G-equivariant Lazard ring, for G a compact abelian Lie group. On the other hand, Hu and Kriz showed that the restriction map induces an isomorphism M ℝC2ρ* ≅ MU2*. In this paper, we blend these stories. We utilize the C2-global spectrum MR defined by Schwede in an unpublished note, which gives rise to a genuine G-spectrum M ℝη for each augmented compact Lie groups η: G→ C2, simultaneously generalizing MUG and M ℝ. In the case of semi-direct product augmentations G \rtimes C2→ C2 with G compact abelian Lie and C2 acting by inversion, we show that the restriction along the inclusion G ⊂ G \rtimes C2 is a split surjection M ℝG \rtimes C2ρ* → MUG2*. Additionally, we propose an evenness conjecture, which implies that this map is an isomorphism. Along the way, we define Real η-orientations, Real global orientations, and corresponding notions of equivariant and global group laws.