Odd primary analogs of Real orientations

Abstract

We define, in $C_p$-equivariant homotopy theory for $p>2$, a notion of ${\mu }_p$-orientation analogous to a $C_2$-equivariant Real orientation. The definition hinges on a $C_p$-space ${\mathbb{CP}}_{{\mu }_p}^{\infty }$, which we prove to be homologically even in a sense generalizing recent $C_2$-equivariant work on conjugation spaces. We prove that the height $p-1$ Morava $E$-theory is ${\mu }_p$-oriented and that $\mathrm{tmf}(2)$ is ${\mu }_3$-oriented. We explain how a single equivariant map $v_1^{{\mu }_p}:S^{2\rho }\to {\Sigma }^{\infty }{\mathbb{CP}}_{{\mu }_p}^{\infty }$ completely generates the homotopy of $E_{p-1}$ and $\mathrm{tmf}(2)$, expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.

arXiv comment31 pages. To appear in Geometry & Topology. Comments still welcome!

References in corpus (13)

• The $C_2$-spectrum $Tm{f}_1(3)$ and its invertible modules
• Gorenstein duality for Real spectra
• Picard groups of higher real $K$-theory spectra at height $p-1$
• The slice spectral sequence of a $C_4$-equivariant height-4 Lubin-Tate theory
• Invertible $K(2)$-Local $E$-Modules in $C_4$-Spectra
• Landweber flat real pairs and ER(n)-cohomology
• Picard groups and duality for Real Morava $E$-theories
• On categories of slices
• Conjugation Spaces are Cohomologically Pure
• Models of Lubin-Tate spectra via Real bordism theory
• Tate blueshift and vanishing for Real oriented cohomology
• Real Wilson Spaces I
• On the $\mathrm{EO}$-orientability of vector bundles