Odd primary analogs of real orientations

Hahn, Jeremy¹ ; Senger, Andrew¹ ; Wilson, Dylan²

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States
² Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 27 (2023) no. 1, pp. 87-129.

Résumé

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 ${ℂ\mathbb{P}}_{{\mu }_p}^{\infty }$, 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 $v_1^{{\mu }_p}:S^{2\rho }\to {\mathrm{\Sigma }}^{\infty }{ℂ\mathbb{P}}_{{\mu }_p}^{\infty }$ completely generates the homotopy of $E_{p-1}$ and $\mathrm{tmf}(2)$, expressing a height-shifting phenomenon pervasive in equivariant chromatic homotopy theory.

Détail

DOI : 10.2140/gt.2023.27.87

Keywords: chromatic homotopy theory, equivariant

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